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Problem 2501

Réflexion
EXERCICE 1: CoTon
L'entreprise CoTon produit du tissu en coton. Celui-ci est fabriqué en 1 mètre de large et pour une longueur $x$ exprimée en kilomètre, $x$ étant compris entre Oet10. Le coût total de production en euros de l'entreprise CoTon est donné en fonction de la longueur $x$ par la formule : $C(x)=15 x^{3}-120 x^{2}+500 x+750$
Le graphique ci-contre donne la représentation graphique de la fonction $\mathbf{C}$ .
Partie A: Étude du bénéfice Si le marché offre un prix $p$ en euros pour un kilomètre de ce tissu, alors la recette de l'entreprise CoTon pour la vente d'une quantité $x$ est égal à $R(x)=p x$ .
1. Tracer sur le graphique la droite $D_{1}$ d'équation : $y=400 x$ . Expliquer, au vu de ce tracé, pourquoi l'entreprise CoTon ne peut pas réaliser un bénéfice si le prix $p$ du marché est égal à 400 euros.
2. Dans cette question on suppose que le prix du marché est égal à 680 euros. Tracer sur le graphique la droite $D_{2}$ d'équation : $y=680 x$ . Déterminer graphiquement, avec la précision permise par le graphique pour quelles quantités produites et vendues, l'entreprise CoTon réalise un bénéfice si le prix $p$ du marché est de 680 euros.
3. On considère la fonction $B$ définie sur l'intervalle $[0 ; 10]$ par: $B(x)=680 x-C(x)$ Démontrer que pour tout $x$ appartenant à l'intervalle $[0 ; 10]$ , on a : $B^{\prime}(x)=-45 x^{2}+240 x+180$
4. Étudier les variations de la fonction B sur $[0 ; 10]$ . En déduire pour quelle quantité produite et vendue le bénéfice réalisé par l'entreprise CoTon est maximum. Donner la valeur de ce bénéfice.
Partie B: Étude du coût moyen On rappelle que le coût moyen de production $C M$ mesure le coût par unité produite. On considère la fonction $C M$ définie sur l'intervalle [0;10] par : $C M(x)=\frac{C(x)}{x}$
1. Démontrer que pour tout $x$ appartenant à l'intervalle $[0 ; 10]$ , $C M^{\prime}(x)=\frac{30(x-5)\left(x^{2}+x+5\right)}{x^{2}}$
2. Montrer que pour tout $x \in[0 ; 10]$ le signe de $C M^{\prime}(x)$ est le même que celui de " $(x-5)$ "
3. Pour quelle quantité de tissu produite le coût moyen de production est-il minimum? Que valent dans ce cas le coût moyen de production et le coût total?

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Problem 2502

Find equations of the tangent lines to the curve $y^{2}-x y-12=0$ at the points $(-1,3)$ , and $(-1,-4)$ .
The tangent line at $(-1,3)$ is $y=$ $\square$ (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)

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Problem 2503

```latex Let $F(x)$ be an antiderivative of $f(x)=(x-1)(x-2)$ such that $F(0)=2$ . Which of the following graphs if the most accurate graph of $F(x)$ ? Circle your choice and explain your answer below.
Option A
Option B
Option C
Option D ```

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Problem 2504

The flow diagram below shows the volume of fluid in a chemostat (a growth chamber for bacteria).
Poll 1: The ODE is: A. $\frac{d V(t)}{d t}=f-d$ B. $\frac{d V(t)}{d t}=f V(t)$

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Problem 2505

8. Integrate: $\int \frac{-\frac{1}{t^{2}}+1}{\sqrt{\frac{1}{t}+t}} d t$

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Problem 2506

A model helicopter takes off from a point $O$ at time $t=0 \mathrm{~s}$ and moves vertically so that its height $y \mathrm{~cm}$ , above $O$ after time $t$ seconds is given by $y=\frac{1}{4} t^{4}-26 t^{2}+96 t, \quad 0 \leq t \leq 4$ a) Find (i) $\frac{d y}{d t}$ (ii) $\frac{d^{2} y}{d t^{2}}$ b) Verify that $y$ has a stationary value when $t=2$ and determine whether this stationary value is a maximum or a minimum value. c) Find the rate of change of $y$ with respect to $t$ when $t=1$ . d) Determine whether the height of the helicopter is increasing or decreasing at the instant when $t=3$ .

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Problem 2507

$\text { solve: } 2 x y y^{\prime}=y^{2}-x^{2}$ the ODE

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Problem 2508

3. Test for exactness. If exact, solve. If not, use an integrating factor to solve it. $2 x y d x+x^{2} d y=0$

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Problem 2509

Find $d y / d x$ in terms of $x$ and $y$ if $x^{3} y-x-4 y-2=0$ . $\frac{d y}{d x}=$ $\square$

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Problem 2510

Kuta Software - Infinite Calculus Slope at a Value Date $\qquad$ Perio
For each problem, find the slope of the function at the given value. 1) $y=x^{2}+6 x+7$ at $x=-2$ 2) $y=x^{3}-3 x^{2}+5$ at $x=3$ 3) $y=x^{3}-6 x^{2}+9 x-4$ at $x=2$ 4) $y=-x^{3}-6 x^{2}-9 x+1$ at $x=-4$

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Problem 2511

21. Design a cylindrical can (with a lid) to contain 1 liter $\left(=1000 \mathrm{~cm}^{3}\right)$ of water, using the minimum amount of meta

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Problem 2512

$f(x)=x^{4}-6 x^{3}+x^{2}+42 x-56$
Find the zero(s) at which $f$ "flattens out". Express the zero(s) as ordered pair(s).

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Problem 2513

The function given by $y=f(x)$ shows the value of $\$ 5000$ invested at $6 \%$ interest compounded continuously, $x$ years after the money was originally invested. (Round your answers to the nearest cent.) Value of $\$ 5000$ with Continuous Compounding at $6 \%$
Part: $0 / 3$
Part 1 of 3 (a) Find the average amount earned per year between the 5 th year and the 10 th year.
The average amount earned between the Sth year and 10 th year is $\$$ per year.

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Problem 2514

$f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18$
Find the zero(s) at which $f$ "flattens out". Express the zero(s) as ordered pair(s).

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Problem 2515

Question
It is estimated that the population of a certain town changes at the rate of $3+t^{3 / 5}$ people per month. If the current population is 30,000 , what will the population be in 7 months? (Round your answer to the nearest whole number.)

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Problem 2516

Question Evaluate the indefinite integral $\int x \sqrt{x-9} d x$

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Problem 2517

Question Evaluate using integration by parts. $\int x e^{-4 x} d x$

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Problem 2518

Let $A(t)=3000 e^{0.03 t}$ be the balance in a savings account after $t$ years. Complete parts (a) through ( $f$ ) below. (Kound to the nearest cent as needed.) (d) What differential equation is satisfied by $y=A(t)$ ?
The differential equation that is satisfied by $y=A(t)$ is $A^{\prime}(t)=0.03 A(t)$ . (e) Use the results from parts (c) and (d) to determine how fast the balance is growing after 11 years.
The balance is growing at approximately $\$ \square$ per year after 11 years. (Round to the nearest cent as needed.)

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Problem 2519

Question Evaluate $\int x e^{5 x} d x$ . Choose $u=x$ and $d v=e^{5 x} d x$ .

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Problem 2520

Evaluate each definite integral if it is known that $\begin{array}{l} \int_{0}^{3} f(x) d x=-5, \text { and } \\ \int_{0}^{3} g(x) d x=3 \end{array}$
Utilize properties of definite integrals to evaluate $\int_{0}^{3}[2 f(x)+3 g(x)] d x$

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Problem 2521

Let $A(t)=3000 e^{0.03 t}$ be the balance in a savings account after $t$ years. Complete parts (a) through (f) below. (e) Use the results from parts (c) and (d) to determine how fast the balance is growing after 11 years.
The balance is growing at approximately $\$ 125.19$ per year after 11 years. (Round to the nearest cent as needed.) (f) How large will the balance be when it is growing at the rate of $\$ 120$ per year?
The balance will be $\$ \square$ when it is growing at the rate of $\$ 120$ per year. (Round to the nearest cent as needed.)

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Problem 2522

Question
Evaluate $\int_{0}^{2}(7 f(x)-6 g(x)) d x$ given that $\int_{0}^{14} f(x) d x=9, \int_{0}^{2} f(x) d x=-9, \int_{0}^{14} g(x) d x=5$ , and $\int_{0}^{2} g(x) d x=3$
Provide your answer below:

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Problem 2523

Question
The marginal cost of production is found to be $C^{\prime}(x)=1200-30 x+x^{2}$ where $x$ is the number of units produced. The fixed cost of production is $\$ 7500$ . The manufacturer sets the price per unit at $\$ 3000$ . Find the cost function.

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Problem 2524

Find the cost function C. Determine where the cost is a minimum. (a) $C^{\prime}(x)=14 x-2800$ (b) $C^{\prime}(x)=20 x-8000$
Fixed cost $=\$ 4300$ Fixed cost $=\$ 500$
Provide your answer below:

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Problem 2525

Four thousand dollars is deposited into a savings account at $4.5 \%$ interest compounded continuously. (a) What is the formula for $A(t)$ , the balance after $t$ years? (b) What differential equation is satisfied by $\mathrm{A}(\mathrm{t})$ , the balance after t years? (c) How much money will be in the account after 9 years? (d) When will the balance reach $\$ 8000$ ? (e) How fast is the balance growing when it reaches $\$ 8000$ ? (b) $A^{\prime}(t)=0.045 A$ (c) $\$ \square$ (Round to the nearest cent as needed.)

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Problem 2526

Question Population Growth It is estimated that the population of a certain town changes at the rate of $2+t^{4 / 5}$ people per month. If the current population is 20,000 , what will the population be in 10 months?
Provide your answer below:
Population in 10 months = $\square$ FEEDBACK MORE INSTRUCTION SUBMIT

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Problem 2527

An object has a constant acceleration of 2 ft per sec every second for a short time of 6 seconds, and then the acceleration decreases to 0 as shown in the graph. Complete parts a through c. (a) Find $\int_{0}^{10} a(t) d t$ . Include units. $\int_{0}^{10} d(x) d t=8 \mathrm{ft/sec}$ per sec (Simplify your answer.)

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Problem 2528

The marginal cost of oil production, in dollars per barrel, is represented by $\mathrm{C}^{\prime}(\mathrm{x})$ , where x is the number of barrels of oil produced. Report the units of $\int_{600} \mathrm{C}^{\prime}(\mathrm{x}) \mathrm{dx}$ and interpret what the integral means.
The units of $\int_{600}^{640} C^{\prime}(x) d x$ are $\square$

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Problem 2529

Use integration by substitution to solve the integral below. Use $C$ for the constant of integration. $\int-y^{3} \sqrt{5 y^{4}-9} d y$

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Problem 2530

During a local high school foothall game, the quarterback for the home team attempts a deep pass to his wide recelver. The ball is launched from 6.25 feet with an initial velocity of 73 feet per second (f/s). a) Write the equation modeling the projectile motion for the football and sketch the graph.

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Problem 2531

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-6x+5$ , with $h \neq 0$ . Simplify your answer.

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Problem 2532

Find the difference quotient of $f(x)=\frac{1}{x+2}$ and simplify.

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Problem 2533

Determine the marginal cost from the cost function $C(x, y) = 240,000 + 6,000x + 4,000y$ .

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Problem 2534

Given the function $f(x)=\left\{\begin{array}{lll}2 x+18 & \text { if } & x<-6 \\ \sqrt{x+42} & \text { if } & x>-6 \\ 2 & \text { if } & x=-6\end{array}\right.$ , determine the truth of the following statements about $f(-6)$ and its limits.

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Problem 2535

Find $h^{\prime}(-5)$ if $h(x)=f(x)-g(x)$ , given $f(-5)=8, f'(-5)=-2, g(-5)=3, g'(-5)=5$ .

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Problem 2536

Find the integral of $\frac{1}{x^{2}-x+1}$ with respect to $x$ .

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Problem 2537

Find the limit of the fish length model $L(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t}$ as $t \to \infty$ .

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Problem 2538

Un cohete "Pioneer" alcanzó $125.000 \mathrm{~km}$ . ¿Cuál es su velocidad al regresar a la atmósfera a $130 \mathrm{~km}$ ?

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Problem 2539

Demostrar que la serie $\sum_{n=1}^{\infty}(\operatorname{sen} n x) / n^{2}$ converge para todo $x$ , y que $f(x)$ es continua en $[0, \pi]$ . Luego, probar que $\int_{0}^{\pi} f(x) d x=2 \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{3}}$ .

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Problem 2540

Find the limit as $x$ approaches 1 for the expression $2 - 3x$ .

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Problem 2541

Evaluate the limit: $107 \lim _{x \rightarrow 1}(2-3 x)=-1$

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Problem 2542

16. The acceleration due to gravity on Earth is $9.8 \mathrm{~m} / \mathrm{s}^{2}$ . A ball is thrown upward at an initial velocity of $15 \mathrm{~m} / \mathrm{s}$ from a height of 1 m above the ground. Round answers to the nearest tenth. a) Write an equation for the height of the ball. b) What is the height of the ball after 1 s ? c) After how many seconds does the ball land? d) What is the maximum height of the 20. ball? When does this occur? e) Repeat parts a) to d) for a ball thrown on the Moon, where $g=1.62 \mathrm{~m} / \mathrm{s}^{2}$ . f) Repeat parts a) to d) for a ball thrown on Jupiter, where $g=23.1 \mathrm{~m} / \mathrm{s}^{2}$ .

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Problem 2543

Encercler une primitive de la fonction $f(x)=\sec ^{2} x+\frac{1}{x}$ . (A) $\tan x+\ln |x|-\pi$ (D) $\tan ^{2} x-\ln |x|+5$ (B) $\tan ^{2} x+\frac{2}{x^{2}}$ (E) $\tan ^{2} x+\frac{x^{2}}{2}$ (C) $2 \sec ^{2} x \tan x-\frac{1}{x^{2}}$ (F) Aucune de ces réponses

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Problem 2544

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the function given below. Use the Taylor series $(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots$ , for $-1<x<1$ . $(1+4 x)^{-2}$
The first term is $\square$

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Problem 2545

Find the equations of the tangent line and normal lines to the graph of the function $f(x)=-6 \sin x$ at $x=\pi / 2$ . a) Tangent line: ? $\square$ b) Normal line: $\square$ $\square$

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Problem 2546

Página Principal Área personal Mis cursos
Una persona que camina por un camino recto tiene su velocidad en millas por hora en el tiempo $t$ dado por la función $v(t)=0.25 t^{3}-1.5 t^{2}+3 t+0.25$ , para tiempos en el intervalo $0 \leq t \leq 2$ . El gráfico de esta función también se propo el diagrama a continuación. Tiempo rest
Determine el valor de $U=C_{1}+C_{2}+C_{3}+C_{4}$ evaluando la función $y=v(t)$ en los valores elegidos adecuadamente y observando el ancho de cada rectángulo. Seleccione una:

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Problem 2547

Challenging: $y=\left[x^{3}+(2 x-1)^{3}\right]^{5}$ find $y^{\prime}$

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Problem 2548

The total cost (in dollars) of producing $x$ coffee machines is $C(x)=1800+50 x-0.8 x^{2}$ (A) Find the exact cost of producing the 21st machine.
Exact cost of 21st machine $=$ $\square$ (B) Use marginal cost to approximate the cost of producing the 21 st machine.
Approx. cost of 21st machine $=$ $\square$

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Problem 2549

The total profit (in dollars) from the sale of $x$ charcoal grills is $P(x)=50 x-0.5 x^{2}-245$ (A) Find the average profit per grill if 40 grills are produced.
Ave. profit = $\square$ Find the marginal average profit at a production level of 40 grills. (B) Marginal avergge profit = $\square$ Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit $=$ $\square$

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Problem 2550

The total profit (in dollars) from the sale of $x$ charcoal grills is $P(x)=50 x-0.5 x^{2}-245$ (A) Find the average profit per grill if 40 grills are produced.
Ave. profit $=23.875$ Find the marginal average profit at a production level of 40 grills. (B) Marginal average profit $=24.125$
Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit $=24.125$

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Problem 2551

The price-demand and cost functions for the production of microwaves are given as $p=235-\frac{x}{90}$ and $C(x)=26000+60 x,$ where $x$ is the number of microwaves that can be sold at a price of $p$ dollars per unit and $C(x)$ is the total cost (in dollars) of producing $x$ units. (A) Find the marginal cost as a function of $x$ . $C^{\prime}(x)=$ (B) Find the revenue function in terms of $x$ . $R(x)=$ $\square$ (C) Find the marginal revenue function in terms of $x$ . $R^{\prime}(x)=$ $\square$ (D) Evaluate the marginal revenue function at $x=1500$ . $R^{\prime}(1500)=$ $\square$ (E) Find the profit function in terms of $x$ . $P(x)=$ $\square$ (F) Evaluate the marginal profit function at $x=1500$ . $P^{\prime}(1500)=\square$

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Problem 2552

A retail store estimates that weekly sales $s$ and weekly advertising costs $x$ (both in dollars) are related by $s=60000-430000 e^{-0.0005 x}$
The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales. Rate of change of sales $=$ $\square$

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Problem 2553

A price $p$ (in dollars) and demand $x$ for a product are related by $2 x^{2}-2 x p+50 p^{2}=16200$
If the price is increasing at a rate of 2 dollars per month when the price is 10 dollars, find the rate of change of the demand. Rate of change of demand = $\square$

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Problem 2554

Suppose that for a company manufacturing calculators, the cost, and revenue equations are given by $C=80000+30 x, \quad R=200-\frac{x^{2}}{40}$ where the production output in one week is $x$ calculators. If the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators, find each of the following:
Rate of change in cost $=$ $\square$
Rate of change in revenue $=$ $\square$
Rate of change in profit = $\square$

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Problem 2555

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. $f(x)=x^{3}+2 x+1$
Find $f^{\prime}(x)$ . $\begin{array}{l} f(x)=x^{3}+2 x+1 \\ f^{\prime}(x)=\square \end{array}$

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Problem 2556

A retail store estimates that weekly sales $s$ and weekly advertising costs $x$ (both in dollars) are related by $s=60000-430000 e^{-0.0005 x}$
The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.
Rate of change of sales = $\square$ 2372.96

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Problem 2557

=autosave\#questio... ns that the particle moved approximately 30.00 meters to the left. that $v(t)=t^{2}-t-20=(t-5)(t+4)$ and so $v(t) \leq v \quad 0$ on the interval $[1,5]$ and $v(t)$ nus, from this equation, the distance traveled is $\begin{aligned} \int_{1}^{7}|v(t)| d t & =\int_{1}^{5}[-v(t)] d t+\int_{5}^{7} v(t) d t \\ & =\int_{1}^{5}\left(-t^{2}+t+20\right) d t+\int_{5}^{7}\left(t^{2}-t-20\right) d t \end{aligned}$ $=\left[-\frac{t^{3}}{3}+\frac{t^{2}}{2}+20 t\right]_{1}^{5}+\left[\frac{t^{3}}{3}-\frac{t^{2}}{2}-20 t\right]$ $=\frac{725}{3}$ Your answer is correct.

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Problem 2558

The function given by $y=f(x)$ shows the value of $\$ 5000$ invested at $5 \%$ interest compounded continuously, $x$ years after the money was originally invested. (Round your answers to the nearest cent.) Value of $\$ 5000$ with Continuous Compounding at $5 \%$
Part: $0 / 3$
Part 1 of 3 (a) Find the average amount earned per year between the 5 th year and the 10 th year.
The average amount earned between the 5 th year and 10 th year is $\$ 364.80$ per year. $\square$
Part: 1 / 3 $\square$
Part 2 of 3 (b) rind the average amount earned per year between the 20 th year and the 25 th year.
The average ampunt earned between the 20 th year and 25 th year is $\$$ per vear. $\square$

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Problem 2559

Evaluate the integral. Use $C$ (upper case) for the constant of integration. $\int \frac{8 x+9}{\sqrt{16-x^{2}}} d x$

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Problem 2560

1. Find $\frac{d y}{d x}$ in terms of $t$ for the following parametric equations. (i) $\quad x=t^{4}+t$ and $y=t^{3}-2 t$ (ii) $x=e^{3 t}+2$ and $y=2 t^{2}+t$ (iii) $x=4 t^{3}-t$ and $y=t^{2}+8 t$
2. Find $y$ for the following, by using the given substitution. (i) $\quad y=\int x \sqrt{x^{2}+2} d x \quad$ , let $u=x^{2}+2$ (ii) $y=\int x \sqrt{1-x^{2}} d x$ , let $u=1-x^{2}$ (iii) $y=\int \frac{-3-3 x^{2}}{2-3 x-x^{3}} d x \quad$ , let $u=2-3 x-x^{3}$
3. Consider the function $x^{3}-x^{2} y+2 y^{2}=8$ (i) Differentiate the function implicitly and show that $\frac{d y}{d x}=\frac{2 x y-3 x^{2}}{4 y-x^{2}}$ (ii) Find the tangent equation to the curve $x^{3}-x^{2} y+2 y^{2}=8$ at the point $(2,0)$ .
4. Consider the function $4 y^{3}-3 x^{2} y+x=1$ (i) Differentiate the function implicitly and show that $\frac{d y}{d x}=\frac{6 x y-1}{12 y^{2}-3 x^{2}}$ (ii) Find the normal equation to the curve $4 y^{3}-3 x^{2} y+x=1$ at the point $(0,1)$ .
5. Find the point of intersection between the curve $y=9-x^{2}$ and the straight line $y-x-3=0$ shown in Figure 1 below.
Hence, find the area bounded by the curve and the straight line.
Figure 1
6. Find the point of intersection between the two curves $y=x^{2}+2 x+2$ and $y=-x^{2}+2 x+10$ . Hence, find the area bounded by the two curves.

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Problem 2561

Find the slope of the tangent line to the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ at the point $(x, y)$ . slope $=$ $\square$
Are there any points where the slope is not defined? (Enter them as comma-separated ordered-pairs, e.g., (1,3), ( $-2,5)$ . Enter none if there are no such points.) slope is undefined at $\square$

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Problem 2562

Докажите, что множество $[0,1) \cap \mathbb{Q}$ не компактно.

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Problem 2563

All changes saved
The power, $P$ , dissipated when a 11 -volt battery is put across a resistance of $R$ ohms is given by $P=\frac{121}{R}$
What is the rate of change of power with respect to resistance? rate of change $=$

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Problem 2564

a. $\lim _{x \rightarrow 2} \frac{x^{4}-5 x-6}{4 x-8}=$

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Problem 2565

(ii) $\int \sqrt{\csc x} \cos ^{3} x d x$

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Problem 2566

b. $\lim _{x \rightarrow 1} \frac{\sqrt{2 x-1}-1}{5 x-5}$

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Problem 2567

5) A rectangular metal sheet shrinks while maintaining its shape, such that its length decreases at a rate of $3 \mathrm{~cm} / \mathrm{second}$ and its width decreases at a rate of $2 \mathrm{~cm} / \mathrm{second}$ . Find the rate of change of its area with respect to time when its length is 6 cm and its width is 5 cm .

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Problem 2568

1. Find the equations of the tangent line to the ellipse, $\frac{(x+2)^{2}}{4}+\frac{(y-3)^{2}}{9}$ when $x=-3$ .

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Problem 2569

Thursday, November 28, 2024 Midterm Exam Calculus I d (0203101 \& 0213105 )
اكتب رمز الإجابة الصحيحة في الجدول بالحروف الكبيزة A, B, C, D \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Question & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline Answer & & & & & & & & & & & & & & & \\ \hline \end{tabular}
If $f(x)=\frac{4}{x-1}, g(x)=4 x$ , then the value of $x$ at which $f \circ g(x)=g \circ f(x)$ is: A) $\frac{1}{4}$ B) $\frac{1}{5}$ C) $-\frac{1}{3}$ D) $\frac{1}{3}$
The graph of the function $f(x)=\frac{\left(x^{2}-7 x+10\right)}{x^{2}-25}$ has at $x=-5$ A) jump B) Hole C) Vertical asymptote D) continuity point

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Problem 2570

7-18 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
7. $g(x)=\int_{1}^{x} \frac{1}{t^{3}+1} d t$
8. $g(x)=\int_{3}^{x} e^{t^{2}-t} d t$
9. $g(s)=\int_{5}^{s}\left(t-t^{2}\right)^{8} d t$
10. $g(r)=\int_{0}^{r} \sqrt{x^{2}+4} d x$
11. $F(x)=\int_{x}^{\pi} \sqrt{1+\sec t} d t$ $\left[\right.$ Hint: $\left.\int_{x}^{\pi} \sqrt{1+\sec t} d t=-\int_{\pi}^{x} \sqrt{1+\sec t} d t\right]$

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Problem 2571

6. Locate all turning points, classifying them, on the curve $y=6 x^{4}-8 x^{3}-24 x^{2}-4$

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Problem 2572

29. $\int_{1}^{9} \frac{x-1}{\sqrt{x}} d x$
30. $\int_{0}^{2}(y-1)(2 y+1) d y$
31. $\int_{0}^{\pi / 4} \sec ^{2} t d t$
32. $\int_{0}^{\pi / 4} \sec \theta \tan \theta d \theta$
33. $\int_{1}^{2}(1+2 y)^{2} d y$
34. $\int_{0}^{3}\left(2 \sin x-e^{x}\right) d x$

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Problem 2573

2.) (5pts) Evaluate the limit, if it exists. $\lim _{x \rightarrow-2} \frac{x+2}{x^{3}+8}$

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Problem 2574

13. Ein U-Boot beginnt eine Tauchfahrt in $\mathrm{P}(100|200| 0)$ mit 11,1 Knoten in Richtung des Peilziels $Z(500|400|-80)$ , bis es eine Tiefe von 80 m erreicht hat. $\left(1 \text { Knoten }=1 \frac{\text { Seemeile }}{\text { Stunde }} \approx 1,852 \frac{\mathrm{~km}}{\mathrm{~h}}\right)$
Anschließend geht es ohne Kurswechsel in eine horizontale Schleichfahrt von 11 Knoten ein. Könnte es zu einer Kollision mit der Tauchkugel T kommen, die zeitgleich vom Forschungsschiff $S(700|800| 0)$ mit einer Geschwindigkeit von $0,5 \frac{\mathrm{~m}}{\mathrm{~s}}$ senkrecht sinkt?

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Problem 2575

```latex \textbf{Flüssigkeiten bei einem Produktionsprozess}
In einem Produktionsprozess werden Flüssigkeiten erhitzt und anschließend abgekühlt. Der Temperaturverlauf kann gezielt gesteuert werden, sodass er sich für den gesamten Erhitzungs- bzw. Abkühlungsvorgang für $t \geq 0$ durch eine der in $\mathbf{R}$ definierten Funktionen $f_{k}$ mit $f_{k}(t)=23+20 \cdot t \cdot e^{-\frac{1}{10} \cdot k \cdot t}$ , wobei $k$ eine positive, reelle Zahl sein soll, beschreiben lässt. Dabei ist $t$ die seit Beginn des Vorgangs vergangene Zeit in Minuten und $f_{k}(t)$ die Temperatur in ${ }^{\circ} C$ .
\begin{enumerate} \item[a)] Die in Abbildung 1 dargestellten Graphen $A, B$ und $C$ gehören jeweils zu einem der Werte $k=0,5; k=2$ und $k=5$ . Ordnen Sie jedem dieser Werte den zugehörigen Graphen zu. \item[b)] Begründen Sie, dass der in Abbildung 1 dargestellte Graph $D$ nicht zu einer der Funktionen $f_{k}$ gehören kann. \item[c)] Zeigen Sie, dass gilt $f_{k}^{\prime}(t)=20 \cdot e^{-\frac{1}{10} \cdot k \cdot t} \cdot\left(1-\frac{1}{10} \cdot k \cdot t\right)$ \item[d)] Ermitteln Sie denjenigen Wert von $k$ , für den die Flüssigkeit im Modell eine Höchsttemperatur von $123^{\circ} \mathrm{C}$ erreicht. \item[e)] Ermitteln Sie die Koordinaten des Wendepunktes des Graphen von $f_{10}$ . Interpretieren Sie anschließend die Bedeutung der x-Koordinate dieses Wendepunkts des Graphen von $f_{10}$ im Sachzusammenhang. \item[f)] Der in der Abbildung 2 dargestellte Graph gibt für einen gesteuerten Temperaturverlauf die Änderungsrate der Temperatur in Abhängigkeit von der Zeit an, die seit Beginn des Vorgangs vergangen ist. Bestimmen Sie einen Näherungswert für die Änderung der Temperatur in den ersten vier Minuten nach Beginn des Vorganges und geben Sie an, ob die Temperatur zu- oder abnimmt. \item[g)] Skizzieren Sie für die ersten zwölf Minuten des in Abbildung 2 dargestellten Vorgangs den Graphen eines möglichen Temperaturverlaufs. \end{enumerate}
\textit{Abbildung 1}
\textit{Abbildung 2} ```

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Problem 2576

13. Ein U-Boot beginnt eine Tauchfahrt in $\mathrm{P}(100|200| 0)$ mit 11,1 Knoten in Richtung des Peilziels $Z(500|400|-80)$ , bis es eine Tiefe von 80 m erreicht hat. $\left(1 \text { Knoten }=1 \frac{\text { Seemeile }}{\text { Stunde }} \approx 1,852 \frac{\mathrm{~km}}{\mathrm{~h}}\right)$
Anschließend geht es ohne Kurswechsel in eine horizontale Schleichfahrt von 11 Knoten ein. Könnte es zu einer Kollision mit der Tauchkugel T kommen, die zeitgleich vom Forschungsschiff $S(700|800| 0)$ mit einer Geschwindigkeit von $0,5 \frac{\mathrm{~m}}{\mathrm{~s}}$ senkrecht sinkt?

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Problem 2577

$f(x)=2 x^{2}+1$ , given that $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ then $f^{\prime}(x)=$

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Problem 2578

```latex $f(x)=\left\{\begin{array}{lr}x-2, & -\pi \leq x \leq 0, \\ 0, & 0<x \leq \pi .\end{array}\right.$
(Omsem: $f(x)=-\frac{\pi+4}{2}+\frac{2}{\pi} \sum_{k=1}^{\infty} \frac{\cos ((2 k-1) x)}{(2 k-1)^{2}}+$
$\left.+\frac{4+\pi}{\pi} \sum_{k=1}^{\infty} \frac{\sin ((2 k-1) x)}{2 k-1}-\sum_{k=1}^{\infty} \frac{\sin (2 k x)}{2 k} .\right)$

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Problem 2579

16 Berechne den Inhalt der Fläche, den der Graph der Funktion $f$ mit $f(x)=x^{2}-x-2$ mit der $x$ -Achse einschließt.

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Problem 2580

(b) Find $\int \frac{1}{1-e^{t}} d t$ .

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Problem 2581

$\lim _{n \rightarrow \infty}\left(1+\frac{2017}{\ln (n)}\right)^{\ln (n)}$

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Problem 2582

Solve the following Differential equations: (1) $\left(1+e^{x}\right) y^{\prime}=e^{x}$

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Problem 2583

5 PROBLEMA SVOLTO $4,0 \mathrm{~mol}$ di un gas perfetto monoatomico si trovano nello stato iniziale $A$ e dopo quattro trasformazioni, rappresentate nel grafico, tornano in $A$ . a) Determina la temperatura in ciascuno stato. b) In relazione a ciascuna trasformazione $A B, B C, C D, D A$ calcola il calore assorbito o ceduto, il lavoro prodotto o subito e la variazione di energia interna. c) Trova infine il calore totale scambiato, il lavoro totale e la variazione totale di energia interna.

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Problem 2584

$\lim _{x \rightarrow-1} \frac{x^{3}+x^{2}+x+1}{x+1}$

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Problem 2585

Q₁ → The value of The value of c that satisfies the conclusion of the mean value theorem for the function f(x)=x-= on the interval [1,4] Momen & Azdeen a) 3 b) 2 c) 1-2,24 d) None Q1 08:18 p

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Problem 2586

3.32. Në qoftë se densiteti i $X$ është $f(x)=1, \quad 0<x<1$ gjeni $E\left[e^{t X}\right]$ . Kryeni llogaritjet për të gjetur $E\left(X^{n}\right)$ .

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Problem 2587

$\int_{1}^{4}\left(2-\frac{x}{3}\right) d x$ using Diemann Sum

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Problem 2588

find the derivative of $(x+c) e^{x}$

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Problem 2589

(1 point)
A table of values for $f, g, f^{\prime}$ , and $g^{\prime}$ is given below: \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $f(x)$ & $g(x)$ & $f^{\prime}(x)$ & $g^{\prime}(x)$ \\ \hline 1 & 3 & 2 & 4 & 6 \\ 2 & 1 & 8 & 5 & 7 \\ 3 & 7 & 2 & 7 & 9 \\ \hline \end{tabular} (a) If $h(x)=f(g(x))$ , find $h^{\prime}(1)$ . (b) If $H(x)=g(f(x))$ , find $H^{\prime}(1)$ . (a) $h^{\prime}(1)=$ $\square$ (b) $H^{\prime}(1)=$ $\square$

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Problem 2590

$f(x)=\left(x^{3}+3 x+8\right)^{3}$ $\begin{array}{l} f^{\prime}(x)= \\ f^{\prime}(1)= \end{array}$

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Problem 2591

A box is to be made out of a 8 cm by 20 cm piece of cardboard. Squares of side length $x \mathrm{~cm}$ 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 $V$ of the box as a function of $x$ . $V=4 x^{3}-56 x^{2}+160 x \mathrm{~cm}^{3}$ (b) Give the domain of $V$ in interval notation. (Use the fact that length and volume must be positive.) $(0,4)$ (c) Find the length $L$ , width $W$ , and height $H$ of the resulting box that maximizes the volume. (Assume that $W \leq L$ ). $\begin{array}{l} L=\square \mathrm{cm} \\ W=\square \mathrm{cm} \\ H=\square \mathrm{cm} \end{array}$ (d) The maximum volume of the box is $\square \mathrm{cm}^{3}$ .

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Problem 2592

Let $f(x)=x+\sqrt{1-x}$ Find the local maximum and minimum values of $f$ 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 $\square$
Note: You can earn partial credit on this problem. Preview My Answers Submit Answers

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Problem 2593

Compute the following limits using L'Hospital's rule if appropriate. Use INF to denote oo and MINF to denote $\begin{array}{l} \lim _{x \rightarrow 1} \frac{7^{\infty}-7}{x^{2}-1}=\square \\ \lim _{x \rightarrow \infty} \frac{\tan ^{-1}(x)}{(1 / x)-7}= \end{array}$ $\square$
Note: You can earn partial credit on this problem.

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Problem 2594

Consider the limit $\lim _{x \rightarrow 0} \frac{\sin ^{2}(8 x)}{1-\cos (8 x)}$
To simplify this limit, we should multiply numerator and denominator by the expression $\square$ After doing this and simplifying the result we find that the value of limit is $\square$
Note: You can earn partial credit on this problem. Preview My Answers Submit Answers

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Problem 2595

Path of the projectile $\mathrm{v}_{i}=20 \mathrm{~m} / \mathrm{s}$
A projectile is fired up an incline (incline angle $\phi$ ) with an initial speed $v_{\mathrm{i}}$ at an angle $\theta_{\mathrm{i}}$ with respect to the horizontal as shown in Figure. (Take $\mathrm{g}=\mathbf{1 0} \mathrm{m} / \mathrm{s}^{2}$ . Please mark the closest answer as correct answer) Find when the projectile's velocity has only $x$ component (a) $\mathrm{t}=1 \mathrm{~s}$ (b) $t=3.9 \mathrm{~s}$ (c) $\mathrm{t}=6.3 \mathrm{~s}$ (d) $t=8.6 \mathrm{~s}$ (e) $t=4 \mathrm{~s}$

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Problem 2596

Path of the projectile $\mathrm{v}_{i}=20 \mathrm{~m} / \mathrm{s}$
A projectile is fired up an incline (incline angle $\phi$ ) with an initial speed $v_{i}$ at an angle $\theta_{\mathrm{i}}$ with respect to the horizontal as shown in Figure. (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ . Please mark the closest answer as correct answer) Find when the projectile's velocity has only $x$ component (a) $\mathrm{t}=1 \mathrm{~s}$ (b) $t=3.9 \mathrm{~s}$ (c) $\mathrm{t}=6.3 \mathrm{~s}$ (d) $t=8.6 \mathrm{~s}$ (e) $t=4 \mathrm{~s}$

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Problem 2597

NYA Module 7: Problem 11
For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second enter the numerical value, and in the third case answer DNE. To discourage blind guessing, this problem is graded on the following scale $\begin{array}{l} 0-9 \text { correct = } 0 \\ 10-13 \text { correct }=.3 \\ 14-16 \text { correct }=.5 \\ 17-19 \text { correct }=.7 \end{array}$
Note that l'Hospital's rule (in some form) may ONLY be applied to indeterminate forms.
1. $\infty^{-\infty}$
2. $\infty^{-0}$
3. $\infty^{\infty}$
4. $\infty^{1}$
5. $\pi^{\infty}$
6. $1^{\infty}$ $7.1^{0}$
8. $\infty^{0}$
9. $\pi^{-\infty}$ $10 . \infty \cdot \infty$
11. $\frac{0}{\infty}$
12. $\frac{1}{-\infty}$ $13.0 \cdot \infty$
14. $\frac{\infty}{0}$
15. $1^{-\infty}$ $16.1 \cdot \infty$
17. $\infty-\infty$ $18.0^{0}$ $\square$ $19.0^{\infty}$ $\square$ 20. $0^{-\infty}$

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Problem 2598

$\begin{array}{l}\text { caushy } \\ y^{\prime \prime}-\frac{3}{x} y^{\prime}+\frac{4}{x^{2}} y=0 \quad{ }^{\prime} \quad x^{\prime}>0\end{array}$

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Problem 2599

$\int_{0}^{\infty} e^{-x^{2}} d x$

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Problem 2600

Q Q If $^{y_{1}}=x$ is solution of $\left(x^{2}-x\right) y^{\prime \prime}+x y^{\prime}+y=0$ find the General solution

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Calculus

Problem 2501

Problem 2502

Problem 2503

```latex Let F(x)F(x)F(x) be an antiderivative of f(x)=(x−1)(x−2)f(x)=(x-1)(x-2)f(x)=(x−1)(x−2) such that F(0)=2F(0)=2F(0)=2. Which of the following graphs if the most accurate graph of F(x)F(x)F(x) ? Circle your choice and explain your answer below.Option AOption BOption COption D ```

Problem 2504

The flow diagram below shows the volume of fluid in a chemostat (a growth chamber for bacteria).Poll 1: The ODE is: A. dV(t)dt=f−d\frac{d V(t)}{d t}=f-ddtdV(t)​=f−d B. dV(t)dt=fV(t)\frac{d V(t)}{d t}=f V(t)dtdV(t)​=fV(t)

Problem 2505

8. Integrate: ∫−1t2+11t+tdt\int \frac{-\frac{1}{t^{2}}+1}{\sqrt{\frac{1}{t}+t}} d t∫t1​+t​−t21​+1​dt

Problem 2506

Problem 2507

solve: 2xyy′=y2−x2\text { solve: } 2 x y y^{\prime}=y^{2}-x^{2} solve: 2xyy′=y2−x2 the ODE

Problem 2508

3. Test for exactness. If exact, solve. If not, use an integrating factor to solve it. 2xydx+x2dy=02 x y d x+x^{2} d y=02xydx+x2dy=0

Problem 2509

Find dy/dxd y / d xdy/dx in terms of xxx and yyy if x3y−x−4y−2=0x^{3} y-x-4 y-2=0x3y−x−4y−2=0. dydx=\frac{d y}{d x}=dxdy​= □\square□

Problem 2510

Problem 2511

21. Design a cylindrical can (with a lid) to contain 1 liter (=1000 cm3)\left(=1000 \mathrm{~cm}^{3}\right)(=1000 cm3) of water, using the minimum amount of meta

Problem 2512

f(x)=x4−6x3+x2+42x−56f(x)=x^{4}-6 x^{3}+x^{2}+42 x-56f(x)=x4−6x3+x2+42x−56Find the zero(s) at which fff "flattens out". Express the zero(s) as ordered pair(s).

Problem 2513

Problem 2514

f(x)=x4+4x3−3x2−24x−18f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18f(x)=x4+4x3−3x2−24x−18Find the zero(s) at which fff "flattens out". Express the zero(s) as ordered pair(s).

Problem 2515

QuestionIt is estimated that the population of a certain town changes at the rate of 3+t3/53+t^{3 / 5}3+t3/5 people per month. If the current population is 30,000 , what will the population be in 7 months? (Round your answer to the nearest whole number.)

Problem 2516

Question Evaluate the indefinite integral ∫xx−9dx\int x \sqrt{x-9} d x∫xx−9​dx

Problem 2517

Question Evaluate using integration by parts. ∫xe−4xdx\int x e^{-4 x} d x∫xe−4xdx

Problem 2518

Problem 2519

Question Evaluate ∫xe5xdx\int x e^{5 x} d x∫xe5xdx. Choose u=xu=xu=x and dv=e5xdxd v=e^{5 x} d xdv=e5xdx.

Problem 2520

Problem 2521

Problem 2522

Problem 2523

Problem 2524

Find the cost function C. Determine where the cost is a minimum. (a) C′(x)=14x−2800C^{\prime}(x)=14 x-2800C′(x)=14x−2800 (b) C′(x)=20x−8000C^{\prime}(x)=20 x-8000C′(x)=20x−8000Fixed cost =$4300=\$ 4300=$4300 Fixed cost =$500=\$ 500=$500Provide your answer below:

Problem 2525

Problem 2526

Problem 2527

Problem 2528

Problem 2529

Use integration by substitution to solve the integral below. Use CCC for the constant of integration. ∫−y35y4−9dy\int-y^{3} \sqrt{5 y^{4}-9} d y∫−y35y4−9​dy

Problem 2530

Problem 2531

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−6x+5f(x)=-6x+5f(x)=−6x+5, with h≠0h \neq 0h=0. Simplify your answer.

Problem 2532

Find the difference quotient of f(x)=1x+2f(x)=\frac{1}{x+2}f(x)=x+21​ and simplify.

Problem 2533

Determine the marginal cost from the cost function C(x,y)=240,000+6,000x+4,000yC(x, y) = 240,000 + 6,000x + 4,000yC(x,y)=240,000+6,000x+4,000y.

Problem 2534

Problem 2535

Find h′(−5)h^{\prime}(-5)h′(−5) if h(x)=f(x)−g(x)h(x)=f(x)-g(x)h(x)=f(x)−g(x), given f(−5)=8,f′(−5)=−2,g(−5)=3,g′(−5)=5f(-5)=8, f'(-5)=-2, g(-5)=3, g'(-5)=5f(−5)=8,f′(−5)=−2,g(−5)=3,g′(−5)=5.

Problem 2536

Find the integral of 1x2−x+1\frac{1}{x^{2}-x+1}x2−x+11​ with respect to xxx.

Problem 2537

Find the limit of the fish length model L(t)=LT−(LT−L0)e−ktL(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t}L(t)=LT​−(LT​−L0​)e−kt as t→∞t \to \inftyt→∞.

Problem 2538

Un cohete "Pioneer" alcanzó 125.000 km125.000 \mathrm{~km}125.000 km. ¿Cuál es su velocidad al regresar a la atmósfera a 130 km130 \mathrm{~km}130 km?

Problem 2539

Problem 2540

Find the limit as xxx approaches 1 for the expression 2−3x2 - 3x2−3x.

Problem 2541

Evaluate the limit: 107lim⁡x→1(2−3x)=−1107 \lim _{x \rightarrow 1}(2-3 x)=-1107limx→1​(2−3x)=−1

Problem 2542

Problem 2543

Problem 2544

Problem 2545

Find the equations of the tangent line and normal lines to the graph of the function f(x)=−6sin⁡xf(x)=-6 \sin xf(x)=−6sinx at x=π/2x=\pi / 2x=π/2. a) Tangent line: ? □\square□ b) Normal line: □\square□ □\square□

Problem 2546

Problem 2547

Challenging: y=[x3+(2x−1)3]5y=\left[x^{3}+(2 x-1)^{3}\right]^{5}y=[x3+(2x−1)3]5 find y′y^{\prime}y′

Problem 2548

Problem 2549

Problem 2550

Problem 2551

Problem 2552

Problem 2553

```latex Let $F(x)$ be an antiderivative of $f(x)=(x-1)(x-2)$ such that $F(0)=2$ . Which of the following graphs if the most accurate graph of $F(x)$ ? Circle your choice and explain your answer below.
Option A
Option B
Option C
Option D ```

The flow diagram below shows the volume of fluid in a chemostat (a growth chamber for bacteria).
Poll 1: The ODE is: A. $\frac{d V(t)}{d t}=f-d$ B. $\frac{d V(t)}{d t}=f V(t)$

8. Integrate: $\int \frac{-\frac{1}{t^{2}}+1}{\sqrt{\frac{1}{t}+t}} d t$

$\text { solve: } 2 x y y^{\prime}=y^{2}-x^{2}$ the ODE

3. Test for exactness. If exact, solve. If not, use an integrating factor to solve it. $2 x y d x+x^{2} d y=0$

Find $d y / d x$ in terms of $x$ and $y$ if $x^{3} y-x-4 y-2=0$ . $\frac{d y}{d x}=$ $\square$

21. Design a cylindrical can (with a lid) to contain 1 liter $\left(=1000 \mathrm{~cm}^{3}\right)$ of water, using the minimum amount of meta

$f(x)=x^{4}-6 x^{3}+x^{2}+42 x-56$
Find the zero(s) at which $f$ "flattens out". Express the zero(s) as ordered pair(s).

$f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18$
Find the zero(s) at which $f$ "flattens out". Express the zero(s) as ordered pair(s).

Question
It is estimated that the population of a certain town changes at the rate of $3+t^{3 / 5}$ people per month. If the current population is 30,000 , what will the population be in 7 months? (Round your answer to the nearest whole number.)

Question Evaluate the indefinite integral $\int x \sqrt{x-9} d x$

Question Evaluate using integration by parts. $\int x e^{-4 x} d x$

Question Evaluate $\int x e^{5 x} d x$ . Choose $u=x$ and $d v=e^{5 x} d x$ .

Find the cost function C. Determine where the cost is a minimum. (a) $C^{\prime}(x)=14 x-2800$ (b) $C^{\prime}(x)=20 x-8000$
Fixed cost $=\$ 4300$ Fixed cost $=\$ 500$
Provide your answer below:

Use integration by substitution to solve the integral below. Use $C$ for the constant of integration. $\int-y^{3} \sqrt{5 y^{4}-9} d y$

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-6x+5$ , with $h \neq 0$ . Simplify your answer.

Find the difference quotient of $f(x)=\frac{1}{x+2}$ and simplify.

Determine the marginal cost from the cost function $C(x, y) = 240,000 + 6,000x + 4,000y$ .

Find $h^{\prime}(-5)$ if $h(x)=f(x)-g(x)$ , given $f(-5)=8, f'(-5)=-2, g(-5)=3, g'(-5)=5$ .

Find the integral of $\frac{1}{x^{2}-x+1}$ with respect to $x$ .

Find the limit of the fish length model $L(t)=L_{T}-\left(L_{T}-L_{0}\right) e^{-k t}$ as $t \to \infty$ .

Un cohete "Pioneer" alcanzó $125.000 \mathrm{~km}$ . ¿Cuál es su velocidad al regresar a la atmósfera a $130 \mathrm{~km}$ ?

Find the limit as $x$ approaches 1 for the expression $2 - 3x$ .

Evaluate the limit: $107 \lim _{x \rightarrow 1}(2-3 x)=-1$

Find the equations of the tangent line and normal lines to the graph of the function $f(x)=-6 \sin x$ at $x=\pi / 2$ . a) Tangent line: ? $\square$ b) Normal line: $\square$ $\square$

Challenging: $y=\left[x^{3}+(2 x-1)^{3}\right]^{5}$ find $y^{\prime}$

Evaluate the integral. Use $C$ (upper case) for the constant of integration. $\int \frac{8 x+9}{\sqrt{16-x^{2}}} d x$

Докажите, что множество $[0,1) \cap \mathbb{Q}$ не компактно.

All changes saved
The power, $P$ , dissipated when a 11 -volt battery is put across a resistance of $R$ ohms is given by $P=\frac{121}{R}$
What is the rate of change of power with respect to resistance? rate of change $=$

a. $\lim _{x \rightarrow 2} \frac{x^{4}-5 x-6}{4 x-8}=$

(ii) $\int \sqrt{\csc x} \cos ^{3} x d x$

b. $\lim _{x \rightarrow 1} \frac{\sqrt{2 x-1}-1}{5 x-5}$

1. Find the equations of the tangent line to the ellipse, $\frac{(x+2)^{2}}{4}+\frac{(y-3)^{2}}{9}$ when $x=-3$ .

6. Locate all turning points, classifying them, on the curve $y=6 x^{4}-8 x^{3}-24 x^{2}-4$

2.) (5pts) Evaluate the limit, if it exists. $\lim _{x \rightarrow-2} \frac{x+2}{x^{3}+8}$