Derivatives

Problem 401

6 Abiturprüfung 2012 (Bayern), Analysis, Aufgabengruppe II, Teil 1, Aufgabe 2 Gegeben ist die in R\mathbb{R} definierte Funktion g:xxe2x\mathrm{g}: \mathrm{x} \mapsto \mathrm{x} \cdot \mathrm{e}^{-2 \mathrm{x}}. a) Bestimmen Sie die Koordinaten des Punkts, in dem der Graph von gg eine waagrechte Tangente hat. b) Geben Sie das Verhalten von g für xx \rightarrow-\infty und x+x \rightarrow+\infty an.

See Solution

Problem 402

Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y -shaped configuation.
Centerville is located at (8,0)(8,0) in the xyx y-plane, Springfield is at (0,5)(0,5), and Shelbyville is at (0,5)(0,-5). The cable runs from Centerville to some point (x,0)(x, 0) on the xx-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0)(x, 0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.
To solve this problem we need to minimize the following function of xx : f(x)=f(x)= \square We find that f(x)f(x) has a critical number at x=x= \square To verify that f(x)f(x) has a minimum at this critical number we compute the second derivative f(x)f^{\prime \prime}(x) and find that its value at the critical number is \square , a positive number.
Thus the minimum length of cable needed is \square Question Help: \square Video Submit Question

See Solution

Problem 403

2,3,4,5,6,7,8,2,10,11,12,13,14,15,16,17,18,12,20,21,22,23,24,25\underline{2}, \underline{3}, \underline{4}, \underline{5}, \underline{6}, 7, \underline{8}, 2, \underline{10}, \underline{11}, \underline{12}, \underline{13}, \underline{14}, \underline{15}, \underline{16}, \underline{17}, \underline{18}, \underline{12}, \underline{20}, \underline{21}, \underline{22}, \underline{23}, \underline{24}, \underline{25}, and 26\underline{26} Differentiate the function.
3. f(x)=ln(x2+3x+5)f(x)=\ln \left(x^{2}+3 x+5\right)

Answer

See Solution

Problem 404

3.2 Implicit Differentiation
Calculus Name:
Find the equation of the tangent line at the given point. (7) x2y2=27x^{2}-y^{2}=27 at (6,3)(6,-3)
8. (xy)24x=20y(x-y)^{2}-4 x=20 y at (4,2)(4,2)

See Solution

Problem 405

Find an equation of the tangent line to the curve y=ex\mathrm{y}=e^{\mathrm{x}} when x=3.57\mathrm{x}=-3.57. y=y= \square (Simplify your answer. Use integers or decimals for any numbers in the equation. Round the final answer as needed.)

See Solution

Problem 406

Submit Answer [-/1 Points] DETAILS MY NOTES TANAPCAL
Find the derivative of the function. f(x)=ln(x22)f(x)=\begin{array}{l} f(x)=\ln \left(\sqrt{x^{2}-2}\right) \\ f^{\prime}(x)=\square \end{array} Need Help? Read it Watch It

See Solution

Problem 407

9. Find dydθ\frac{d y}{d \theta} for y=cscθcotθy=\csc \theta-\cot \theta. (a) 0 (b) cot2θ+cscθcotθ-\cot ^{2} \theta+\csc \theta \cot \theta (c) secθtanθsec2θ\sec \theta \tan \theta-\sec ^{2} \theta (d) cscθcotθ+csc2θ-\csc \theta \cot \theta+\csc ^{2} \theta (e) None of these

See Solution

Problem 408

Given x+y=y3\sqrt{x+y}=y^{3}, find dydx\frac{d y}{d x} in terms of xx and yy.

See Solution

Problem 409

yx=xyy=??\begin{array}{l}y^{x}=x^{y} \\ y^{\prime}=? ?\end{array}

See Solution

Problem 410

2. g(x)=x1+x2g(x)=\frac{x}{1+x^{2}}, so g(x)=(1+x2)(1)x(2x)(1+x2)2=1x2(1+x2)2g^{\prime}(x)=\frac{\left(1+x^{2}\right)(1)-x(2 x)}{\left(1+x^{2}\right)^{2}}=\frac{1-x^{2}}{\left(1+x^{2}\right)^{2}} and g(x)=(1+x2)2(2x)(1x2)2(1+x2)(2x)(1+x2)4=2x(1+x2)(1+x2+22x2)(1+x2)4=2x(3x2)(1+x2)3g^{\prime \prime}(x)=\frac{\left(1+x^{2}\right)^{2}(-2 x)-\left(1-x^{2}\right) 2\left(1+x^{2}\right)(2 x)}{\left(1+x^{2}\right)^{4}}=\frac{-2 x\left(1+x^{2}\right)\left(1+x^{2}+2-2 x^{2}\right)}{\left(1+x^{2}\right)^{4}}=-\frac{2 x\left(3-x^{2}\right)}{\left(1+x^{2}\right)^{3}} The sign diagram for gg^{\prime \prime} shows that gg is concave downward on (,3)(-\infty,-\sqrt{3}) and (0,3)(0, \sqrt{3}) and concave upward on (3,0)(-\sqrt{3}, 0) and (3,)(\sqrt{3}, \infty).

See Solution

Problem 411

y=ln(ex+e2x+1)y=??y=??\begin{array}{c}y=\ln \left(e^{x}+\sqrt{e^{2 x}+1}\right) \\ y^{\prime}=? ? \\ y^{\prime \prime}=? ?\end{array}

See Solution

Problem 412

If h(x)=ln(3x2)h(x)=\ln \left(3 x^{2}\right), then h(2)=h^{\prime \prime}(2)=

See Solution

Problem 413

5. Consider the function h(x)=x+1xh(x)=x+\frac{1}{x}. Use the first derivative to: a) Identify (exactly) a relative maximum (x,h(x))(x, h(x)). b) Identify (exactly) a relative minimum (x,h(x))(x, h(x)).

See Solution

Problem 414

Remaining Time: 01:45:20
Aladder 18 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1ft/s1 \mathrm{ft} / \mathrm{s}, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 10 ft from the wall?
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

See Solution

Problem 415

A water tank has the shape of a right circular cone, with the tip pointing downwards. The tank is 22 cm tall, and the radius (at the top) is 10 cm . If water is being drained from the tank at a rate of 20 cm3/s20 \mathrm{~cm}^{3} / \mathrm{s}, find the rate at which the water level is changing when the water is 12 cm deep.
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

See Solution

Problem 416

Remaining Time: 00:23:20
A ladder 18 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 9ft/s9 \mathrm{ft} / \mathrm{s}, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 9 ft from the wall?
Do not include units in your answer. Your answer can be exact or approximate. If it is approximate, round to three decimal places.

See Solution

Problem 417

2. If y=sinxcosxy=\sin x \cos x, then at x=π6,dydx=x=\frac{\pi}{6}, \frac{d y}{d x}=

See Solution

Problem 418

(1 point)
Let f(x)=x+1xf(x)=x+\sqrt{1-x} Find the local maximum and minimum values of ff using both the first and second derivative tests. Which method do you prefer? (That last question can be treated as rhetorical)
Below, type none if there are none. Points with local maximum values \square Points with local minimum values \square

See Solution

Problem 419

5. Review. A bead slides without fricAMT tion around a loop-the-loop (Fig. M P8.5). The bead is released from rest at a height h=3.50Rh=3.50 R. (a) What
Figure P8.5 is its speed at point (A)? (b) How large is the normal force on the bead at point (A) if its mass is 5.00 g ?

See Solution

Problem 420

if y=ln(ex+e2x+1)\quad y=\ln \left(e^{x}+\sqrt{e^{2 x}+1}\right) then Prove in (ex21)y=y\left(e^{x^{2}}-1\right) y^{\prime \prime}=y^{\prime}

See Solution

Problem 421

1. Исследовать функции на локальный экстремум. Указать точки экстремума, вид экстремума, значение функции в точках экстремума. (по 1 баллу)
Ответы: 1.1z=3x+6yx2xy+y21.1 z=3 x+6 y-x^{2}-x y+y^{2} экстремумов нет

See Solution

Problem 422

ДЗ 13 ответы
1. Исследовать функции на локальный экстремум. Указать точки экстремума, вид экстремума, значение функции в точках экстремума. (по 1 баллу)

Ответы: 1.1z=3x+6yx2xy+y21.2z=2x3xy2+5x2+y2zmin(0;0)=01.3z=ex2(x+y2)zmin(2;0)=2/e\begin{array}{ll} 1.1 z=3 x+6 y-x^{2}-x y+y^{2} & \\ 1.2 z=2 x^{3}-x y^{2}+5 x^{2}+y^{2} & \boldsymbol{z}_{\min }(\mathbf{0} ; \mathbf{0})=\mathbf{0} \\ 1.3 z=e^{\frac{x}{2}}\left(x+y^{2}\right) & \boldsymbol{z}_{\min }(-\mathbf{2} ; \mathbf{0})=-\mathbf{2} / \boldsymbol{e} \end{array}

See Solution

Problem 423

Quiz-4: Problem 1 (1 point)
Suppose that f(x)=6exexef(x)=6 e^{x}-e x^{e}. Find f(3)f^{\prime}(3).

See Solution

Problem 424

2. Исследовать функции на условный экстремум. (по 1 баллу) 2.1z=x4y32.1 z=\sqrt[4]{x} \sqrt[3]{y} \quad при условии 2x+5y=1002 x+5 y=100

See Solution

Problem 425

Find the derivative of the function f(x)=6x2+3 f(x) = 6^{\sqrt{x^2 + 3}} .

See Solution

Problem 426

\text{If } y^{x} = x^{y} \text{ then prove } y' = \frac{y^{2}(1-\ln x)}{x^{2}(1-\ln y)} \\

See Solution

Problem 427

Übung 5 Untersuchen Sie die Funktion f auf lokale Extrempunkte. Skizzieren Sie den Graphen oder stellen Sie ihn mit dem TR/Computer dar. a) f(x)=x2+exf(x)=x-2+e^{-x} b) f(x)=x2ex+1f(x)=x^{2} \cdot e^{x+1}
Übung 6 Untersuchen Sie die Funktion f auf Wendepunkte. Skizzieren Sie den Graphen oder stellen Sie ihn mit dem TR/Computer dar. a) f(x)=2exexf(x)=2 \cdot e^{x}-e^{-x} b) f(x)=(x21)e0,5xf(x)=\left(x^{2}-1\right) \cdot e^{-0,5 x}

See Solution

Problem 428

Übung 5 Untersuchen Sie die Funktion f auf lokale Extrempunkte. Skizzieren Sie den Graphen oder stellen Sie ihn mit dem TR/Computer dar. a) f(x)=x2+exf(x)=x-2+e^{-x} b) f(x)=x2ex+1f(x)=x^{2} \cdot e^{x+1}
Übung 6 Untersuchen Sie die Funktion f auf Wendepunkte. Skizzieren Sie den Graphen oder stellen Sie ihn mit dem TR/Computer dar. a) f(x)=2exexf(x)=2 \cdot e^{x}-e^{-x} b) f(x)=(x21)e0,5xf(x)=\left(x^{2}-1\right) \cdot e^{-0,5 x}

See Solution

Problem 429

5. Kurvendiskussionen
Führen Sie eine Kurvendiskussion durch. Überprüfen Sie hierzu f auf Nullstellen, Extrema und Wendepunkte. Untersuchen Sie, wie f sich für x±x \rightarrow \pm \infty verhält. Skizzieren Sie den Graphen von ff in einem sinnvollen Bereich. Überprüfen Sie Ihre Skizze mit dem TR/Computer. a) f(x)=(2x+2)e0,5xf(x)=(2 x+2) \cdot e^{-0,5 x} b) f(x)=(1x)e2xf(x)=(1-x) \cdot e^{2-x} c) f(x)=ex2exf(x)=e^{x}-2 e^{-x}

See Solution

Problem 430

f(x)=4x10f^{\prime}(x)=4 x-10
Finding the Derivative of a Simple Rational function Que if f(x)=1x(x0)f(x)=\frac{1}{x}(x \neq 0) find f(x)f^{\prime}(x) Que. If f(x)=xf(x)=\sqrt{x}, ford f(x)f^{\prime}(x) Now f(x)=1xf(x)=\frac{1}{x}

See Solution

Problem 431

The function h(x)h(x) is defined in terms of a differentiable f(x)f(x). When h(x)=f(x5)xh(x)=\frac{f\left(x^{5}\right)}{x}, find an expression for h(x)h^{\prime}(x).
Choose the correct answer below. A. h(x)=5x5f(x5)f(x5)x2h^{\prime}(x)=\frac{5 x^{5} f^{\prime}\left(x^{5}\right)-f\left(x^{5}\right)}{x^{2}} Rh(x)=5x4f(x4)f(x5)\cap R \quad h^{\prime}(x)=5 x^{4} f^{\prime}\left(x^{4}\right)-f\left(x^{5}\right)

See Solution

Problem 432

If y334xy+4x=0-y^{3}-3-4 x y+4 x=0 then find dydx\frac{d y}{d x} in terms of xx and yy.

See Solution

Problem 433

A box is to be made out of a 10 cm by 16 cm piece of cardboard. Squares of side length x cmx \mathrm{~cm} will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. (A) Express the volume VV of the box as a function of xx. V=cm3V=\square \mathrm{cm}^{3} (B) Give the domain of VV in interval notation. (Use the fact that length and volume must be positive.) domain == \square (C) Find the length LL, width WW, and height HH of the resulting box that maximizes the volume. (Assume that WLW \leq L ). L=cmW=cmH=cm\begin{array}{l} L=\square \mathrm{cm} \\ W=\square \mathrm{cm} \\ H=\square \mathrm{cm} \end{array} (D) The maximum volume of the box is \square cm3\mathrm{cm}^{3}.

See Solution

Problem 434

If f(x)=sin4xf(x)=\sin ^{4} x, find f(x)f^{\prime}(x)
Find f(5)f^{\prime}(5) \qquad >> Next question

See Solution

Problem 435

Find f(x)f^{\prime}(x) and f(x)f^{\prime \prime}(x) for f(x)=x1/3(x+4)f(x)=x^{1 / 3}(x+4).

See Solution

Problem 436

The function ff is defined by f(x)=x2ex2f(x)=x^{2} e^{-x^{2}}. At what values of xx does ff have a relative maximum? (A) -2 (B) 0 (C) 1 only (D) -1 and 1

See Solution

Problem 437

If g(x)=lnxg(x)=\ln x and ff is a differentiable function of xx, which of the following is equivalent to the derivative of f(g(x))f(g(x)) with respect to xx ? (A) f(1x)f^{\prime}\left(\frac{1}{x}\right) (B) f(x)x\frac{f^{\prime}(x)}{x} (C) f(lnx)f^{\prime}(\ln x) (D) f(lnx)x\frac{f^{\prime}(\ln x)}{x}

See Solution

Problem 438

1998 AP Calculus AB Scoring Guidelines
5. Consider the curve defined by 2y3+6x2y12x2+6y=12 y^{3}+6 x^{2} y-12 x^{2}+6 y=1. (a) Show that dydx=4x2xyx2+y2+1\frac{d y}{d x}=\frac{4 x-2 x y}{x^{2}+y^{2}+1}. (b) Write an equation of each horizontal tangent line to the curve. (c) The line through the origin with slope -1 is tangent to the curve at point PP. Find the xx- and yy-coordinates of point PP.

See Solution

Problem 439

Find the derivative of y=x1/2(x2+3)2/3(3x4)4y=x^{-1 / 2}\left(x^{2}+3\right)^{2 / 3}(3 x-4)^{4}

See Solution

Problem 440

6. Find the points on the given curves where the tangent is horizontal or vertical. Check your answers with your grapher. (a) x=10t2,y=t312tx=10-t^{2}, \quad y=t^{3}-12 t (b) x=secθ,y=tanθx=\sec \theta, \quad y=\tan \theta

See Solution

Problem 441

Question 11 (1 point) \checkmark Saved
If f(x)>0f^{\prime}(x)>0 for all xx, then a) f(x1)>f(x2)f\left(x_{1}\right)>f\left(x_{2}\right) for every x1>x2x_{1}>x_{2} b) f(x1)=f(x2)f\left(x_{1}\right)=f\left(x_{2}\right) for every x1<x2x_{1}<x_{2} c) f(x1)<f(x2)f\left(x_{1}\right)<f\left(x_{2}\right) for every x1>x2x_{1}>x_{2} d) f(x1)>f(x2)f\left(x_{1}\right)>f\left(x_{2}\right) for every x1<x2x_{1}<x_{2}

See Solution

Problem 442

12. Find the derivative of y=xlog3xy=x^{\log _{3} x}

See Solution

Problem 443

Question Watch Video Show Examples Given the function y=1x3y=\frac{1}{x^{3}}, find dydx\frac{d y}{d x}. Express your answer in simplest form without using negative exponents.

See Solution

Problem 444

The graph of y=g(x)y=g^{\prime}(x) is shown below. Note: this is the graph of the DERIVATIVE. (a) Find the critical points of gg. x=x= \square (b) Determine the intervals where gg is increasing and the intervals where gg is decreasing.
Increasing: \square Decreasing: \square

See Solution

Problem 445

Let g(x)=3x440x3+96x2+17g(x)=3 x^{4}-40 x^{3}+96 x^{2}+17. (a) Find the critical points of gg. x=x= \square (b) Determine the intervals where gg is increasing and the intervals where gg is decreasing.
Increasing: \square
Decreasing: \square (c) Find the coordinates (x,y)(x, y) of the local extrema of gg.
Local max: \square Local min: \square

See Solution

Problem 446

Let g(x)=3x440x3+96x2+17g(x)=3 x^{4}-40 x^{3}+96 x^{2}+17. (a) Find the critical points of gg. x=0,2,8\boldsymbol{x}=0,2,8 (b) Determine the intervals where gg is increasing and the intervals where gg is decreasing.
Increasing: (0,2)(8,)(0,2) \cup(8, \infty)
Decreasing: (,0)(2,8)(-\infty, 0) \cup(2,8) (c) Find the coordinates (x,y)(x, y) of the local extrema of gg.
Local max: \square Local min: \square

See Solution

Problem 447

Differentiate the function with respect to the independent variable f(x)=lnx236x3216f(x)=\ln \frac{x^{2}-36}{x^{3}-216}

See Solution

Problem 448

Evaluate dydx\frac{d y}{d x} when x=0x=0 for each of the following (i) y=ln(x2+x2+1)y=\ln \left(x^{2}+\sqrt{x^{2}+1}\right). dydm=(1x2+x2+x)(2m+1\frac{d y}{d m}=\left(\frac{1}{x^{2}+\sqrt{x^{2}+x}}\right)(2 m+1 (ii) y=ex(2x2+1)x+1y=\frac{e^{x}\left(2 x^{2}+1\right)}{\sqrt{x+1}}.

See Solution

Problem 449

If f(x)=0xt3dtf(x)=\int_{0}^{x} t^{3} d t then f(x)=f(5)=\begin{array}{l} f^{\prime}(x)= \\ f^{\prime}(-5)= \end{array}

See Solution

Problem 450

2. (a) If AA is the area of a circle with radius rr and the circle expands as time passes, find dA/dtd A / d t in terms of dr/dtd r / d t. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s1 \mathrm{~m} / \mathrm{s}, how fast is the area of the spill increasing when the radius is 30 m ?

See Solution

Problem 451

At what point does the curve have maximum curvature? y=3ex(x,y)=()\begin{array}{c} y=3 e^{x} \\ (x, y)=(\square) \end{array}
What happens to the curvature as xx \rightarrow \infty ? x(x)x(x) approaches \square as xx \rightarrow \infty,

See Solution

Problem 452

10. A triangle with base bb and height hh is expanding such that its area is increasing at the rate 4 m2/s4 \mathrm{~m}^{2} / \mathrm{s}. (A) dAdt=4\frac{d A}{d t}=4 B) dbdt=4\frac{d b}{d t}=4 C) dhdt=4\frac{d h}{d t}=4 D) dtdA=4\frac{d t}{d A}=4 dAdt=4 m2/s\frac{d A}{d t}=4 \mathrm{~m}^{2} / \mathrm{s}
11. A spherical rock erodes over time due to winds and rain. The radius of the rock is changing at a rate of 3in/3 \mathrm{in} / year. Find the rate that the volume of the rock is changing when the surface area (A) of the rock is 400in2400 \mathrm{in}^{2}. A) dAdt=3\frac{d A}{d t}=3 B) drdt=3\frac{d r}{d t}=3 (C) dVdt=3\frac{d V}{d t}=-3 D) drdt=3\frac{d r}{d t}=-3

Directions: Solve the following problems. Show all work clearly and neatly. (3r+1)1/2(3 r+1)^{1 / 2}
12. Let ss and rr be variables such that 2s=3r+12 s=\sqrt{3 r+1}. Find drdt\frac{d r}{d t} when dsdt=5\frac{d s}{d t}=5 and s=2s=2. 2dsdt=12(3r+1)123drdt(22)2=3r+1225=123r+13drdt16=3r+1203r+1=3drdt15=3r20r\begin{array}{ll} 2 \frac{d s}{d t}=\frac{1}{2}(3 r+1)^{-\frac{1}{2}} \cdot 3 \frac{d r}{d t} & (2 \cdot 2)^{2}=\sqrt{3 r+1}^{2} \\ 2 \cdot 5=\frac{1}{2 \sqrt{3 r+1}} \cdot 3 \frac{d r}{d t} & 16=3 r+1 \\ 20 \sqrt{3 r+1}=3 \frac{d r}{d t} & 15=3 r \\ 20 r \end{array} 3r+1=3drdt203r+13=drdt=20163=2043=803dx\begin{array}{l} \sqrt{3 r+1}=3 \frac{d r}{d t} \\ \frac{20 \sqrt{3 r+1}}{3}=\frac{d r}{d t}=\frac{20 \sqrt{16}}{3}=\frac{20 \cdot 4}{3}=\frac{80}{3} d x \end{array}
13. A particle is moving along the graph of xy2=36x y^{2}=36 where dydt=2\frac{d y}{d t}=-2. Find dxdt\frac{d x}{d t} when y=3y=-3. x(3)2=36dxdt2ydydt=0x(9)=36dxdt232x=4dxdt12=0\begin{array}{ll} x(-3)^{2}=36 & \frac{d x}{d t} \cdot 2 y \frac{d y}{d t}=0 \\ x(9)=36 & \frac{d x}{d t} \cdot 2 \cdot-3 \cdot-2 \\ x=4 & \frac{d x}{d t} \cdot 12=0 \end{array}
14. The radius of a circle is decreasing at a rate of 3 inches per second. At what rate is the area of the circle changing when the circumference is 16π16 \pi inches?

See Solution

Problem 453

9. Find all critical points of the function. (a) f(x)=x22x+4f(x)=x^{2}-2 x+4 (b) f(x)=x392x254x+2f(x)=x^{3}-\frac{9}{2} x^{2}-54 x+2 (c) f(t)=8t3t2f(t)=8 t^{3}-t^{2} (d) f(x)=1x1x2f(x)=\frac{1}{x}-\frac{1}{x^{2}} (e) f(x)=x2x24x+8f(x)=\frac{x^{2}}{x^{2}-4 x+8} (f) f(t)=t4t+1f(t)=t-4 \sqrt{t+1} (g) f(t)=4tt2+1f(t)=4 t-\sqrt{t^{2}+1} (h) g(θ)=sin2θg(\theta)=\sin ^{2} \theta

See Solution

Problem 454

Question Watch Video Show Examples
The area of a circle is increasing at a constant rate of 178 square feet per second. At the instant when the radius of the circle is 4 feet, what is the rate of change of the radius? Round your answer to three decimal places.
Answer Attempt 1 out of 2 ftsec\frac{\mathrm{ft}}{\mathrm{sec}} Submit Answer

See Solution

Problem 455

oblem 3. (1 point) Find a function f(x)f(x) that satisfies f(x)=x38xf^{\prime}(x)=x^{3}-8 x and f(1)=4f(1)=4.
Answer: \square

See Solution

Problem 456

Find the derivative of the function g(x)=4x+5sin(x) g(x) = \sqrt{4x + 5\sin(x)} .

See Solution

Problem 457

2 3- جد مركبات شعاع الوحدة المماسي

See Solution

Problem 458

5. Найти наибольшее и наименьшее значения функции z=10xy2+x2+10x+1z=-10 x y^{2}+x^{2}+10 x+1 на замкнутом множестве D:x7+y21D: \frac{|x|}{7}+\frac{|y|}{2} \leq 1. (4 балла)

See Solution

Problem 459

Question: Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector. (a) f(x,y)=xy,(x0,y0)=(e,e),d=20i+21jf(x, y)=x^{y},\left(x_{0}, y_{0}\right)=(e, e), d=20 i+21 j (b) f(x,y,z)=ex+yz,(x0,y0,z0)=(1,1,1),df(x, y, z)=e x+y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,1,1), d

See Solution

Problem 460

44. The discriminant fxxfyyfxy2f_{x x} f_{y y}-f_{x y}{ }^{2} is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z=f(x,y)z=f(x, y) looks like. Describe your reasoning in each case. a. f(x,y)=x2y2f(x, y)=x^{2} y^{2} b. f(x,y)=1x2y2f(x, y)=1-x^{2} y^{2} c. f(x,y)=xy2f(x, y)=x y^{2} d. f(x,y)=x3y2f(x, y)=x^{3} y^{2} e. f(x,y)=x3y3f(x, y)=x^{3} y^{3} f. f(x,y)=x4y4f(x, y)=x^{4} y^{4}

See Solution

Problem 461

13. Consider the curve defined by x2+xy+y2=27x^{2}+x y+y^{2}=27. a) Show that dydx=2xyx+2y(2pts)\frac{d y}{d x}=\frac{-2 x-y}{x+2 y} \cdot(2 \mathrm{pts}) b) Write an equation for the line tangent to the curve at the point (3,6)(3,-6). (3 pts) c) Where is the curve not differentiable? Justify your answer. (3 pts) x+2y=0x+2 y=0 d) Find d2ydx2\frac{d^{2} y}{d x^{2}} in terms of xx and yy. (2 pts)

See Solution

Problem 462

14. Use the table below to find the following values. \begin{tabular}{|c|c|c|c|c|} \hlinexx & f(x)f(x) & g(x)g(x) & f(x)f^{\prime}(x) & g(x)g^{\prime}(x) \\ \hline \hline-1 & -5 & 1 & 3 & 0 \\ \hline 0 & -2 & 0 & 1 & 1 \\ \hline 1 & 0 & -3 & 0 & 0.5 \\ \hline 2 & 5 & -1 & 5 & 2 \\ \hline \end{tabular} (a) Find h(2)h^{\prime}(2) whenh (x)=3g(x)(x)=-3 g(x) (2 pts) (b) Find h(1)h^{\prime}(-1) when h(x)=f(13x)e3xh(x)=f(-1-3 x)-e^{3 x} (2 pts) (c) Find h(2)h^{\prime}(2) when h(x)=x3g(x)(2pts)h(x)=x^{3} g(x)(2 \mathrm{pts}) (d) Find h(0)h^{\prime}(0) whenh (x)=g(12x)f(x)(2pts)(x)=g\left(-\frac{1}{2} x\right) \cdot f(x)(2 \mathrm{pts}) (e) Find h(1)h^{\prime}(1) when h(x)=g(x)(x24)(2pts)h(x)=\frac{g(x)}{\left(x^{2}-4\right)}(2 \mathrm{pts}) (f) Find (f1)(5)\left(f^{-1}\right)^{\prime}(-5) (2 pts)

See Solution

Problem 463

If exyy2=e4e^{x y}-y^{2}=e-4, then at x=12x=\frac{1}{2} and y=2,dydx=y=2, \frac{d y}{d x}=

See Solution

Problem 464

Let H(t)=at40betH(t)=\frac{a}{t^{40}}-b \cdot e^{t} where aa and bb are both fixed constants. What is H(t)H^{\prime}(t) ?

See Solution

Problem 465

PREVIOUS ANSWERS PRACTICE ANOTHER
Find the equation for the plane tangent to each surface z=f(x,y)z=f(x, y) at the indicated point. (a) z=x3+y35xyz=x^{3}+y^{3}-5 x y, at the point (1,2,1)(1,2,-1)

See Solution

Problem 466

14. Given h(x)=f(g(x))h(x)=f(g(x)), use the graph to the right to find h(4)h^{\prime}(4). (A) 12\frac{1}{2} (B) 54\frac{5}{4} f(g(x))g(x)f^{\prime}(g(x)) g^{\prime}(x) (C) 58\frac{5}{8} (D) 5 f(g(4))g(4)f(5)(12)=54(12)\begin{array}{l} f^{\prime}(g(4)) g^{\prime}(4) \\ f^{\prime}(5)\left(\frac{1}{2}\right)=\frac{5}{4}\left(\frac{1}{2}\right) \end{array}

See Solution

Problem 467

If the figure below is the graph of the derivative f f^{\prime} , answer the following:
Where do the points of inflection of f f occur? \square
On which interval(s) is f f concave down? \square
Note: You can earn partial credit on this problem.

See Solution

Problem 468

26. What is wrong with the following argument? Suppose w=f(x,y,z)w=f(x, y, z) and z=g(x,y)z=g(x, y). By the chain rule, wx=wxxx+wyyx+wzzx=wx+wzzx\frac{\partial w}{\partial x}=\frac{\partial w}{\partial x} \frac{\partial x}{\partial x}+\frac{\partial w}{\partial y} \frac{\partial y}{\partial x}+\frac{\partial w}{\partial z} \frac{\partial z}{\partial x}=\frac{\partial w}{\partial x}+\frac{\partial w}{\partial z} \frac{\partial z}{\partial x}
Hence, 0=(w/z)(z/x)0=(\partial w / \partial z)(\partial z / \partial x), and so w/z=0\partial w / \partial z=0 or z/x=0\partial z / \partial x=0, which is, in general, absurd.

See Solution

Problem 469

Suppose f(π3)=4f\left(\frac{\pi}{3}\right)=4 and f(π3)=5f^{\prime}\left(\frac{\pi}{3}\right)=-5. Let g(x)=f(x)sin(x)g(x)=f(x) \sin (x) and h(x)=cos(x)f(x)h(x)=\frac{\cos (x)}{f(x)}. Find the following. (a) g(π3)g^{\prime}\left(\frac{\pi}{3}\right) \square (b) h(π3)h^{\prime}\left(\frac{\pi}{3}\right) \square

See Solution

Problem 470

Determine whether the statement is true or false. The derivative of a polynomial is a polynomial. True False Submit Answer

See Solution

Problem 471

Suppose f(π3)=2f\left(\frac{\pi}{3}\right)=2 and f(π3)=3f^{\prime}\left(\frac{\pi}{3}\right)=-3. Let g(x)=f(x)sin(x)g(x)=f(x) \sin (x) and h(x)=cos(x)f(x)h(x)=\frac{\cos (x)}{f(x)}. Find the following. (a) g(π3)g^{\prime}\left(\frac{\pi}{3}\right) \square (b) h(π3)h^{\prime}\left(\frac{\pi}{3}\right) \square Submit Answer

See Solution

Problem 472

Let f(x)=1x24f(x)=\frac{1}{x^{2}-4} Where does ff have critical points? Choose all answers that apply:
A x=2x=-2 B x=0x=0 c. x=2x=2
D ff has no critical points.

See Solution

Problem 473

Let g(x)=sin(2x)g(x)=\sin (2 x), for π2xπ2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}. Where does gg have critical points?

See Solution

Problem 474

Let h(x)=e2xx3h(x)=\frac{e^{2 x}}{x-3} Where does hh have critical points?

See Solution

Problem 475

For 14x13-14 \leq x \leq 13 the function ff is defined by f(x)=x7(x+8)2f(x)=x^{7}(x+8)^{2} On which two intervals is the function increasing (enter intervals in ascending order)? x=x= \square \square and \square to x=x= x=x= to x=x= \square Find the interval on which the function is positive: xx - \square to x=x= \square Where does the function achieve its minimum? x=x= \square

See Solution

Problem 476

Let hh be a polynomial function and let hh^{\prime}, its derivative, be defined as h(x)=x2(x2)2(x1)2h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}.
At how many points does the graph of hh have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

See Solution

Problem 477

7. Consider the curve given by the equation y33xy=2y^{3}-3 x y=2. a. Find dydx\frac{d y}{d x}.

See Solution

Problem 478

Find a general formula for the average rate of change on the interval [4,4+h][4,4+h]. Given f(x)=1x+11f(x)=\frac{1}{x+11}, use the Difference Quotient on [4,4+h][4,4+h]. (Note: Your answer will be an expression involving hh and must be simplified.) \square The interval [4,4+h][4,4+h] gets smaller when h0h \rightarrow 0. If we substitute h=0h=0 into our formula, the A.R.O.C simplies to what? \square This will be our estimate for the instanteous rate of change for x=4x=4.

See Solution

Problem 479

Question 7
Let f(x)=3x+4f(x)=\sqrt{3 x+4}. Calculate the difference quotient: f(x+h)f(x)h\frac{f(x+h)-f(x)}{h}

See Solution

Problem 480

Prove that sin2x+cos2x=1\sin ^{2} x+\cos ^{2} x=1 using Calculus by: (1) Find the derivative of f(x)=sin2x+cos2xf(x)=\sin ^{2} x+\cos ^{2} x (2) Use corollay to show that f(x)=1f(x)=1 on all of [0,2π][0,2 \pi].

See Solution

Problem 481

Let f(x)=x2tan1(8x)f(x)=x^{2} \tan ^{-1}(8 x) f(x)=f^{\prime}(x)=

See Solution

Problem 482

Current Attempt in Progress Let C(q)C(q) represent the cost, R(q)R(q) the revenue, and π(q)\pi(q) the total profit, in dollars, of producing qq items. (a) If C(50)=77C^{\prime}(50)=77 and R(50)=83R^{\prime}(50)=83, approximately how much profit is earned by the 51st 51^{\text {st }} item?
The profit earned from the 51st 51^{\text {st }} item will be approximately $\$ i \square (b) If C(90)=73C^{\prime}(90)=73 and R(90)=69R^{\prime}(90)=69, approximately how much profit is earned by the 91st 91^{\text {st }} item?
The profit earned from the 91st 91^{\text {st }} item will be approximately \ \squarei.(c)If i . (c) If \pi(q)isamaximumwhen is a maximum when q=78,howdoyouthink, how do you think C^{\prime}(78)and and R^{\prime}(78)compare? compare? C^{\prime}(78) \square R^{\prime}(78)$ eTextbook and Media

See Solution

Problem 483

6) An observer stands 700 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 900ft/sec900 \mathrm{ft} / \mathrm{sec}. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 2400 ft from the ground?

See Solution

Problem 484

(1 point) Let f(x)=(ln(x))sed(x)f(x)=(\ln (x))^{\operatorname{sed}(x)}. Find f(x)f^{\prime}(x). f(x)=f^{\prime}(x)=

See Solution

Problem 485

1. If f(x)=2x2+4f(x)=2 x^{2}+4, which of the following will calculate the derivative of f(x)f(x) ? (a) [2(x+Δx)2+4](2x2+4)Δx\frac{\left[2(x+\Delta x)^{2}+4\right]-\left(2 x^{2}+4\right)}{\Delta x} (b) limΔx0(2x2+4+Δx)(2x2+4)Δx\lim _{\Delta x \rightarrow 0} \frac{\left(2 x^{2}+4+\Delta x\right)-\left(2 x^{2}+4\right)}{\Delta x} (c) limΔx0[2(x+Δx)2+4](2x2+4)Δx\lim _{\Delta x \rightarrow 0} \frac{\left[2(x+\Delta x)^{2}+4\right]-\left(2 x^{2}+4\right)}{\Delta x} (d) (2x2+4+Δx)(2x2+4)Δx\frac{\left(2 x^{2}+4+\Delta x\right)-\left(2 x^{2}+4\right)}{\Delta x} (e) None of these

See Solution

Problem 486

For problems 11-14, use proper notation throughout. 11.) Consider the function f(t)f(t) Int a.) Calculate the instantancous rate of change of the function at t=12t=\frac{1}{2}, b.) Find the equation of the tangent line at the point where t=3t=3. Leave your answer in terms of the natural logarithm.

See Solution

Problem 487

12.) Find the following using the Limit Definition of Derivative. You should be able to do this with very little computation. \begin{tabular}{|l|l|} \hline a.) limΔx0sin(x+Δx)sin(x)Δx\lim _{\Delta x \rightarrow 0} \frac{\sin (x+\Delta x)-\sin (x)}{\Delta x} & b.) limh0(x+h)2x2h\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h} \\ c.) limxπ4sinxsinπ4xπ4\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\sin \frac{\pi}{4}}{x-\frac{\pi}{4}} & d.) limh0sin(π6+h)12h\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\frac{1}{2}}{h} \\ \hline e.) limΔx0(2+Δx)38Δx\lim _{\Delta x \rightarrow 0} \frac{(2+\Delta x)^{3}-8}{\Delta x} & f.) limxπcosx+1xπ\lim _{x \rightarrow \pi} \frac{\cos x+1}{x-\pi} \\ \hline & \\ \hline \end{tabular}

See Solution

Problem 488

5. Find d2ydx2\frac{d^{2} y}{d x^{2}} for y=x+3x1y=\frac{x+3}{x-1} (a) 0 (b) 8(x1)3\frac{-8}{(x-1)^{3}} (c) 4(x1)3\frac{-4}{(x-1)^{3}} (d) 8(x1)3\frac{8}{(x-1)^{3}} (e) None of these

See Solution

Problem 489

3. Find the a bsolute max and min f(x)=x323x,0x4f(x)=x^{\frac{3}{2}}-3 \sqrt{x}, 0 \leq x \leq 4 f(x)=x323x12f(x)=32x1232x12\begin{array}{l} f(x)=x^{\frac{3}{2}}-\frac{3 x^{\frac{1}{2}}}{} \\ f^{\prime}(x)=\frac{3}{2} x^{\frac{1}{2}}-\frac{3}{2} x^{-\frac{1}{2}} \end{array}

See Solution

Problem 490

7. Find dydx\frac{d y}{d x} if y23xy+x2=7y^{2}-3 x y+x^{2}=7. (a) 2x+y3x2y\frac{2 x+y}{3 x-2 y} (b) 3y2x2y3x\frac{3 y-2 x}{2 y-3 x} (c) 2x32y\frac{2 x}{3-2 y} (d) 2xy\frac{2 x}{y} (e) None of these

See Solution

Problem 491

8. Find yy^{\prime} if y=sin(x+y)y=\sin (x+y). (a) 0 (b) cos(x+y)1cos(x+y)\frac{\cos (x+y)}{1-\cos (x+y)} (c) cos(x+y)\cos (x+y) (d) 1 (c) None of these

See Solution

Problem 492

Compute the derivatives of the given functions. a) f(r)=10r.f(r)=f(r)=10^{r} . \quad f^{\prime}(r)= \square . b) g(s)=179.g(s)=g(s)=17^{9} . \quad g^{\prime}(s)= \square . b) h(t)=5t6th(t)=h(t)=\frac{5^{t}}{6^{t}} \quad h^{\prime}(t)= \square .

See Solution

Problem 493

10. [0/1 Points] DETAILS
MYNOTES
Find the absolute maximum and absolute minimum values of ff on the given interval. f(x)=4x36x2144x+1,[4,5]f(x)=4 x^{3}-6 x^{2}-144 x+1, \quad[-4,5] absolute minimum \square \square absolute maximum Need Help? Readit \square Watchlt \square \square Masterlit

See Solution

Problem 494

points) f(x)=sin(sin(x))f(x)=\sin (\sin (x)) then f(x)=f^{\prime}(x)=\square Preview My Answers Submit Answers

See Solution

Problem 495

(1 point) Please answer the following questions about the function f(x)=2x2x225f(x)=\frac{2 x^{2}}{x^{2}-25}
Instructions: - If you are asked for a function, enter a function. - If you are asked to find xx - or yy-values, enter either a number or a list of numbers separated by commas. If there are no solutions, enter None. - If you are asked to find an interval or union of intervals, use interval notation. Enter \{ \} if an interval is empty. - If you are asked to find a limit, enter either a number, I for ,I\infty,-I for -\infty, or DNED N E if the limit does not exist. (a) Calculate the first derivative of ff. Find the critical numbers of ff, where it is increasing and decreasing, and its local extrema. f(x)=f^{\prime}(x)= \square Critical numbers x=x= \square Union of the intervals where f(x)f(x) is increasing \square Union of the intervals where f(x)f(x) is decreasing \square Local maxima x=x= \square Local minima x=x= \square (b) Find the following left- and right-hand limits at the vertical asymptote x=5x=-5. limx52x2x225=\lim _{x \rightarrow-5^{-}} \frac{2 x^{2}}{x^{2}-25}= \square limx5+2x2x225=?\lim _{x \rightarrow-5^{+}} \frac{2 x^{2}}{x^{2}-25}=? \square Find the following left-and right-hand limits at the vertical asymptote x=5x=5. limx52x2x225=?\lim _{x \rightarrow 5^{-}} \frac{2 x^{2}}{x^{2}-25}=? \square limx5+2x2x225=\lim _{x \rightarrow 5^{+}} \frac{2 x^{2}}{x^{2}-25}= \square Find the following limits at infinity to determine any horizontal asymptotes. limx2x2x225=?limx+2x2x225=?\lim _{x \rightarrow-\infty} \frac{2 x^{2}}{x^{2}-25}=? \quad \vee \quad \lim _{x \rightarrow+\infty} \frac{2 x^{2}}{x^{2}-25}=? \square \square (c) Calculate the second derivative of ff. Find where ff is concave up, concave down, and has inflection points. f(x)=f^{\prime \prime}(x)=\square
Union of the intervals where f(x)f(x) is concave up \square Union of the intervals where f(x)f(x) is concave down \square \square Inflection points x=x=

See Solution

Problem 496

Suppose that f(x)=(5ln(x))3f(x)=(5-\ln (x))^{3}. Find f(1)f^{\prime}(1). f(1)=f^{\prime}(1)=

See Solution

Problem 497

Suppose that f(x)=3ln(x2+2)f(x)=\frac{3}{\ln \left(x^{2}+2\right)}
Find f(1)f^{\prime}(1). f(1)=f^{\prime}(1)=

See Solution

Problem 498

If f(x)=(5x+4)1f(x)=(5 x+4)^{-1}
Find f(x)f^{\prime}(x). Then f(x)=f^{\prime}(x)= \square Find f(3)f^{\prime}(3). Then f(3)=f^{\prime}(3)= \square

See Solution

Problem 499

(1 point) Let f(x)=e7x2f(x)=e^{-7 x^{2}}. Then f(x)f(x) has a relative minimum at x=x= a relative maximum at x=x=\square and inflection points at x=x=\square and at x=x=\square
Write DNE if any of the above do not exist. Write the inflection points (if ar

See Solution

Problem 500

Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. (If an answer is undefined, enter UNDEFINED.) y=25(x+5)2,(0,1)y=\frac{25}{(x+5)^{2}},(0,1) \square Need Help? Read It Submit Answer

See Solution
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord