Calculus

Problem 9501

Find the absolute extrema of $f(x)=\sqrt{x}-3$ on the interval $[0,9]$ . Give minimum and maximum points $(x, y)$ .

See Solution

Problem 9502

Find the relationship between $B(p)$ , $E(p)$ , $C(k)$ , and $T(p)$ . Which expression estimates extra cookies sold with one more employee?

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Problem 9503

Check if Rolle's Theorem applies to $f(x)=x^{3}-48 x-6$ on $[-4,8]$ . Select all that apply. $c=$

See Solution

Problem 9504

Find the limit as $x$ approaches 0 for $g(x)$ , given $10 x^{2}+7 x+8<g(x)<7 x^{2}-3 x+8$ .

See Solution

Problem 9505

Find the average rate of change of $f(x)=4x^{2}-8$ on $[3, a]$ . Express your answer in terms of $a$ .

See Solution

Problem 9506

Find the limit as $x$ approaches 0 for $g(x)$ , given $9 x^{2}+5 x+6<g(x)<9 x^{2}-8 x+6$ .

See Solution

Problem 9507

Find the tangent line of $y=\sin x$ at $x=0$ . Is it a good approximation for the curve near the origin? Approximate $\sin(0.2)$ .

See Solution

Problem 9508

Find the linear approximation of $f(x)=\frac{1}{\sqrt{6+x}}$ at $x=0$ .

See Solution

Problem 9509

Find the derivative of the function $y=\left(5 x^{3}-2 x^{2}+8 x-6\right) e^{x^{3}}$ .

See Solution

Problem 9510

Estimate $\sqrt{4.7}$ using linear approximation at $a=4$ .

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Problem 9511

Find the differential, $d y$ , of the function $y=f(x)=\tan(4x^{2})$ .

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Problem 9512

Find the derivative $\frac{dy}{dx}$ for the equation $\sin x + 2 \cos(2y) = 1$ .

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Problem 9513

Find the average rate of change of $f(x)=2x^{2}-3$ from $x=1$ to $x=a$ . Express your answer in terms of $a$ .

See Solution

Problem 9514

Find $\frac{d y}{d x}$ if $\frac{d y}{d x}[(4-x) \cdot y^{2}] = x^{3}$ .

See Solution

Problem 9515

Find the average value $f_{\text{ave}}$ of $f(x, y)$ over region $D$ given $\iint_{D} f(x, y) dA = \frac{128}{9}$ and area $D = 2\pi$ .

See Solution

Problem 9516

Find the linear function for $g(x)=\sec x$ at $x=\frac{\pi}{3}$ , approximate $g\left(\frac{\pi}{3}-0.01\right)$ , and calculate the percent error.

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Problem 9517

Estimate $18^{1/4}$ using linear approximation, knowing $(16)^{1/4}=2$ .

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Problem 9518

Find the differential $d y$ for the function $y=\frac{1+\sin x}{4-\sin x}$ .

See Solution

Problem 9519

Find the derivative $d y / d x$ from the equation $\frac{d y}{d x}\left[x e^{y}-10 x+3 y\right]=0$ .

See Solution

Problem 9520

Find the limit as $x$ approaches infinity for $\frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}$ .

See Solution

Problem 9521

What is the temperature of hot chocolate after 2 minutes in a freezer at 0°F if it starts at 190°F? Use $k=0.12$ .

See Solution

Problem 9522

Estimate paint needed for a sphere (diameter $12 \mathrm{cms}$ ) with a coat thickness of $1 / 6 \mathrm{cms}$ .

See Solution

Problem 9523

Determine if these integrals converge or diverge: (a) $\int_{e}^{\infty} \frac{d x}{x^{2} \ln x}$ , (b) $\int_{e}^{\infty} \frac{\ln x}{\sqrt{x}} d x$ .

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Problem 9524

Given the cost function $C(x)=19600+500 x+x^{2}$ , find:
a) Cost at $x=1750$ b) Average cost at $x=1750$ c) Marginal cost at $x=1750$ d) Production level minimizing average cost e) Minimal average cost

See Solution

Problem 9525

Find the linearization of $f(x) = 2 \cos x$ at $x = -\frac{\pi}{4}$ .

See Solution

Problem 9526

Estimate the increase in hamburgers sold when advertising spending goes from \ $14 to \$14.70, using$ N=1200+210x-5x^2$.

See Solution

Problem 9527

Find the linear approximation of $f(x)=2 \cos x$ at $x=-\pi / 4$ .

See Solution

Problem 9528

Find critical points of $f(x)=x(2-x)^{4}$ .

See Solution

Problem 9529

Find the instantaneous rate of change of $f(x)=\sqrt{x-2}$ at $x=2$ using the limit as $h$ approaches $0$ .

See Solution

Problem 9530

Find the derivative of the function $f(x)=x^{3}$ using the limit definition of a derivative. No need to simplify.

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Problem 9531

Find the critical points of the function $f(x)=x(2-x)^{3/5}$ .

See Solution

Problem 9532

Find the derivative of $f(x)=5 x^{5}$ using the limit: $\lim _{h \rightarrow 0}$ . No need to simplify.

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Problem 9533

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+\pi^{2}}-\pi}{x^{2}}$ . Options: 0, $\frac{1}{\pi}$ , does not exist, $\frac{1}{2 \pi}$ .

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Problem 9534

Find the derivative of $f(x)=3 \cos x$ at $x=\pi$ using the limit definition: $f'(x) = \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 9535

Find values of $a$ and $b$ so that $\lim _{x \rightarrow 2} \frac{(a x-2)(x+b)}{x^{2}-x-2}$ exists.

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Problem 9536

Approximate the zero of $f$ using the tangent line at $x=-2$ given $f(-2)=1$ and $f^{-1}(-2)=5$ . Choices: A) -0.3 B) -2.6 C) -2.2 D) -2.4

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Problem 9537

Find the limit: $\lim _{x \rightarrow \infty} \frac{(2 x-5)^{3}(3 x-1)^{2}}{(2 x-1)^{2}(3 x-5)^{3}}$ . Options: A) 1, E) $\frac{3}{5}$ , I) $\frac{2}{3}$ , U) does not exist.

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Problem 9538

Find the limit: $\lim _{x \rightarrow-\infty} \frac{2^{x}+\cos x}{x^{2}+\sin x}$ . What is the value?

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Problem 9539

Find the critical points of $f(x)=\sin ^{2} x+\sqrt{3} \cos x$ in the interval $(-\pi / 2, \pi / 2)$ .

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Problem 9540

Find the dimensions of a box with volume $62500 \mathrm{~cm}^{3}$ that minimizes surface area.
1. Surface area formula: $A(x)=...$
2. Derivative: $A^{\prime}(x)=...$
3. Set $A^{\prime}(x)=0$ : $x=...$
4. Second derivative: $A^{\prime \prime}(x)=...$
5. Evaluate $A^{\prime \prime}(x)$ at the $x$ from step 3.

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Problem 9541

Find the derivative of the function $f(x)=x \sqrt{1-x^{2}}-3$ on the interval $[-1,1]$ .

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Problem 9542

Sketch a function $f$ with these properties:
1. $f' > 0$ , $f'' < 0$ for $x < -6$
2. $f' > 0$ , $f'' > 0$ for $-6 < x < -4$
3. $f' > 0$ , $f'' < 0$ for $-4 < x < 2$
4. $f' < 0$ , $f'' < 0$ for $2 < x < 4$
5. $f' < 0$ , $f'' > 0$ for $x > 4$

See Solution

Problem 9543

Find the derivative of the function $h(x)=8^{x^{2}-x}$ . What is $h^{\prime}(x)=?$

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Problem 9544

Find the dimensions of a box with volume $237276 \mathrm{~cm}^{3}$ that minimizes surface area.
1. Surface area formula: $A(x)=...$
2. Derivative: $A^{\prime}(x)=...$
3. Solve $A^{\prime}(x)=0$ for $x=...$
4. Second derivative: $A^{\prime \prime}(x)=...$ and evaluate at found $x$ .

See Solution

Problem 9545

Find the absolute maximum of $f(x)=\frac{6+2x}{x^{2}+16}$ on the interval $[-2,4]$ .

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Problem 9546

Find dimensions of an open-top box with volume $219488 \mathrm{~cm}^{3}$ that minimizes surface area.
1. Surface area formula: $A(x)=x^{2}+4 \cdot \frac{219488}{x}$ .
2. Derivative: $A^{\prime}(x)=2 \cdot x-4 \cdot \frac{219488}{x^{2}}$ .
3. Solve $A^{\prime}(x)=0$ for $x$ .
4. Find $A^{\prime \prime}(x)$ for the second derivative test.

See Solution

Problem 9547

Find the maximum product $y = x(50 - x)$ for positive $x$ such that $x + (50 - x) = 50$ .

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Problem 9548

Find the average price $p(8)$ and rate of change $p'(8)$ of a two-bedroom apartment from 1994 to 2004 using $p(t)=0.16 e^{0.10 t}$ .

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Problem 9549

Find the day $t$ (1 ≤ $t$ ≤ 50) when the maximum growth rate of car sales $N(t) = 2000 + 54t^2 - t^3$ happens.

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Problem 9550

Given the profit function $P(x)=-1.5 x^{2}+1075 x-7500$ , find:
a) Profit for 80 units. b) Average profit per unit for 80 units. c) Rate of profit change at 80 units. d) Average profit change from 80 to 160 units. e) Units sold when profit stops increasing.

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Problem 9551

Find the cylinder dimensions for a 550 cm³ soup cup that minimize costs: sides at 0.04¢/cm², top at 0.06¢/cm².

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Problem 9552

Find the maximum value of $f(x)=5+4 \sin (x) \cos (x)$ on the interval $[0, \pi]$ .

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Problem 9553

Check if Rolle's Theorem applies to $f(x)=\frac{x^{2}-3 x-18}{x-3}$ on $[-3,6]$ and find all $c$ that satisfy it.

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Problem 9554

Check if $f(x)=x \sqrt{12-x}$ meets Rolle's Theorem on $[0,12]$ and find all $c$ that satisfy its conclusion.

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Problem 9555

Cut squares of side length $x$ from a 32 in x 32 in cardboard to make an open-top box. Find $V(x)$ and max volume.

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Problem 9556

What is the maximum speed limit Fred can set to catch Natalie speeding on her 30-mile trip to Wal-Mart in under 42 minutes? Use the MVT.

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Problem 9557

Bestimmen Sie mit der zweiten Ableitung die Intervalle für Links- und Rechtskurven der Funktionen: a) $f(x)=\frac{3}{20} x^{5}-2 x^{3}+x$ b) $f(x)=\frac{1}{4} x^{4}+3 x^{3}-2$ c) $f(x)=x^{4}-2 x^{3}$ Analysieren Sie den Graphen von $f^{\prime}$ .

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Problem 9558

Maximize the area of a rectangular corral with 200 ft of fencing for 2 equal pens. Use $2l + 3w = 200$ .

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Problem 9559

Find the derivative of $f(x)=\sin ^{2}(x)-\cos (x)$ on $[0,2 \pi]$ and all critical points in $(0,2 \pi)$ .

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Problem 9560

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Extremwerte der Summe und Differenz in $x \in [0; 4]$ .

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Problem 9561

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde $x \in [0; 4]$ für max/min der Summe und Differenz von $f$ und $g$ .

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Problem 9562

Untersuche die Funktion $f_{1}(x)=(x+1) \cdot e^{-x}$ auf Nullstellen, Extrempunkte und Wendepunkte. Finde die Gleichung für Wendepunkte der Funktionenschar.

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Problem 9563

Find a variable change to transform the Ricker model $\frac{d N}{d t}=r N e^{-\beta N}$ into $\frac{d n}{d \tau}=n e^{-n}$ .

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Problem 9564

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}$ .
a) Untersuchen Sie Nullstellen, Extrempunkte und Wendepunkte. b) Finden Sie die Gleichung für Wendepunkte. c) Zeichnen Sie $f_{-1}$ und $f_{0,5}$ . d) Berechnen Sie die Fläche $A(k)$ für $k>0$ und $\lim_{k \rightarrow \infty} A(k)$ .

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Problem 9565

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Maxima/Minima der Summe und Differenz in $x \in[0; 4]$ .

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Problem 9566

If 10000 zlotych is invested at an $8\%$ continuous interest rate, what is the average annual income for 5 and 20 years?

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Problem 9567

Berechne die Fläche A(k) zwischen der Funktion $f_{0,5}$ , den Achsen und der Linie $x=k$ . Bestimme $\lim_{k \rightarrow \infty} A(k)$ .

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Problem 9568

Explain how to use l'Hôpital's Rule for limits of the form $\frac{0}{0}$ . Choose the correct method: A, B, C, or D.

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Problem 9569

How to change a limit $0 \cdot \infty$ to $0 / 0$ or $\infty / \infty$ ? Choose the correct option from A, B, C, D.

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Problem 9570

Find the derivative of $y=5x(x^{3}-1)^{4}$ .

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Problem 9571

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{4}{2 x-\pi}\right)}$ using l'Hôpital's Rule.

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Problem 9572

Solve the integral $\int_{1}^{k}(x-2) d x=\frac{3}{2}$ for the variable $k$ .

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Problem 9573

Convert the limit $\lim _{x \rightarrow 1-}(x-1) \tan \left(\frac{\pi x}{2}\right)$ from $0 \cdot \infty$ to $\frac{0}{0}$ and evaluate it.

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Problem 9574

Find $a + b$ for the function $f(x) = \{ a x^3 + 2, x < \frac{1}{2}; b x^2 + 1, x \geq \frac{1}{2} \}$ if it's differentiable at $x = \frac{1}{2}$ .

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Problem 9575

Find the derivative of $y = 3x^{2} \ln(3x+1)$ .

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Problem 9576

Leiten Sie die Funktion $f$ ab: a) $f(x)=x \cdot \sin (x)$ b) $f(x)=3 x \cdot \cos (x)$ c) $f(x)=(3 x+2) \cdot \sqrt{x}$

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Problem 9577

Convert a limit of the form $0 \cdot \infty$ to $0 / 0$ or $\infty / \infty$ . Choose the correct option below.

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Problem 9578

Gegeben ist $f(x)=-x^{2}+2 a x$ mit Nullstellen 0 und $2 a$ . a) Flächeninhalt ist $\frac{4}{3} a^{3}$ . b) Bestimme $a$ für Quadratfläche gleich Flächeninhalt.

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Problem 9579

Find the derivative of $y=3 x^{2}-2 x+1$ . What is $\frac{d y}{d x}$ ?

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Problem 9580

Find the derivative of $y=\frac{1}{2} x^{3}+3 x+\sqrt{x}$ .

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Problem 9581

Find the derivative of $y=-4 x^{-7}-2 x+\sqrt{3}$ . What is $\frac{d y}{d x}$ ?

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Problem 9582

Determine the horizontal asymptotes for these functions:
1. $y=\frac{x-10}{3 x^{2}+x+1}$
2. $y=\frac{x^{2}-10}{3 x^{2}+x+1}$

See Solution

Problem 9583

Find the marginal cost function $C'(z)$ for the total cost $C(z)=2 z^{2}+25 z-100$ . Choose the correct option.

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Problem 9584

Find the marginal cost function $C'(z)$ for the total cost $C(z)=2z^2+25z-100$ . Choose the correct option.

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Problem 9585

Calculate the limit: $=\frac{1}{4}\left(-\frac{1}{2} \cdot \lim _{x \rightarrow \frac{\pi}{2}-}\left(\sec ^{2}(x)(2 x-\pi)^{2}\right)\right)$

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Problem 9586

Calculate the limit: $=\frac{1}{4}\left(-\frac{1}{2} \cdot \lim _{x \rightarrow \frac{\pi}{2}-}\left(\sec ^{2}(x)(2 x-\pi)^{2}\right)\right)$ .

See Solution

Problem 9587

Differentiate $f(x)=2^{x}$ using logarithmic differentiation. Find $f^{\prime}(x)$ from $\ln(f(x))=x \ln 2$ .

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Problem 9588

Evaluate the expression: $=\frac{1}{4}\left(-\frac{1}{2} \cdot \lim _{x \rightarrow \frac{\pi}{2}}\left(\sec ^{2}(x)(2 x-\pi)^{2}\right)\right)$ .

See Solution

Problem 9589

Differentiate the function $f(x) = x^x$ using logarithmic differentiation.

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Problem 9590

Find the derivative of $g(t)=\csc(t)-\sin(t)+\tan(t)$ .

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Problem 9591

A particle is launched at $12 \mathrm{~ms}^{-1}$ on an incline with $5 \mathrm{~ms}^{-2}$ down. Find time for displacement > 8 m.

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Problem 9592

Find the derivative of the function $y=2e^x\sin(x)$ .

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Problem 9593

Bestimme die Tangente an die Wendepunkte der Funktion $f(x)=\frac{1}{6} x^{3}-\frac{3}{4} x^{2}+2$ .

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Problem 9594

Bestimme den Grenzwert $\lim _{x \rightarrow 4} \frac{2 x^{2}-32}{x-4}$ durch Umformung oder die h-Methode.

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Problem 9595

Find critical points of $f(x)=x \mathrm{e}^{-x^{2}}$ and analyze their nature using the first and second derivatives.

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Problem 9596

Find a formula for the $n^{\text{th}}$ derivative of $f(x)=\frac{x}{2 e^{x}}$ and compute $f^{(12345)}(0)$ .

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Problem 9597

Find the velocity of a particle at $t=4$ seconds given $s(t)=9 t^{2}+3$ . What is $v(4)$ in feet per second?

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Problem 9598

Analyze the function $f(x) = x^n$ as $x$ approaches $\infty$ or $-\infty$ for even and odd $n$ . Explain the behavior.

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Problem 9599

Find the velocity of a particle at $t=6$ seconds given $s(t)=3 t^{2}+41 t$ .

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Problem 9600

Design a rectangular container with a square base and volume $2673 \mathrm{ft}^{3}$ . Minimize cost given material prices: top \$8/ft², sides \$3/ft², bottom \$14/ft². Find dimensions.

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Calculus

Problem 9501

Find the absolute extrema of f(x)=x−3f(x)=\sqrt{x}-3f(x)=x​−3 on the interval [0,9][0,9][0,9]. Give minimum and maximum points (x,y)(x, y)(x,y).

Problem 9502

Find the relationship between B(p)B(p)B(p), E(p)E(p)E(p), C(k)C(k)C(k), and T(p)T(p)T(p). Which expression estimates extra cookies sold with one more employee?

Problem 9503

Check if Rolle's Theorem applies to f(x)=x3−48x−6f(x)=x^{3}-48 x-6f(x)=x3−48x−6 on [−4,8][-4,8][−4,8]. Select all that apply. c=c= c=

Problem 9504

Find the limit as xxx approaches 0 for g(x)g(x)g(x), given 10x2+7x+8<g(x)<7x2−3x+810 x^{2}+7 x+8<g(x)<7 x^{2}-3 x+810x2+7x+8<g(x)<7x2−3x+8.

Problem 9505

Find the average rate of change of f(x)=4x2−8f(x)=4x^{2}-8f(x)=4x2−8 on [3,a][3, a][3,a]. Express your answer in terms of aaa.

Problem 9506

Find the limit as xxx approaches 0 for g(x)g(x)g(x), given 9x2+5x+6<g(x)<9x2−8x+69 x^{2}+5 x+6<g(x)<9 x^{2}-8 x+69x2+5x+6<g(x)<9x2−8x+6.

Problem 9507

Find the tangent line of y=sin⁡xy=\sin xy=sinx at x=0x=0x=0. Is it a good approximation for the curve near the origin? Approximate sin⁡(0.2)\sin(0.2)sin(0.2).

Problem 9508

Find the linear approximation of f(x)=16+xf(x)=\frac{1}{\sqrt{6+x}}f(x)=6+x​1​ at x=0x=0x=0.

Problem 9509

Find the derivative of the function y=(5x3−2x2+8x−6)ex3y=\left(5 x^{3}-2 x^{2}+8 x-6\right) e^{x^{3}}y=(5x3−2x2+8x−6)ex3.

Problem 9510

Estimate 4.7\sqrt{4.7}4.7​ using linear approximation at a=4a=4a=4.

Problem 9511

Find the differential, dyd ydy, of the function y=f(x)=tan⁡(4x2)y=f(x)=\tan(4x^{2})y=f(x)=tan(4x2).

Problem 9512

Find the derivative dydx\frac{dy}{dx}dxdy​ for the equation sin⁡x+2cos⁡(2y)=1\sin x + 2 \cos(2y) = 1sinx+2cos(2y)=1.

Problem 9513

Find the average rate of change of f(x)=2x2−3f(x)=2x^{2}-3f(x)=2x2−3 from x=1x=1x=1 to x=ax=ax=a. Express your answer in terms of aaa.

Problem 9514

Find dydx\frac{d y}{d x}dxdy​ if dydx[(4−x)⋅y2]=x3\frac{d y}{d x}[(4-x) \cdot y^{2}] = x^{3}dxdy​[(4−x)⋅y2]=x3.

Problem 9515

Find the average value favef_{\text{ave}}fave​ of f(x,y)f(x, y)f(x,y) over region DDD given ∬Df(x,y)dA=1289\iint_{D} f(x, y) dA = \frac{128}{9}∬D​f(x,y)dA=9128​ and area D=2πD = 2\piD=2π.

Problem 9516

Find the linear function for g(x)=sec⁡xg(x)=\sec xg(x)=secx at x=π3x=\frac{\pi}{3}x=3π​, approximate g(π3−0.01)g\left(\frac{\pi}{3}-0.01\right)g(3π​−0.01), and calculate the percent error.

Problem 9517

Estimate 181/418^{1/4}181/4 using linear approximation, knowing (16)1/4=2(16)^{1/4}=2(16)1/4=2.

Problem 9518

Find the differential dyd ydy for the function y=1+sin⁡x4−sin⁡xy=\frac{1+\sin x}{4-\sin x}y=4−sinx1+sinx​.

Problem 9519

Find the derivative dy/dxd y / d xdy/dx from the equation dydx[xey−10x+3y]=0\frac{d y}{d x}\left[x e^{y}-10 x+3 y\right]=0dxdy​[xey−10x+3y]=0.

Problem 9520

Find the limit as xxx approaches infinity for 9x4+1x2−3x+5\frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}x2−3x+59x4+1​​.

Problem 9521

What is the temperature of hot chocolate after 2 minutes in a freezer at 0°F if it starts at 190°F? Use k=0.12k=0.12k=0.12.

Problem 9522

Estimate paint needed for a sphere (diameter 12cms12 \mathrm{cms}12cms) with a coat thickness of 1/6cms1 / 6 \mathrm{cms}1/6cms.

Problem 9523

Determine if these integrals converge or diverge: (a) ∫e∞dxx2ln⁡x\int_{e}^{\infty} \frac{d x}{x^{2} \ln x}∫e∞​x2lnxdx​, (b) ∫e∞ln⁡xxdx\int_{e}^{\infty} \frac{\ln x}{\sqrt{x}} d x∫e∞​x​lnx​dx.

Problem 9524

Given the cost function C(x)=19600+500x+x2C(x)=19600+500 x+x^{2}C(x)=19600+500x+x2, find:a) Cost at x=1750x=1750x=1750 b) Average cost at x=1750x=1750x=1750 c) Marginal cost at x=1750x=1750x=1750 d) Production level minimizing average cost e) Minimal average cost

Problem 9525

Find the linearization of f(x)=2cos⁡xf(x) = 2 \cos xf(x)=2cosx at x=−π4x = -\frac{\pi}{4}x=−4π​.

Problem 9526

Estimate the increase in hamburgers sold when advertising spending goes from \14to$14.70,using14 to \$14.70, using 14to$14.70,usingN=1200+210x-5x^2$.

Problem 9527

Find the linear approximation of f(x)=2cos⁡xf(x)=2 \cos xf(x)=2cosx at x=−π/4x=-\pi / 4x=−π/4.

Problem 9528

Find critical points of f(x)=x(2−x)4f(x)=x(2-x)^{4}f(x)=x(2−x)4.

Problem 9529

Find the instantaneous rate of change of f(x)=x−2f(x)=\sqrt{x-2}f(x)=x−2​ at x=2x=2x=2 using the limit as hhh approaches 000.

Problem 9530

Find the derivative of the function f(x)=x3f(x)=x^{3}f(x)=x3 using the limit definition of a derivative. No need to simplify.

Problem 9531

Find the critical points of the function f(x)=x(2−x)3/5f(x)=x(2-x)^{3/5}f(x)=x(2−x)3/5.

Problem 9532

Find the derivative of f(x)=5x5f(x)=5 x^{5}f(x)=5x5 using the limit: lim⁡h→0\lim _{h \rightarrow 0}limh→0​. No need to simplify.

Problem 9533

Find the limit: lim⁡x→0x2+π2−πx2\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+\pi^{2}}-\pi}{x^{2}}limx→0​x2x2+π2​−π​. Options: 0, 1π\frac{1}{\pi}π1​, does not exist, 12π\frac{1}{2 \pi}2π1​.

Problem 9534

Find the derivative of f(x)=3cos⁡xf(x)=3 \cos xf(x)=3cosx at x=πx=\pix=π using the limit definition: f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​.

Problem 9535

Find values of aaa and bbb so that lim⁡x→2(ax−2)(x+b)x2−x−2\lim _{x \rightarrow 2} \frac{(a x-2)(x+b)}{x^{2}-x-2}limx→2​x2−x−2(ax−2)(x+b)​ exists.

Problem 9536

Approximate the zero of fff using the tangent line at x=−2x=-2x=−2 given f(−2)=1f(-2)=1f(−2)=1 and f−1(−2)=5f^{-1}(-2)=5f−1(−2)=5. Choices: A) -0.3 B) -2.6 C) -2.2 D) -2.4

Problem 9537

Find the limit: lim⁡x→∞(2x−5)3(3x−1)2(2x−1)2(3x−5)3\lim _{x \rightarrow \infty} \frac{(2 x-5)^{3}(3 x-1)^{2}}{(2 x-1)^{2}(3 x-5)^{3}}limx→∞​(2x−1)2(3x−5)3(2x−5)3(3x−1)2​. Options: A) 1, E) 35\frac{3}{5}53​, I) 23\frac{2}{3}32​, U) does not exist.

Problem 9538

Find the limit: lim⁡x→−∞2x+cos⁡xx2+sin⁡x\lim _{x \rightarrow-\infty} \frac{2^{x}+\cos x}{x^{2}+\sin x}limx→−∞​x2+sinx2x+cosx​. What is the value?

Problem 9539

Find the critical points of f(x)=sin⁡2x+3cos⁡xf(x)=\sin ^{2} x+\sqrt{3} \cos xf(x)=sin2x+3​cosx in the interval (−π/2,π/2)(-\pi / 2, \pi / 2)(−π/2,π/2).

Problem 9540

Find the absolute extrema of $f(x)=\sqrt{x}-3$ on the interval $[0,9]$ . Give minimum and maximum points $(x, y)$ .

Find the relationship between $B(p)$ , $E(p)$ , $C(k)$ , and $T(p)$ . Which expression estimates extra cookies sold with one more employee?

Check if Rolle's Theorem applies to $f(x)=x^{3}-48 x-6$ on $[-4,8]$ . Select all that apply. $c=$

Find the limit as $x$ approaches 0 for $g(x)$ , given $10 x^{2}+7 x+8<g(x)<7 x^{2}-3 x+8$ .

Find the average rate of change of $f(x)=4x^{2}-8$ on $[3, a]$ . Express your answer in terms of $a$ .

Find the limit as $x$ approaches 0 for $g(x)$ , given $9 x^{2}+5 x+6<g(x)<9 x^{2}-8 x+6$ .

Find the tangent line of $y=\sin x$ at $x=0$ . Is it a good approximation for the curve near the origin? Approximate $\sin(0.2)$ .

Find the linear approximation of $f(x)=\frac{1}{\sqrt{6+x}}$ at $x=0$ .

Find the derivative of the function $y=\left(5 x^{3}-2 x^{2}+8 x-6\right) e^{x^{3}}$ .

Estimate $\sqrt{4.7}$ using linear approximation at $a=4$ .

Find the differential, $d y$ , of the function $y=f(x)=\tan(4x^{2})$ .

Find the derivative $\frac{dy}{dx}$ for the equation $\sin x + 2 \cos(2y) = 1$ .

Find the average rate of change of $f(x)=2x^{2}-3$ from $x=1$ to $x=a$ . Express your answer in terms of $a$ .

Find $\frac{d y}{d x}$ if $\frac{d y}{d x}[(4-x) \cdot y^{2}] = x^{3}$ .

Find the average value $f_{\text{ave}}$ of $f(x, y)$ over region $D$ given $\iint_{D} f(x, y) dA = \frac{128}{9}$ and area $D = 2\pi$ .

Find the linear function for $g(x)=\sec x$ at $x=\frac{\pi}{3}$ , approximate $g\left(\frac{\pi}{3}-0.01\right)$ , and calculate the percent error.

Estimate $18^{1/4}$ using linear approximation, knowing $(16)^{1/4}=2$ .

Find the differential $d y$ for the function $y=\frac{1+\sin x}{4-\sin x}$ .

Find the derivative $d y / d x$ from the equation $\frac{d y}{d x}\left[x e^{y}-10 x+3 y\right]=0$ .

Find the limit as $x$ approaches infinity for $\frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}$ .

What is the temperature of hot chocolate after 2 minutes in a freezer at 0°F if it starts at 190°F? Use $k=0.12$ .

Estimate paint needed for a sphere (diameter $12 \mathrm{cms}$ ) with a coat thickness of $1 / 6 \mathrm{cms}$ .

Determine if these integrals converge or diverge: (a) $\int_{e}^{\infty} \frac{d x}{x^{2} \ln x}$ , (b) $\int_{e}^{\infty} \frac{\ln x}{\sqrt{x}} d x$ .

Given the cost function $C(x)=19600+500 x+x^{2}$ , find:
a) Cost at $x=1750$ b) Average cost at $x=1750$ c) Marginal cost at $x=1750$ d) Production level minimizing average cost e) Minimal average cost

Find the linearization of $f(x) = 2 \cos x$ at $x = -\frac{\pi}{4}$ .

Estimate the increase in hamburgers sold when advertising spending goes from \ $14 to \$14.70, using$ N=1200+210x-5x^2$.

Find the linear approximation of $f(x)=2 \cos x$ at $x=-\pi / 4$ .

Find critical points of $f(x)=x(2-x)^{4}$ .

Find the instantaneous rate of change of $f(x)=\sqrt{x-2}$ at $x=2$ using the limit as $h$ approaches $0$ .

Find the derivative of the function $f(x)=x^{3}$ using the limit definition of a derivative. No need to simplify.

Find the critical points of the function $f(x)=x(2-x)^{3/5}$ .

Find the derivative of $f(x)=5 x^{5}$ using the limit: $\lim _{h \rightarrow 0}$ . No need to simplify.

Find the limit: $\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+\pi^{2}}-\pi}{x^{2}}$ . Options: 0, $\frac{1}{\pi}$ , does not exist, $\frac{1}{2 \pi}$ .

Find the derivative of $f(x)=3 \cos x$ at $x=\pi$ using the limit definition: $f'(x) = \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Find values of $a$ and $b$ so that $\lim _{x \rightarrow 2} \frac{(a x-2)(x+b)}{x^{2}-x-2}$ exists.

Approximate the zero of $f$ using the tangent line at $x=-2$ given $f(-2)=1$ and $f^{-1}(-2)=5$ . Choices: A) -0.3 B) -2.6 C) -2.2 D) -2.4

Find the limit: $\lim _{x \rightarrow \infty} \frac{(2 x-5)^{3}(3 x-1)^{2}}{(2 x-1)^{2}(3 x-5)^{3}}$ . Options: A) 1, E) $\frac{3}{5}$ , I) $\frac{2}{3}$ , U) does not exist.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{2^{x}+\cos x}{x^{2}+\sin x}$ . What is the value?

Find the critical points of $f(x)=\sin ^{2} x+\sqrt{3} \cos x$ in the interval $(-\pi / 2, \pi / 2)$ .

Find the derivative of the function $f(x)=x \sqrt{1-x^{2}}-3$ on the interval $[-1,1]$ .

Find the derivative of the function $h(x)=8^{x^{2}-x}$ . What is $h^{\prime}(x)=?$

Find the absolute maximum of $f(x)=\frac{6+2x}{x^{2}+16}$ on the interval $[-2,4]$ .

Find the maximum product $y = x(50 - x)$ for positive $x$ such that $x + (50 - x) = 50$ .

Find the average price $p(8)$ and rate of change $p'(8)$ of a two-bedroom apartment from 1994 to 2004 using $p(t)=0.16 e^{0.10 t}$ .

Find the day $t$ (1 ≤ $t$ ≤ 50) when the maximum growth rate of car sales $N(t) = 2000 + 54t^2 - t^3$ happens.

Find the maximum value of $f(x)=5+4 \sin (x) \cos (x)$ on the interval $[0, \pi]$ .

Check if Rolle's Theorem applies to $f(x)=\frac{x^{2}-3 x-18}{x-3}$ on $[-3,6]$ and find all $c$ that satisfy it.

Check if $f(x)=x \sqrt{12-x}$ meets Rolle's Theorem on $[0,12]$ and find all $c$ that satisfy its conclusion.

Cut squares of side length $x$ from a 32 in x 32 in cardboard to make an open-top box. Find $V(x)$ and max volume.

Maximize the area of a rectangular corral with 200 ft of fencing for 2 equal pens. Use $2l + 3w = 200$ .

Find the derivative of $f(x)=\sin ^{2}(x)-\cos (x)$ on $[0,2 \pi]$ and all critical points in $(0,2 \pi)$ .

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Extremwerte der Summe und Differenz in $x \in [0; 4]$ .

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde $x \in [0; 4]$ für max/min der Summe und Differenz von $f$ und $g$ .

Untersuche die Funktion $f_{1}(x)=(x+1) \cdot e^{-x}$ auf Nullstellen, Extrempunkte und Wendepunkte. Finde die Gleichung für Wendepunkte der Funktionenschar.

Find a variable change to transform the Ricker model $\frac{d N}{d t}=r N e^{-\beta N}$ into $\frac{d n}{d \tau}=n e^{-n}$ .

Gegeben sind $f(x)=0,5 x^{2}+2$ und $g(x)=x^{2}-2 x+2$ . Finde Maxima/Minima der Summe und Differenz in $x \in[0; 4]$ .

If 10000 zlotych is invested at an $8\%$ continuous interest rate, what is the average annual income for 5 and 20 years?

Berechne die Fläche A(k) zwischen der Funktion $f_{0,5}$ , den Achsen und der Linie $x=k$ . Bestimme $\lim_{k \rightarrow \infty} A(k)$ .

Explain how to use l'Hôpital's Rule for limits of the form $\frac{0}{0}$ . Choose the correct method: A, B, C, or D.

How to change a limit $0 \cdot \infty$ to $0 / 0$ or $\infty / \infty$ ? Choose the correct option from A, B, C, D.

Find the derivative of $y=5x(x^{3}-1)^{4}$ .

Evaluate the limit: $\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{4}{2 x-\pi}\right)}$ using l'Hôpital's Rule.

Solve the integral $\int_{1}^{k}(x-2) d x=\frac{3}{2}$ for the variable $k$ .

Convert the limit $\lim _{x \rightarrow 1-}(x-1) \tan \left(\frac{\pi x}{2}\right)$ from $0 \cdot \infty$ to $\frac{0}{0}$ and evaluate it.

Find $a + b$ for the function $f(x) = \{ a x^3 + 2, x < \frac{1}{2}; b x^2 + 1, x \geq \frac{1}{2} \}$ if it's differentiable at $x = \frac{1}{2}$ .