Browse Derivatives problems Studdy Solved!

Problem 701

Differentiate the following function. $\begin{array}{l} y=(\ln x)^{10}+\ln \left(x^{10}\right) \\ \frac{d y}{d x}=10 \end{array}$

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Problem 702

Find an equation of the tangent line to the curve $\mathrm{y}=e^{\mathrm{x}}$ when $\mathrm{x}=-3.73$ . $y=$ $\square$ (Simplify your answer. Use integers or decimals for any numbers in the equation. as needed.)

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Problem 703

Sign in What's new New tab Login / Virgi About / The 10.5 Hmwk https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=35488821\&tags=autosave\#question5097773_4 Mathway Mail Carlse The Coconu $A^{\prime \prime}$
6. [-/1 Points]
DETAILS MY NOTES TANAPMATH7 10.5.010.EP. PRACTICE ANOTHER
Consider the following closed rectangular box that has a square cross section, a capacity of $112 \mathrm{in}^{3}$ , and is constructed using the least amount of material.
Let $x$ denote the length (in inches) of the sides of the box and let $y$ denote the height (in inches) of the box. Utilize the given volume to write an equation for $y$ in terms of $x$ . $y=\square 1$
Write a function $f$ in terms of $x$ that describes the amount of material needed to create the box. $f(x)=$ $\square$ Find $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ . $\begin{array}{c} f^{\prime}(x)=\square \\ f^{\prime \prime}(x)=\square \end{array}$
What are the dimensions of the box if it is constructed using the least amount of material? $\begin{array}{ll} x=\square & \text { in } \\ y=\square & \text { in } \end{array}$

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Problem 704

The table below gives selected values for the differentiable and increasing function $f$ and its derivative. If $g(x)=f^{-1}(x)$ , what is the value of $g^{\prime}(-2) ?$ \begin{tabular}{|c|c|c|} \hline $x$ & $f(x)$ & $f^{\prime}(x)$ \\ \hline-2 & -5 & 4 \\ \hline 1 & -2 & 7 \\ \hline 4 & 3 & 3 \\ \hline 6 & 4 & 10 \\ \hline 7 & 6 & 9 \\ \hline \end{tabular}

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Problem 705

Given that $v=4 x^{2}+2$ , find $\frac{d}{d x}\left(v^{5}-4 \sin x\right)$ in terms of only $x$ .

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Problem 706

A point moves along the circle $x^{2}+y^{2}=121$ so that $\frac{\mathrm{d} x}{\mathrm{~d} t}=5 \mathrm{~cm} / \mathrm{min}$ . Find $\frac{\mathrm{d} y}{\mathrm{~d} t}$ at the point $(-8, \sqrt{57})$ . Use the squrt function or an exponent of $1 / 2$ to represent a square root in your answer. For example to write $\sqrt{5}$ you should either enter $5^{\circ}(1 / 2)$ or squrti(5). Also, don't forget to use * to represent any multiplication.
Numeric part Units

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Problem 707

Remaining Tim
A very small car (apparently a VW Beetle) is driving back and forth along the $x$ -axis with velocity given by the function $v(t)=3 t^{2}-18 t+24$ . At time $t=1$ second, the car's postion is $x(1)=20 \mathrm{~cm}$ . (a) Give an expression for the acceleration of the car at any time $t$ . $a(t)=$ $a^{b}$ $\frac{a}{b}$ $\sqrt{a}$ $|a|$ $\pi$ $\sin (a)$

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Problem 708

A point moves along the circle $x^{2}+y^{2}=25$ so that $\frac{\mathrm{d} x}{\mathrm{~d} t}=2 \mathrm{~cm} / \mathrm{min}$ . Find $\frac{\mathrm{d} y}{\mathrm{~d} t}$ at the point $(-2, \sqrt{21})$ Use the sqrt function or an exponent of $1 / 2$ to represent a square root in your answer. For example to write $\sqrt{5}$ you should either enter $5^{\wedge}(1 / 2)$ or sqrt(5). Also, don't forget to use * to represent any multiplication.
Numeric part Units

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Problem 709

he second derivative of the function. $f(x)=\frac{1}{13 x-4}$

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Problem 710

Use logarithmic differentiation to find the derivative. $y=\sqrt[5]{\frac{x^{2}+3}{x^{2}+4}}$

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Problem 711

* Consider the function $f(x)=x\left(\left(x^{2}-1\right)^{2}+\frac{1}{2}\right)$ - Find the transition points. - Find the intervals where the function is increasing / decreasing and concave down. - Find the local minima and maxima. - Find the horizontal and vertical asymptotes, if there are any. - Sketch the graph.

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Problem 712

* Consider the function $f(x)=x\left(\left(x^{2}-1\right)^{2}+\frac{1}{2}\right)$ - Find the transition points.

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Problem 713

$\frac{d y}{d x}(6 y+2 x)=4 x-2 y$ $\frac{d y}{d x}=\frac{4 x-2 y}{6 y+2 x}$
27. Let $f$ be the function defined by $f(x)=x^{3}+x$ . If $g(x)=f^{-1}(x)$ and $g(2)=1$ , what is the value of $g^{\prime}(2)$ ? (D) 4 (E) 13 $\begin{array}{l} 1=x^{3}+x \\ 0=x^{2}+x-1 \end{array}$

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Problem 714

Question 2
Plot the function $f(x)=\frac{2 x}{x^{2}-5}$ . How many inflection points does this function have? 0 2 1 3

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Problem 715

Question 7
If $x^{2}+x y^{3}=1$ , find $y^{\prime}$ at $(2,-1)$ . Answer this in decimal form, rounded to one decimal place.

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Problem 716

Implicit diffrenciation $y=x^{2}+x y$

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Problem 717

Let $C(q)$ represent the cost, $R(q)$ the revenue, and $\pi(q)$ the total profit, in dollars, of producing $q$ items. (a) If $C^{\prime}(50)=76$ and $R^{\prime}(50)=84$ , approximately how much profit is earned by the $51^{\text {st }}$ item?
The profit earned from the $51^{\text {st }}$ item will be approximately $\$$ $\square$ . (b) If $C^{\prime}(90)=71$ and $R^{\prime}(90)=66$ , approximately how much profit is earned by the $91^{\text {st }}$ item?
The profit earned from the $91^{\text {st }}$ item will be approximately $\$$ $\square$ . (c) If $\pi(q)$ is a maximum when $q=78$ , how do you think $C^{\prime}(78)$ and $R^{\prime}(78)$ compare? $C^{\prime}(78)$ $\square$ $R^{\prime}(78)$

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Problem 718

Question 11
Find the critical numbers of the following function $f(x)=\frac{1}{4} x^{4}-\frac{5}{3} x^{3}+x^{2}+8 x$ . How many critical numbers does this function have? 1 4 2 3

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Problem 719

Question 12 1 pts
Consider the function $f(x)=e^{x} \cos x$ . Using a linear approximation of this function centered at the point $x=2$ , estimate $f(x)$ at $x=2$ . (round to one decimal place) $\square$

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Problem 720

2. Differentiate the following function. (Do not simplify your a) [8pts] $f(x)=\cos (2 x) \tan (3 x)-e^{x^{2}-3 x}$

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Problem 721

Harvesting Due to over-harvesting, the population of western Atlantic cod collapsed in the early 1990s. The government has since declared a moratorium on the Northern Cod fishery, which drastically changed the fishing industry in Atlantic Canada.
Suppose that the cod population off the eastern coast of Canada recovers enough to support fishing. The Department of Fisheries and Oceans will want to decide on a safe fixed proportion of fish, $h$ , to authorize for harvest each year, following the reproduction season. This means that $h$ is a non-negative real number and the size of the catch is determined by multiplying $h$ times the cod population at the end of the previous year.
The modified discrete-time dynamical system will be: $N_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}-h N_{t}$
Use this for problem 3 and onward. $N$ will still represent fish in tens of millions and $t$ will still be in years. We will work to find the maximal number of fish that can be harvested, while maintaining a stable population.
3. Again suppose that for Atlantic cod $r=2.5$ , and find the equilibria points for this new system algebraically (the expression for one of the equilibrium points will contain the parameter $h$ ).
4. To prevent another collapse in the population of Atlantic cod, we need to harvest from a stable equilibrium population. Assume for the moment that the positive equilibrium point that you found in problem 3 is stable. The annual yield of the cod population at equilibrium will be the product of two terms: - the positive equilibrium value and - the safe proportion value $h$ .
Write down the formula for the annual yield, $Y(h)$ , a function of $h$ .
5. The Department of Fisheries and Oceans wants to maximize the annual yield. Find the value for $h$ that will give you a maximal annual yield, i.e. find the local maximum value of the function you found in problem 4 , and show that it is indeed maximal using the 1st derivative test - we will see that this is actually a global max. (remember that $h$ is non-negative, so the domain of $Y(h)$ is $[0, \infty))$ . Please include the first two decimal places and round the answer you get from the calculator.
6. Using the expressions for the positive equilibrium population from problem 3 and the annual yield from problem 4, find the specific values by substituting the value $h$ you found in problem 5. Please include the first two decimal places and round the answer you get from the calculator. (Give the answer also in terms of numbers of fish.)
7. Use the slope criterion from the derivative test to determine whether this equilibrium population (the one referenced in problem 6) is indeed stable. You will want to first substitute the value for $h$ from problem 5 before differentiating. The zero population represents a collapse of the fishery. Determine if the zero solution is stable using the slope criterion. 2

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Problem 722

Consider the function $f(x)=e^{x} \cos x$ . Using a linear approximation of this function centered point $x=2$ , estimate $f(x)$ at $x=2$ . (round to one decimal place) Not saved

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Problem 723

5. (15 pts) Sketch the graph of the following function. Include the following: $f(x)=\left(x^{2}-x\right) e^{-x}$ a. Domain: $\qquad$ b. Vertical Asymptote: $\qquad$ c. Intervals Increasing: $\qquad$ d. Intervals Decreasing: $\qquad$ e. Local Extrema: $\qquad$ f. Inflection Points: $\qquad$ g. Intervals Concave Up: $\qquad$ h. Intervals Concave Down: $\qquad$

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Problem 724

Find the coordinates of the maximum point. $y=\frac{4 \ln x+4}{\sqrt{x}}$
The maximum point is $\square$ (Type an ordered pair. Type your answer using exponential notation.)

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Problem 725

Find the derivative of the function $f(x) = (5x - 5)^2(4x^2 + 1)^3$ .

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Problem 726

Differentiate. $\begin{array}{c} y=\ln \left[e^{6 x}\left(x^{5}+4\right)\left(x^{8}+3 x\right)\right] \\ \frac{d}{d x}\left[\ln \left[e^{6 x}\left(x^{5}+4\right)\left(x^{8}+3 x\right)\right]\right]= \end{array}$

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Problem 727

Harvesting Due to over-harvesting, the population of western Atlantic cod collapsed in the early 1990s. The government has since declared a moratorium on the Northern Cod fishery, which drastically changed the fishing industry in Atlantic Canada.
Suppose that the cod population off the eastern coast of Canada recovers enough to support fishing. The Department of Fisheries and Oceans will want to decide on a safe fixed proportion of fish, $h$ , to authorize for harvest each year, following the reproduction season. This means that $h$ is a non-negative real number and the size of the catch is determined by multiplying $h$ times the cod population at the end of the previous year.
The modified discrete-time dynamical system will be: $N_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}-h N_{t}$
Use this for problem 3 and onward. $N$ will still represent fish in tens of millions and $t$ will still be in years. We will work to find the maximal number of fish that can be harvested, while maintaining a stable population.
3. Again suppose that for Atlantic cod $r=2.5$ , and find the equilibria points for this new system algebraically (the expression for one of the equilibrium points will contain the parameter $h$ ).
4. To prevent another collapse in the population of Atlantic cod, we need to harvest from a stable equilibrium population. Assume for the moment that the positive equilibrium point that you found in problem 3 is stable. The annual yield of the cod population at equilibrium will be the product of two terms: - the positive equilibrium value and - the safe proportion value $h$ .
Write down the formula for the annual yield, $Y(h)$ , a function of $h$ .
5. The Department of Fisheries and Oceans wants to maximize the annual yield. Find the value for $h$ that will give you a maximal annual yield, i.e. find the local maximum value of the function you found in problem 4, and show that it is indeed maximal using the 1st derivative test - we will see that this is actually a global max. (remember that $h$ is non-negative, so the domain of $Y(h)$ is $[0, \infty)$ ). Please include the first two decimal places and round the answer you get from the calculator.
6. Using the expressions for the positive equilibrium population from problem 3 and the annual yield from problem 4, find the specific values by substituting the value $h$ you found in problem 5. Please include the first two decimal places and round the answer you get from the calculator. (Give the answer also in terms of numbers of fish.)
7. Use the slope criterion from the derivative test to determine whether this equilibrium population (the one referenced in problem 6) is indeed stable. You will want to first substitute the value for $h$ from problem 5 before differentiating. The zero population represents a collapse of the fishery. Determine if the zero solution is stable using the slope criterion. 2

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Problem 728

Harvesting Due to over-harvesting, the population of western Atlantic cod collapsed in the early 1990s. The government has since declared a moratorium on the Northern Cod fishery, which drastically changed the fishing industry in Atlantic Canada.
Suppose that the cod population off the eastern coast of Canada recovers enough to support fishing. The Department of Fisheries and Oceans will want to decide on a safe fixed proportion of fish, $h$ , to authorize for harvest each year, following the reproduction season. This means that $h$ is a non-negative real number and the size of the catch is determined by multiplying $h$ times the cod population at the end of the previous year.
The modified discrete-time dynamical system will be: $N_{t+1}=\left(\frac{r}{1+0.1 N_{t}}\right) N_{t}-h N_{t}$
Use this for problem 3 and onward. $N$ will still represent fish in tens of millions and $t$ will still be in years. We will work to find the maximal number of fish that can be harvested, while maintaining a stable population.
3. Again suppose that for Atlantic cod $r=2.5$ , and find the equilibria points for this new system algebraically (the expression for one of the equilibrium points will contain the parameter $h$ ).
4. To prevent another collapse in the population of Atlantic cod, we need to harvest from a stable equilibrium population. Assume for the moment that the positive equilibrium point that you found in problem 3 is stable. The annual yield of the cod population at equilibrium will be the product of two terms: - the positive equilibrium value and - the safe proportion value $h$ .
Write down the formula for the annual yield, $Y(h)$ , a function of $h$ .
5. The Department of Fisheries and Oceans wants to maximize the annual yield. Find the value for $h$ that will give you a maximal annual yield, i.e. find the local maximum value of the function you found in problem 4, and show that it is indeed maximal using the lst derivative test - we will see that this is actually a global max. (remember that $h$ is non-negative, so the domain of $Y(h)$ is $[0, \infty)$ ). Please include the first two decimal places and round the answer you get from the calculator.
6. Using the expressions for the positive equilibrium population from problem 3 and the annual yield from problem 4 , find the specific values by substituting the value $h$ you found in problem 5. Please include the first two decimal places and round the answer you get from the calculator. (Give the answer also in terms of numbers of fish.)
7. Use the slope criterion from the derivative test to determine whether this equilibrium population (the one referenced in problem 6) is indeed stable. You will want to first substitute the value for $h$ from problem 5 before differentiating. The zero population represents a collapse of the fishery. Determine if the zero solution is stable using the slope criterion. 2

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Problem 729

Find the tangent line equation for $f(x)=\sqrt{3-x}$ at a given point.

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Problem 730

Find the derivative of the function $y=x^{4}+12x$ . What is $\frac{dy}{dx}$ ?

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Problem 731

A ball is thrown with a velocity of $41 \mathrm{ft} / \mathrm{s}$ . Its height is $y=41 t-22 t^{2}$ . Find average velocity from $t=2$ and instantaneous velocity at $t=2$ .

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Problem 732

Find the rate of change of the city's population modeled by $P(t)=22 e^{0.08 t}$ from 2025 to 2034.

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Problem 733

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Height after $t$ seconds is $y=41t-22t^2$ . Find average velocity from $t=2$ for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at $t=2$ ?

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Problem 734

Encuentra la derivada de $f(x) = 3x^3 - x$ .

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Problem 735

Approximate the average rate of change of COVID-19 infections in Minnesota from 0 to 8 days after April 1, 2020.

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Problem 736

Find the rate of change of the city's population, modeled by $P(t)=10 e^{0.04 t}$ , from 2021 to 2030 in thousand persons/year.

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Problem 737

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using $\frac{\Delta y}{\Delta x}$ :
1. 0 to 8 days: (calculate)
2. 8 to 16 days: 0.59 thousand/day
3. 0 to 16 days: 0.46 thousand/day

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Problem 738

Two engines start at the origin and meet at $t=e-1$ . Thomas' equation is $x=300 \ln(t+1)$ and Henry's is $x=kt$ .
a. Sketch the graphs. b. Show $k=\frac{300}{e-1}$ . c. Find the max distance between them in the first $e-1$ minutes and when it occurs.

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Problem 739

A particle $P$ moves along a line with displacement $x = \frac{1}{3} t^{3} - 4 t^{2} + 15 t$ . Find: (a) when $P$ is at rest, (b) distance in 5 seconds.

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Problem 740

Find the distance from the origin when the particle stops, given $v=9t-3t^{2}$ m/s and starts at $t=0$ .

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Problem 741

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height after $t$ seconds is $y=41t-22t^2$ . Find average velocity for $t=2$ over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at $t=2$ .

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Problem 742

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height is $y=41t-22t^2$ . Find average velocity from $t=2$ for given intervals and the instantaneous velocity at $t=2$ .

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Problem 743

Find the limit: $\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h}$ .

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Problem 744

Relatel Ratas
17. The volume of a cube decreases at a rate of $10 \mathrm{~m}^{3} / \mathrm{s}$ . Find the rate at which the side of the cube changes when the side of the cube is 2 m .

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Problem 745

A conical tank (vertex down) is 7 meters across the top and 9 meters deep. If the depth of the water (the height) is decreasing at 6.6 meters per minute, what is the change in the volume of the water in the tank when the height of the water in the tank is 4 meters?
Include units on your final answer, and your answer must be entered as number (not 5*7+3).

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Problem 746

A conical tank (vertex down) is 7 meters across the top and 9 meters deep. If the depth of the water (the height) is decreasing at 6.6 meters per minute, what is the change in the volume of the water in the tank when the height of the water in the tank is 4 meters?
Include units on your final answer, and your answer must be entered as number (not 5*7+3).

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Problem 747

Find $d y / d x$ by implicit differentiation. $d y / d x=\square \quad \sin x+\cos y=\sin x \cos y$

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Problem 748

Problem 5. (1 point)
Find the equation of the line tangent to the graph of $y=2 \ln (x)$ at $x=1$ . Tangent Line: $y=$ $\square$

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Problem 749

$f$ and $g$ are twice differentiable functions such that $g(x)=e^{f(x)}$ and $g^{\prime \prime}(x)=h(x) e^{f(x)}$ , then $h(x)=$ (A) $f^{\prime}(x)+f^{\prime \prime}(x)$ (B) $f^{\prime}(x)+\left(f^{\prime \prime}(x)\right)^{2}$
C $\left(f^{\prime}(x)+f^{\prime \prime}(x)\right)^{2}$ (D) $\left(f^{\prime}(x)\right)^{2}+f^{\prime \prime}(x)$ (E) $2 f^{\prime}(x)+f^{\prime \prime}(x)$

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Problem 750

Calculate the derivative of $y$ with respect to $x$ . Express derivative in terms of $x$ and $y$ . $e^{2 x y}=\sin \left(y^{7}\right)$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $\frac{d y}{d x}=$ $\square$

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Problem 751

$: \operatorname{lR} \omega_{c}=20$ -II $b \in \mathbb{R}, \alpha \in \mathbb{R}\left(g(x)=-x^{3}+\alpha x+b\right.$ $A(0 ; 2)$ - (II $D_{f}$ (6) (

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Problem 752

Trova i punti di massimo e di minimo della funzione $y = \sin 3x + \sqrt{3} \cos 3x$ e traccia il grafico.

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Problem 753

find $y^{\prime}$ given: $y^{3}-x \sin y+\frac{y^{2}}{x}=8$ $y^{\prime}=\frac{x^{2} \cdot \sin y+y^{2}}{3 x^{2} y^{2}+x^{3} \cos y+2 x y}$ $y^{\prime}=\frac{x^{2} \cdot \sin y+y^{2}}{3 x^{2} y^{2}-x^{3} \cos y+2 x y}$ $y^{\prime}=\frac{3 x^{2} y^{2}+x^{3} \cos y+2 x y}{x^{2} \cdot \sin y+y^{2}}$ $y^{\prime}=\frac{3 x^{2} y^{2}-x^{3} \cos y+2 x y}{x^{2} \cdot \sin y+y^{2}}$

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Problem 754

Find $y^{\prime \prime}$ if $x^{3} y^{3}-4=0$ $\frac{y}{x^{2}}$ $\frac{2 x}{y^{2}}$ $\frac{2 y}{x^{2}}$

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Problem 755

Find $f$ such that $f^{\prime}(x)=8 x-5, f(9)=0$ . $f(x)=$

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Problem 756

Find $y^{\prime}$ . $y=\sqrt{(2 x-3)^{3}\left(x^{2}+5\right)^{4}}$ $+$
Choose the correct answer below. A. $y^{\prime}=\frac{\left(x^{2}+5\right)\left(11 x^{2}-12 x+15\right)}{\sqrt{2 x-3}}$ B. $y^{\prime}=\sqrt{2 x-3}\left(x^{2}+5\right)\left(11 x^{2}-12 x+15\right)$ C. $y^{\prime}=\sqrt{x^{2}+5}(2 x-3)\left(10 x^{2}+6 x-30\right)$ D. $y^{\prime}=\frac{(2 x-3)\left(10 x^{2}+6 x-30\right)}{\sqrt{x^{2}+5}}$

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Problem 757

Differentiate. $\begin{array}{c} y=\frac{\ln x}{x^{5}} \\ \frac{d y}{d x}=\square \end{array}$ $\square$

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Problem 758

32. What is the minimum value of $f(x)=x \ln (x)$ ?

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Problem 759

(b) Given $y=3^{2 x-5}$ . Find $\frac{d^{2} y}{d x^{2}}$ when $x=3$ . Give the answer in the form of logarithm.
3. The function $f(x)=x^{3}-6 x^{2}+9 x-3$ is defined on the interval $[0,5]$ . Find the [4 marks] critical points of $f(x)$ on this interval and determine whether the critical points are local minimum or maximum. [7 marks]

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Problem 760

2. (a) Consider the parametric equations, $x=t^{2}$ , and $y=t^{3}-3 t$ . [3 marks] Evaluate $\frac{d y}{d x}$ when $t=\sqrt{3}$ . [5 marks] (b) Given $y=3^{2 x-5}$ . Find $\frac{d^{2} y}{d x^{2}}$ when $x=3$ . Give the answer in the form of logarithm. [4 marks]

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Problem 761

Given the function $f$ and point a below, complete parts (a)-(c). $f(x)=7-6 x, a=\frac{1}{6}$
1 (x) - 6 b. Graph $f(x)$ and $f^{-1}(x)$ together. Choose the correct graph below. A. B. C. D. c. Evaluate $\frac{d f}{d x}$ at $x=a$ and $\frac{d f^{-1}}{d x}$ at $x=f(a)$ to show that $\left.\frac{d f^{-1}}{d x}\right|_{x=f(a)}=\frac{1}{\left.(d f / d x)\right|_{x=a}}$ $\left.\frac{d f}{d x}\right|_{x=\frac{1}{6}}=$ $\square$ $\left.\frac{d f^{-1}}{d x}\right|_{x=f\left(\frac{1}{6}\right)}=$ $\square$ (Simplify your answers. Use integers or fractions for any numbers in the expressions.)

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Problem 762

45. Let $f$ be the function with derivative given by $f^{\prime}(x)=x^{2}-\frac{2}{x}$ . On which of the following intervals is $f$ decreasing?

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Problem 763

Consider the function $f(x)=x e^{-5 x}, \quad 0 \leq x \leq 2$ . This function has an absolute minimum value equal to: $\square$ which is attained at $x=$ $\square$ and an absolute maximum value equal to: which is attained at $x=$ $\square$

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Problem 764

Given that $u=v^{2}-4$ , find $\frac{d}{d v}\left(3 u^{5}-2 \sin v\right)$ in terms of only $v$ .

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Problem 765

54. The function $f$ is defined by $f(x)=e^{-x}\left(x^{2}+2 x\right)$ . At what values of $x$ does $f$ have a relative maximum? (A) $x=-2+\sqrt{2}$ and $x=-2-\sqrt{2}$

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Problem 766

21 Mark for Review
The function $m$ is given bs $m(x)=\log _{10} e+\log _{10}\left(x^{-1}\right)$ . Which of the following statements about $m$ is true? (A) $m$ is increasing, the graph of $m$ is concave up, and $\lim _{x \rightarrow-\infty} m(x)=\log _{10} e$ . (B) $m$ is increasing, the graph of $m$ is concave down, and $\lim _{x \rightarrow 0^{+}} m(x)=-\infty$ . (C) $m$ is decreasing, the graph of $m$ is concave up, and $\lim _{x \rightarrow 0^{+}} m(x)=\infty$ . (D) $m$ is decreasing, the graph of $m$ is concave down, and $\lim _{x \rightarrow-\infty} m(x)=-\log _{10} e$

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Problem 767

Bestimme die erste und zweite Ableitung der Funktionen: a) $f(x)=e^{x}+1$ , b) $f(x)=e^{x}+x$ , c) $f(x)=e^{x}+2 x^{2}$ , d) $f(x)=-e^{x}+1$ , e) $f(x)=2 e^{x}+3 x^{2}$ , f) $f(x)=-5 e^{x}-0,5 x^{3}$ , g) $f(x)=-\frac{1}{2}(e^{x}-x^{3})$ , h) $f(x)=\frac{1}{4}e^{x}+\sin(x)$ .

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Problem 768

Bestimmen Sie die erste Ableitung für die folgenden Funktionen und überprüfen Sie mit dem GTR: a) $2 x \cdot(4 x-1)$ , b) $(5 x+3) \cdot(x+2)$ , c) $(2-5 x) \cdot(x+2)$ , d) $2 x \cdot e^{x}$ , e) $(4 x+2) \cdot e^{x}$ , f) $(6 x+1) \cdot e^{x}$ .

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Problem 769

Find the marginal cost $M C$ and average cost $A C$ from the total cost function $T C=300 \ln (q+30)+150$ .

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Problem 770

Find the tangent line equation for $f(x) = \sqrt{x}$ at the point (1,1).

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Problem 771

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$ .

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Problem 772

Find the function $f(x)$ and the number $c$ given the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$

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Problem 773

Find $F^{\prime}(x)$ using the first principle if $F(x)=x^{2}-3x$ .

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Problem 774

Find the average rate of change of $h(t)=\cot t$ over the intervals: a. $\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right]$ b. $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$

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Problem 775

Calculate the average rate of change of $h(t)=\cot t$ over the intervals: a. $\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right]$ , b. $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$ .

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Problem 776

Find the average rate of change of $h(t)=\cot t$ over $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$ .

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Problem 777

Differentiate $3xy = \sqrt{16x} - \frac{3}{x^3}$ with respect to $x$ .

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Problem 778

Differentiate $y=\frac{6 x^{4}-x+5}{3 x^{2}}$ and express the answer with positive exponents.

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Problem 779

Find the derivative of $y$ with respect to $x$ . $\begin{array}{l} y=(1+5 x) e^{-5 x} \\ \frac{d y}{d x}=\square \end{array}$ $\square$

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Problem 780

Find the $2019^{\text {th }}$ derivative of $y=\sin (2 x+1)+5 x^{6}+20 x^{100}$

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Problem 781

Suppose $f(x)=x^{3}+4, x \in[0,1]$ . (a) Find the slope of the secant line connecting the points $(x, y)=(0,4)$ and $(1,5)$ . (b) Find a number $c \in(0,1)$ such that $f^{\prime}(c)$ is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in ( 0,1 ).

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Problem 782

Find the derivative of $y$ with respect to $x$ . $\begin{array}{l} y=\ln \left(x^{12}\right) \\ \frac{d y}{d x}=\square \end{array}$

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Problem 783

Find $y^{\prime \prime}$ for $y=\left(2+\frac{1}{x}\right)^{3}$ $y^{\prime \prime}=$

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Problem 784

The figure shows the electric field inside a cylinder of radius $R=3.3 \mathrm{~mm}$ . The field strength is increasing with time as $E=1.0 \times 10^{8} t^{2} \mathrm{~V} / \mathrm{m}$ , where $t$ is in s . The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t<0$ . (Figure 1)
Part A
Part B
Find an expression for the magnetic field strength as a function of time at a distance $r<R$ from the center. Express your answer in teslas as a multiple of product of distance $r$ and time $t$ .

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Problem 785

The function $f(x)=(4 x-1) e^{-3 x}$ has one critical number. Find it.

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Problem 786

Let $f(x)=x^{3}-12 x^{2}+45 x+6$ . Find the open intervals on which $f$ is increasing (decreasing). Then determine the $x$ -coordinates of all relative maxima (minima).
1. $f$ is increasing on the intervals $\square$

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Problem 787

If the tangent line equation to the curve $\sin (x y)=\sqrt{y}-x-\frac{1}{\sqrt{2}}$ at $\left(0, \frac{1}{2}\right)$ is $y=a x+\frac{1}{2}$ then $a=$

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Problem 788

If $x^{2}+y^{2}=13$ and $\frac{\mathrm{d} y}{\mathrm{~d} t}=8$ , find $\frac{\mathrm{d} x}{\mathrm{~d} t}$ when $y=3$ and $x=2$ . $\left.\frac{\mathrm{d} x}{\mathrm{~d} t}\right|_{(x, y)=(2,3)}=$

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Problem 789

Evaluate the infinite series by identifying it as the value of a derivative of a geometric series. $\sum_{n=1}^{\infty} \frac{n}{4^{n}}=$ $\square$ Hint: Write it as $f^{\prime}\left(\frac{1}{4}\right)$ where $f(x)=\sum_{n=0}^{\infty} x^{n}$ . Question Help: D Post to forum

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Problem 790

Find the derivative of the function $y = \sqrt{1-x^{2}} \, \operatorname{arcCos}(x)$ .

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Problem 791

Find the derivative of the function $f(x) = \sin^2(2-x^2) + 3 \tan(e^x)$ .

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Problem 792

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R},(x, y) \mapsto x^{2}-y^{2}$ , and let $S$ be the circle of radius 1 around the orig Find the extrema of $f \mid S$ .

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Problem 793

Exercises - find the derivative of $y=f(x)$ at $x=0$ where $y$ is determined by $y^{5}+2 y-x-3 x^{7}=0$ - find the tangent line of $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ at $\left(2, \frac{3 \sqrt{3}}{2}\right)$

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Problem 794

Find the derivative of the function $g(x) = \ln \left| x^2 - 4 \right|$ .

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Problem 795

Leiten Sie die Funktion ab und klammern Sie aus.

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Problem 796

23. $f(x)=\frac{x+1}{x-1}$ on $[2,4]$
HIDE ANSWER
Answer: Absolute maximum value: 3 ; absolute minimum value: $\frac{5}{3}$

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Problem 797

Find the derivative of the function $y=(\csc x+\cot x)^{-1}$ . $\frac{d y}{d x}=$ $\square$

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Problem 798

Find the derivative of the function below. $\begin{array}{l} h(x)=x \cot (10 \sqrt{x})+17 \\ \frac{d h}{d x}=\square \end{array}$ $\square$

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Problem 799

Aufgabe 1 Wandern a) Lukas unternimmt eine Wanderung.
Zu Beginn wandert er für 1 h 15 min mit einer konstanten Geschwindigkeit von $4 \mathrm{~km} / \mathrm{h}$ . Dann wandert er mit einer konstanten Geschwindigkeit von $2 \mathrm{~km} / \mathrm{h}$ weiter. Er benötigt für die gesamte Wanderung 3 h 45 min . 1) Berechnen Sie die mittlere Geschwindigkeit für die gesamte Wanderung. $[0 / 1$ P. $]$ b) Lena unternimmt eine Wanderung.
Der von ihr zurückgelegte Weg kann dabei in Abhängigkeit von der Zeit näherungsweise durch die Funktion s beschrieben werden. $s(t)=0,32 \cdot t^{3}-2,32 \cdot t^{2}+7,08 \cdot t \text { mit } 0 \leq t \leq 4,5$ $t$ ... Zeit seit Beginn der Wanderung in h $s(t)$ ... zurückgelegter Weg zur Zeit $t$ in km
In der nebenstehenden Abbildung ist der Graph der Funktion s dargestellt. 1) Bestimmen Sie die Durchschnittsgeschwindigkeit von Lena in den ersten 2 Stunden ihrer Wanderung. $[0 / 1 \mathrm{P}$ . 2) Ermitteln Sie, nach welcher Zeit Lena mit der geringsten Geschwindigkeit wandert. [0 / 1 P.

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Problem 800

6. Aşağıdaki sorulardan kaçı aynı yanıta sahiptir? I. $y=\cos (2 t), x=\sin t \Rightarrow \frac{d y}{d x}=$ ? II. $f(t)=\cos t, \lim _{h \rightarrow \infty} h\left[f\left(t+\frac{4}{h}\right)-f(t)\right]=$ ? III. $y=4 \sin t, x=t^{2} \Rightarrow \frac{d^{2} y}{d x^{2}}=$ ? IV. $f(x)=\cos x$ ve $t \in \mathbb{R}^{+}$ . $\lim _{x \rightarrow t} 2\left[\frac{f(x)-f(t)}{\sqrt{x t}-t}\right]=?$ V. $y=\cos ^{2} t$ eğrisine $(t, y)=\left(-\frac{\pi}{4}, \frac{1}{2}\right)$ noktasında teğet olan doğrunun denklemi $y=b t-\frac{\pi}{a}+\frac{1}{2}$ ise $a \sin (b t)=$ ? (a) 2 (b) 3 (c) 4 (d) 5 (e) Hiç

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Derivatives

Problem 701

Differentiate the following function. y=(ln⁡x)10+ln⁡(x10)dydx=10\begin{array}{l} y=(\ln x)^{10}+\ln \left(x^{10}\right) \\ \frac{d y}{d x}=10 \end{array}y=(lnx)10+ln(x10)dxdy​=10​

Problem 702

Find an equation of the tangent line to the curve y=ex\mathrm{y}=e^{\mathrm{x}}y=ex when x=−3.73\mathrm{x}=-3.73x=−3.73. y=y=y= □\square□ (Simplify your answer. Use integers or decimals for any numbers in the equation. as needed.)

Problem 703

Problem 704

Problem 705

Given that v=4x2+2v=4 x^{2}+2v=4x2+2, find ddx(v5−4sin⁡x)\frac{d}{d x}\left(v^{5}-4 \sin x\right)dxd​(v5−4sinx) in terms of only xxx.

Problem 706

Problem 707

Problem 708

Problem 709

he second derivative of the function. f(x)=113x−4f(x)=\frac{1}{13 x-4}f(x)=13x−41​

Problem 710

Use logarithmic differentiation to find the derivative. y=x2+3x2+45y=\sqrt[5]{\frac{x^{2}+3}{x^{2}+4}}y=5x2+4x2+3​​

Problem 711

Problem 712

* Consider the function f(x)=x((x2−1)2+12)f(x)=x\left(\left(x^{2}-1\right)^{2}+\frac{1}{2}\right)f(x)=x((x2−1)2+21​) - Find the transition points.

Problem 713

Problem 714

Question 2Plot the function f(x)=2xx2−5f(x)=\frac{2 x}{x^{2}-5}f(x)=x2−52x​. How many inflection points does this function have? 0 2 1 3

Problem 715

Question 7If x2+xy3=1x^{2}+x y^{3}=1x2+xy3=1, find y′y^{\prime}y′ at (2,−1)(2,-1)(2,−1). Answer this in decimal form, rounded to one decimal place.

Problem 716

Implicit diffrenciation y=x2+xyy=x^{2}+x yy=x2+xy

Problem 717

Problem 718

Question 11Find the critical numbers of the following function f(x)=14x4−53x3+x2+8xf(x)=\frac{1}{4} x^{4}-\frac{5}{3} x^{3}+x^{2}+8 xf(x)=41​x4−35​x3+x2+8x. How many critical numbers does this function have? 1 4 2 3

Problem 719

Question 12 1 ptsConsider the function f(x)=excos⁡xf(x)=e^{x} \cos xf(x)=excosx. Using a linear approximation of this function centered at the point x=2x=2x=2, estimate f(x)f(x)f(x) at x=2x=2x=2. (round to one decimal place) □\square□

Problem 720

2. Differentiate the following function. (Do not simplify your a) [8pts] f(x)=cos⁡(2x)tan⁡(3x)−ex2−3xf(x)=\cos (2 x) \tan (3 x)-e^{x^{2}-3 x}f(x)=cos(2x)tan(3x)−ex2−3x

Problem 721

Problem 722

Consider the function f(x)=excos⁡xf(x)=e^{x} \cos xf(x)=excosx. Using a linear approximation of this function centered point x=2x=2x=2, estimate f(x)f(x)f(x) at x=2x=2x=2. (round to one decimal place) Not saved

Problem 723

Problem 724

Find the coordinates of the maximum point. y=4ln⁡x+4xy=\frac{4 \ln x+4}{\sqrt{x}}y=x​4lnx+4​The maximum point is □\square□ (Type an ordered pair. Type your answer using exponential notation.)

Problem 725

Find the derivative of the function f(x)=(5x−5)2(4x2+1)3 f(x) = (5x - 5)^2(4x^2 + 1)^3 f(x)=(5x−5)2(4x2+1)3.

Problem 726

Problem 727

Problem 728

Problem 729

Find the tangent line equation for f(x)=3−xf(x)=\sqrt{3-x}f(x)=3−x​ at a given point.

Problem 730

Find the derivative of the function y=x4+12xy=x^{4}+12xy=x4+12x. What is dydx\frac{dy}{dx}dxdy​?

Problem 731

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft} / \mathrm{s}41ft/s. Its height is y=41t−22t2y=41 t-22 t^{2}y=41t−22t2. Find average velocity from t=2t=2t=2 and instantaneous velocity at t=2t=2t=2.

Problem 732

Find the rate of change of the city's population modeled by P(t)=22e0.08tP(t)=22 e^{0.08 t}P(t)=22e0.08t from 2025 to 2034.

Problem 733

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Height after ttt seconds is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity from t=2t=2t=2 for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at t=2t=2t=2?

Problem 734

Encuentra la derivada de f(x)=3x3−xf(x) = 3x^3 - xf(x)=3x3−x.

Problem 735

Approximate the average rate of change of COVID-19 infections in Minnesota from 0 to 8 days after April 1, 2020.

Problem 736

Find the rate of change of the city's population, modeled by P(t)=10e0.04tP(t)=10 e^{0.04 t}P(t)=10e0.04t, from 2021 to 2030 in thousand persons/year.

Problem 737

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using ΔyΔx\frac{\Delta y}{\Delta x}ΔxΔy​:1. 0 to 8 days: (calculate)2. 8 to 16 days: 0.59 thousand/day3. 0 to 16 days: 0.46 thousand/day

Problem 738

Problem 739

A particle PPP moves along a line with displacement x=13t3−4t2+15tx = \frac{1}{3} t^{3} - 4 t^{2} + 15 tx=31​t3−4t2+15t. Find: (a) when PPP is at rest, (b) distance in 5 seconds.

Problem 740

Find the distance from the origin when the particle stops, given v=9t−3t2v=9t-3t^{2}v=9t−3t2 m/s and starts at t=0t=0t=0.

Problem 741

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Its height after ttt seconds is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity for t=2t=2t=2 over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at t=2t=2t=2.

Problem 742

A ball is thrown with a velocity of 41ft/s41 \mathrm{ft/s}41ft/s. Its height is y=41t−22t2y=41t-22t^2y=41t−22t2. Find average velocity from t=2t=2t=2 for given intervals and the instantaneous velocity at t=2t=2t=2.

Problem 743

Find the limit: lim⁡h→0(x+h)2−x2h\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h}limh→0​h(x+h)2−x2​.

Problem 744

Relatel Ratas17. The volume of a cube decreases at a rate of 10 m3/s10 \mathrm{~m}^{3} / \mathrm{s}10 m3/s. Find the rate at which the side of the cube changes when the side of the cube is 2 m .

Problem 745

Problem 746

Problem 747

Find dy/dxd y / d xdy/dx by implicit differentiation. dy/dx=□sin⁡x+cos⁡y=sin⁡xcos⁡yd y / d x=\square \quad \sin x+\cos y=\sin x \cos ydy/dx=□sinx+cosy=sinxcosy

Problem 748

Differentiate the following function. $\begin{array}{l} y=(\ln x)^{10}+\ln \left(x^{10}\right) \\ \frac{d y}{d x}=10 \end{array}$

Find an equation of the tangent line to the curve $\mathrm{y}=e^{\mathrm{x}}$ when $\mathrm{x}=-3.73$ . $y=$ $\square$ (Simplify your answer. Use integers or decimals for any numbers in the equation. as needed.)

Given that $v=4 x^{2}+2$ , find $\frac{d}{d x}\left(v^{5}-4 \sin x\right)$ in terms of only $x$ .

he second derivative of the function. $f(x)=\frac{1}{13 x-4}$

Use logarithmic differentiation to find the derivative. $y=\sqrt[5]{\frac{x^{2}+3}{x^{2}+4}}$

* Consider the function $f(x)=x\left(\left(x^{2}-1\right)^{2}+\frac{1}{2}\right)$ - Find the transition points.

Question 2
Plot the function $f(x)=\frac{2 x}{x^{2}-5}$ . How many inflection points does this function have? 0 2 1 3

Question 7
If $x^{2}+x y^{3}=1$ , find $y^{\prime}$ at $(2,-1)$ . Answer this in decimal form, rounded to one decimal place.

Implicit diffrenciation $y=x^{2}+x y$

Question 11
Find the critical numbers of the following function $f(x)=\frac{1}{4} x^{4}-\frac{5}{3} x^{3}+x^{2}+8 x$ . How many critical numbers does this function have? 1 4 2 3

Question 12 1 pts
Consider the function $f(x)=e^{x} \cos x$ . Using a linear approximation of this function centered at the point $x=2$ , estimate $f(x)$ at $x=2$ . (round to one decimal place) $\square$

2. Differentiate the following function. (Do not simplify your a) [8pts] $f(x)=\cos (2 x) \tan (3 x)-e^{x^{2}-3 x}$

Consider the function $f(x)=e^{x} \cos x$ . Using a linear approximation of this function centered point $x=2$ , estimate $f(x)$ at $x=2$ . (round to one decimal place) Not saved

Find the coordinates of the maximum point. $y=\frac{4 \ln x+4}{\sqrt{x}}$
The maximum point is $\square$ (Type an ordered pair. Type your answer using exponential notation.)

Find the derivative of the function $f(x) = (5x - 5)^2(4x^2 + 1)^3$ .

Find the tangent line equation for $f(x)=\sqrt{3-x}$ at a given point.

Find the derivative of the function $y=x^{4}+12x$ . What is $\frac{dy}{dx}$ ?

A ball is thrown with a velocity of $41 \mathrm{ft} / \mathrm{s}$ . Its height is $y=41 t-22 t^{2}$ . Find average velocity from $t=2$ and instantaneous velocity at $t=2$ .

Find the rate of change of the city's population modeled by $P(t)=22 e^{0.08 t}$ from 2025 to 2034.

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Height after $t$ seconds is $y=41t-22t^2$ . Find average velocity from $t=2$ for 0.01, 0.005, and 0.002 seconds. What is the instantaneous velocity at $t=2$ ?

Encuentra la derivada de $f(x) = 3x^3 - x$ .

Find the rate of change of the city's population, modeled by $P(t)=10 e^{0.04 t}$ , from 2021 to 2030 in thousand persons/year.

Calculate the average rates of change for COVID-19 infections in Minnesota over these intervals using $\frac{\Delta y}{\Delta x}$ :
1. 0 to 8 days: (calculate)
2. 8 to 16 days: 0.59 thousand/day
3. 0 to 16 days: 0.46 thousand/day

A particle $P$ moves along a line with displacement $x = \frac{1}{3} t^{3} - 4 t^{2} + 15 t$ . Find: (a) when $P$ is at rest, (b) distance in 5 seconds.

Find the distance from the origin when the particle stops, given $v=9t-3t^{2}$ m/s and starts at $t=0$ .

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height after $t$ seconds is $y=41t-22t^2$ . Find average velocity for $t=2$ over 0.01, 0.005, 0.002, and 0.001 seconds, then determine instantaneous velocity at $t=2$ .

A ball is thrown with a velocity of $41 \mathrm{ft/s}$ . Its height is $y=41t-22t^2$ . Find average velocity from $t=2$ for given intervals and the instantaneous velocity at $t=2$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h}$ .

Relatel Ratas
17. The volume of a cube decreases at a rate of $10 \mathrm{~m}^{3} / \mathrm{s}$ . Find the rate at which the side of the cube changes when the side of the cube is 2 m .

Find $d y / d x$ by implicit differentiation. $d y / d x=\square \quad \sin x+\cos y=\sin x \cos y$

Problem 5. (1 point)
Find the equation of the line tangent to the graph of $y=2 \ln (x)$ at $x=1$ . Tangent Line: $y=$ $\square$

Calculate the derivative of $y$ with respect to $x$ . Express derivative in terms of $x$ and $y$ . $e^{2 x y}=\sin \left(y^{7}\right)$ (Express numbers in exact form. Use symbolic notation and fractions where needed.) $\frac{d y}{d x}=$ $\square$

$: \operatorname{lR} \omega_{c}=20$ -II $b \in \mathbb{R}, \alpha \in \mathbb{R}\left(g(x)=-x^{3}+\alpha x+b\right.$ $A(0 ; 2)$ - (II $D_{f}$ (6) (

Trova i punti di massimo e di minimo della funzione $y = \sin 3x + \sqrt{3} \cos 3x$ e traccia il grafico.

Find $y^{\prime \prime}$ if $x^{3} y^{3}-4=0$ $\frac{y}{x^{2}}$ $\frac{2 x}{y^{2}}$ $\frac{2 y}{x^{2}}$

Find $f$ such that $f^{\prime}(x)=8 x-5, f(9)=0$ . $f(x)=$

Differentiate. $\begin{array}{c} y=\frac{\ln x}{x^{5}} \\ \frac{d y}{d x}=\square \end{array}$ $\square$

32. What is the minimum value of $f(x)=x \ln (x)$ ?

45. Let $f$ be the function with derivative given by $f^{\prime}(x)=x^{2}-\frac{2}{x}$ . On which of the following intervals is $f$ decreasing?

Consider the function $f(x)=x e^{-5 x}, \quad 0 \leq x \leq 2$ . This function has an absolute minimum value equal to: $\square$ which is attained at $x=$ $\square$ and an absolute maximum value equal to: which is attained at $x=$ $\square$

Given that $u=v^{2}-4$ , find $\frac{d}{d v}\left(3 u^{5}-2 \sin v\right)$ in terms of only $v$ .

54. The function $f$ is defined by $f(x)=e^{-x}\left(x^{2}+2 x\right)$ . At what values of $x$ does $f$ have a relative maximum? (A) $x=-2+\sqrt{2}$ and $x=-2-\sqrt{2}$

Find the marginal cost $M C$ and average cost $A C$ from the total cost function $T C=300 \ln (q+30)+150$ .

Find the tangent line equation for $f(x) = \sqrt{x}$ at the point (1,1).

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$ .

Find the function $f(x)$ and the number $c$ given the limit: $\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}$

Find $F^{\prime}(x)$ using the first principle if $F(x)=x^{2}-3x$ .

Find the average rate of change of $h(t)=\cot t$ over the intervals: a. $\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right]$ b. $\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]$