Calculus

Problem 10501

Find $f^{\prime}(x)$ for $f(x)=x^{2}-4x$ , then find slopes at $x=-2,0,4$ and sketch tangent lines.

See Solution

Problem 10502

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and evaluate at $x=1,3,5$ sec.

See Solution

Problem 10503

Find the average velocity of $y=f(x)=x^{2}+x$ for $x$ from 2 to 5, then from 2 to $2+h$ , and the instantaneous velocity at $x=2$ .

See Solution

Problem 10504

An object moves along the $y$ axis with $y=f(x)=x^{2}+x$ . Find: (A) average velocity from $x=4$ to $6$ ; (B) from $4$ to $4+h$ ; (C) instantaneous velocity at $x=4$ .

See Solution

Problem 10505

Find the cylinder dimensions for maximum volume that fits inside a sphere of radius $R$ .

See Solution

Problem 10506

Find the limit: $\lim _{x \rightarrow e^{+}}(\ln x)^{\frac{1}{x-e}}$ .

See Solution

Problem 10507

Find $f^{\prime}(2)$ given $f(x)=g(h(x))$ , $g^{\prime}(3)=\frac{1}{2}$ , $h(2)=3$ , and $h^{\prime}(2)=10$ .

See Solution

Problem 10508

Differentiate the equation $x^{4}=300+4 y^{2}$ implicitly and find $y'$ . Choose the correct option from a to f.

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Problem 10509

Find the $y$ -intercept of the tangent line to $y=\frac{3.1}{\sqrt{4+4 x}}$ at the point $(2,0.8949)$ .

See Solution

Problem 10510

Find the derivative of $g(x)=3^{5 x}$ . What is it? a. $\ln (3) 3^{5 x}$ b. $3 \ln (5) 3^{x}$ c. $\ln (3) 3^{x}$ d. $5 \ln (3) 3^{5 x}$ e. $5 \ln (3) 3^{x}$ f. $3^{4 x}$

See Solution

Problem 10511

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and evaluate it at $x=1,3,5$ .

See Solution

Problem 10512

Find the derivative $f'(x)$ for $f(x)=\pi \sin (x) e^{x} x^{4}$ . Choose the correct option.

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Problem 10513

Differentiate $\sin^{3}(5 \cos x)$ with respect to $x$ .

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Problem 10514

Find the velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and calculate it at $x=1, 3, 5$ seconds.

See Solution

Problem 10515

Determine the true statement about inflection points for $f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}$ .

See Solution

Problem 10516

Find the absolute maximum of $f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{13}{6}$ on $[-2,2]$ .

See Solution

Problem 10517

Identify conclusions about $f(x)$ given critical points at $x=8$ , $x=10$ and its behavior in intervals.

See Solution

Problem 10518

Find the minimum value of $f(7)$ given $f(4)=9$ and $f^{\prime}(x) \geq 3$ for all $x$ .

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Problem 10519

Find the minimum value of $f(\gamma)$ given that $f(4)=9$ and $f^{\prime}(x) \geq 3$ for all $x$ .

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Problem 10520

Given $\int_{-2}^{1} f(x) d x=6$ , $\int_{-2}^{-1} f(x) d x=4$ , $\int_{0}^{1} f(x) d x=2$ , find $\int_{-1}^{0} f(x) d x=$ and $\int_{0}^{-1} 6 f(x)-4 d x=$ .

See Solution

Problem 10521

For $f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}$ , which statements about inflection points are true?

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Problem 10522

Let $F(x)=\int_{0}^{x} \frac{1-t}{t^{2}+8} d t$ .
(a) Find $x$ where $F$ has a maximum: $x=$
(b) Determine intervals where $F$ is increasing and decreasing (use "none" if applicable).
Increasing intervals: Decreasing intervals:
(c) Find intervals where $F$ is concave up and concave down (use "none" if applicable).
Concave up intervals: Concave down intervals:

See Solution

Problem 10523

Calculate the area of the surface formed by rotating $y=\frac{1}{3} x^{3 / 2}$ , for $0 \leq x \leq 12$ , around the $y$ -axis.

See Solution

Problem 10524

Find the marginal cost at a production level of 60 items for the functions $C(q)=q^{3}-10 q^{2}+53 q+5000$ and $R(q)=-3 q^{2}+2500 q$ .

See Solution

Problem 10525

Find the velocity of a projectile at $t = 2.6$ seconds given the height function $f(t) = -16 t^{2} + 301 t + 5$ .

See Solution

Problem 10526

Cost function: $C(q)=100q+98$ , Revenue function: $R(q)=100q+\frac{51q}{\ln q}$ . Find marginal cost, profit function, and profit for 9 units sold when 8 units are sold.

See Solution

Problem 10527

Find the oblique asymptote for $f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}$ . Answer: $y=$

See Solution

Problem 10528

Find the limit using I'Hopital's Rule: $\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t \, dt}{x^{2}}$

See Solution

Problem 10529

Evaluate the limit of $f(x) = \frac{3x^2 - 2}{x^2 + 2}$ as $x \to -\infty$ . What is the limit?

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Problem 10530

Find the oblique asymptote for $f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}$ . Answer: $y=$

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Problem 10531

Evaluate $f(x)=\frac{3 x^{2}-2}{x^{2}+2}$ as $x \to -\infty$ . What is the limit? $\lim _{x \rightarrow-\infty} f(x) =$

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Problem 10532

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=2x+\sqrt{4x^{2}+1}$ .

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Problem 10533

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-(x-1)(x+1)^{2}(2x^{2}+1)$ .

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Problem 10534

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{-4+x^{4}-2 x^{3}}{-5 x+x^{2}+1}$ .

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Problem 10535

Find the derivative $f^{\prime}(x)$ and evaluate it at $c=0$ for the function $f(x)=(x^{4}+3x)(3x^{4}+3x-3)$ .

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Problem 10536

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=6 \cos \left(\frac{1}{-4 x-8}\right)$ .

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Problem 10537

Find the derivative $f^{\prime}(x)$ for $f(x)=x \cos (x)$ and evaluate it at $c=\frac{\pi}{4}$ .

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Problem 10538

Find the rate of change of the ant population given by $P(t)=(t+120) \ln (t+1)$ on days 4 and 7.

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Problem 10539

Find the derivatives $f^{\prime}(x)$ and $f^{\prime}(c)$ for $f(x)=x \cos(x)$ where $c=\frac{\pi}{4}$ .

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Problem 10540

Evaluate the function $f(x)=-(\sqrt{3}) x+\sqrt{3 x^{2}+1}-2$ as $x \rightarrow \infty$ and find its limit.

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Problem 10541

Find the integral: $y=\int \frac{6}{\sqrt{3 x+1}} \, dx$

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Problem 10542

Find the limit: $\lim _{x \rightarrow \infty} \frac{10 x^{3}+x^{2}-5}{8-4 x-4 x^{3}}$ . Choose the answer: $\infty$ , 0, $\frac{-5}{2}$ , $-\infty$ .

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Problem 10543

Find the rate of change of the current $A(t)=5 \sin(120 \pi t)$ at $t=2 \mathrm{~s}$ .

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Problem 10544

Use Newton's method to find a root of $f(x)=7 x^{2}-\sqrt{x}-2$ starting with $x_{0}=1$ . Show your work in a table.

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Problem 10545

Differentiate the function $F(X, Y)=(2 X^{2}-2 X+4)(-Y+6)$ with respect to $X$ : $\frac{\partial F}{\partial X}=$

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Problem 10546

Find the derivative of $f(x)=\frac{3}{x^{2}}$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 10547

Find the first and second derivatives of the function $f(x)=\sec(x)$ .

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Problem 10548

Find the integral $\int \cos (\ln (x)) d x$ using integration by parts.

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Problem 10549

Find the first and second derivatives of the function $f(x)=\sec(x)$ . First derivative: $f'(x)=\sec(x) \cdot \tan(x)$ .

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Problem 10550

Find the first and second derivatives of the function $f(x)=\sec(x)$ . First: $f'(x)=\sec(x) \cdot \tan(x)$ ; Second: $f''(x)=2 \sec^2(x) - \tan(x)$ .

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Problem 10551

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-5 x+\sqrt{25 x^{2}+3 x}$ .

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Problem 10552

Find the volume of soup in a hemispherical bowl with radius $5 \mathrm{~cm}$ filled to a height of $4 \mathrm{~cm}$ .

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Problem 10553

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x) = 2x + \sqrt{4x^2 + 1}$ .

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Problem 10554

Find the maximum revenue for the function $R(x)=x(400-0.4x)$ .

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Problem 10555

Find the integral for the area between $y=x^{2}+5$ and the $x$ -axis from $x=-2$ to $x=5$ .

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Problem 10556

Find the derivative of $y=\sqrt{\frac{x-1}{x^{8}+1}}$ using logarithmic differentiation. What is $y^{\prime}(x)=?$

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Problem 10557

(a) Find $\frac{d A}{d t}$ for a circle's area $A$ in terms of radius change $\frac{d r}{d t}$ . $\frac{d A}{d t}=(\square) \frac{d r}{d t}$ (b) If the oil spill radius increases at $2 \mathrm{~m/s}$ , how fast is the area increasing when $r=39 \mathrm{~m}$ ? $\mathrm{m}^{2} / \mathrm{s}$

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Problem 10558

A culture starts with 500 bacteria and grows as $P(t)=500\left(5+\frac{3 t}{44+t^{2}}\right)$ . Find $P^{\prime}(t)$ and $P^{\prime}(3)$ (rounded to two decimal places).

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Problem 10559

Find the volume $V$ of a sphere with radius $r$ in terms of $r$ , then determine $r$ when diameter is $60 \mathrm{~mm}$ . Also, find the rate of volume increase when radius grows at $3 \mathrm{~mm/s}$ and diameter is $60 \mathrm{~mm}$ .

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Problem 10560

Find the period $T$ of a dwarf planet with a semimajor axis of $454 \mathrm{AU}$ .

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Problem 10561

How fast (in $\mathrm{m}^{2} / \mathrm{s}$ ) is the area of an oil spill increasing when the radius is $39 \mathrm{~m}$ ?

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Problem 10562

A baseball diamond is a square (90 ft sides). A batter runs to first base at 24 ft/s. Find rates to second and third bases halfway.

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Problem 10563

Find the producer surplus for the supply function $p(x)=3+0.01 x^{2}$ when $X=10$ .

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Problem 10564

Find the correct limits of integration for the region $R$ bounded by $y=(x+5)^{2}-2$ and $y=2$ when rotated about $x=7$ .

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Problem 10565

Find the derivative $\frac{dy}{dx}$ for the parametric equations $x=t^{3}+1$ and $y=t^{2}-t$ .

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Problem 10566

Find the correct limits of integration for the region $R$ bounded by $y=(x+5)^{2}-2$ and $y=2$ when rotated about $x=7$ .

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Problem 10567

Find the linear approximation $L(x)$ of $g(x)=\sqrt[5]{1+x}$ at $a=0$ . Use it to approximate $\sqrt[5]{0.95}$ and $\sqrt[5]{1.1}$ .

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Problem 10568

Find the derivative of the function $y=e^{4 x}$ .

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Problem 10569

Find the differential of the function $y=e^{4 x}$ . What is $d y$ ?

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Problem 10570

Find the limit: $\lim _{x \rightarrow \infty} g(x)=\frac{2-x}{|x|}$ .

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Problem 10571

A circular disk has a radius of $18 \mathrm{~cm}$ and a measurement error of $0.2 \mathrm{~cm}$ . Find the max area error (a) and relative & percentage errors (b).

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Problem 10572

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=-5 x^{4}-3 x+2 \cos(3 x^{3})$ .

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Problem 10573

Find $\Delta y$ and $d y$ for $y=\sqrt{x-3}$ at $x=4$ and $\Delta x=0.8$ . Round to three decimal places.

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Problem 10574

Find the second derivative of $g(x)=x^{10}-10 x^{5}+16$ .

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Problem 10575

What is true about the function $g(x)$ if its derivative is $100 x^{94}+50 x^{4}+28 x^{6}+13$ and $g(0) = 5$ ?

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Problem 10576

How many years will it take to become a millionaire by investing \ $1,000 at$ 9\% $continuous interest?$ \mathrm{yr}$

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Problem 10577

Bestimmen Sie die Ableitungen $f^{\prime}, g^{\prime}$ und $h^{\prime}$ für die Funktionen $f(x) = x^{2}-4$ , $g(x) = x^{2}+1$ , $h(x) = x^{2}+4$ .

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Problem 10578

Find the derivative of $f(x)=\tan \sqrt{x}$ . Choices: A) $1+\tan ^{2} \sqrt{x}$ B) $\frac{1+\tan ^{2} \sqrt{x}}{\sqrt{x}}$ C) $\frac{1+\tan ^{2} \sqrt{x}}{2 \sqrt{x}}$ D) $\sqrt{x}\left(1+\tan ^{2} \sqrt{x}\right)$

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Problem 10579

Find the derivative of $f(x)=\frac{-1}{\sqrt{x^{2}-2}}$ . Choose from the options A, B, C, or D.

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Problem 10580

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-10 \csc \left(\frac{1}{3 x}\right)$ .

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Problem 10581

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-10 \csc \left(\frac{1}{3 x}\right)$ .

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Problem 10582

Evaluate the limits: $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x) = \frac{-5 + 2 e^{-x}}{5 e^{-x} + 6}$ .

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Problem 10583

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\cot^{-1}(14x + 2x^3)$ .

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Problem 10584

Find the derivative of the function $y=(\sec 2 x)^{2}-(\tan 2 x)^{2}$ .

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Problem 10585

Find the instantaneous velocity $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and calculate it at $x=1,3,5$ seconds.

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Problem 10586

Is the function $f(x)=5-|x|$ differentiable at $x=0$ ? Consider the limit $\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}$ .

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Problem 10587

How long for \$5,000 to grow to \$7,700 at a continuous compound rate of 2.50%?

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Problem 10588

How many years for \$1,000 to grow to \$1,700 at a continuous 3% interest rate?

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Problem 10589

An object is dropped from a height of $1600 \mathrm{ft}$ . Find the time to hit the ground and its impact velocity.

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Problem 10590

Find the point where the line $y=-4x+12$ is tangent to the curve $y=\frac{100}{x+7}$ .

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Problem 10591

Find the derivative of the function $(\cot \sqrt{1+x^{2}})$ .

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Problem 10592

Exercice 7 : 1) Calculez la circulation de $\vec{V}=y^{2} \vec{e}_{x}+x^{2} \vec{e}_{y}$ sur $y^{2}=x+1$ entre A(1,1) et B(-1,0). 2) a- Circulation de $\vec{F}=\left(x^{2}-2 y\right) \vec{e}_{y}+\left(y^{2}-2 x\right) \vec{e}_{y}$ sur [A B] avec A(-4,0) et B(0,2). b- Montrez que $\vec{F}$ dérive d'un potentiel. c- Trouvez le potentiel et la circulation de $\vec{F}$ .

See Solution

Problem 10593

An object drops from a tower at $1600 \mathrm{ft}$ . Height after $x$ seconds is $s(x)=1600-16 x^{2}$ . Find time to hit ground and impact velocity.

See Solution

Problem 10594

Find the average revenue change from producing 1,000 to 1,050 car seats using $R(x)=32x-0.010x^2$ . Also, find $R'(x)$ and evaluate it at $x=1000$ .

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Problem 10595

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{9 x-6}{3 x-8}$ and evaluate it at $x=3$ .

See Solution

Problem 10596

Find the surface area of a parabolic antenna modeled by $y=\sqrt{\frac{K}{4}} x^{2}$ , $0 \leq x \leq R$ .

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Problem 10597

Given that $f\left(\frac{\pi}{6}\right)=-8$ and $f^{\prime}\left(\frac{\pi}{6}\right)=8$ , find $g^{\prime}(\pi / 6)$ and $h^{\prime}(\pi / 6)$ for $g(x)=f(x) \sin x$ and $h(x)=\frac{\cos x}{f(x)}$ .

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Problem 10598

Given $f\left(\frac{\pi}{6}\right)=-8$ and $f^{\prime}\left(\frac{\pi}{6}\right)=8$ , find $g^{\prime}(\frac{\pi}{6})$ and $h^{\prime}(\frac{\pi}{6})$ for $g(x)=f(x) \sin x$ and $h(x)=\frac{\cos x}{f(x)}$ .

See Solution

Problem 10599

Find the integral for the surface area of a tent modeled by the curve $y=R \cosh \left(1-\frac{x}{R}\right)$ rotated around the $y$ -axis.

See Solution

Problem 10600

Find the tangent line to $y=4 x \cos x$ at $(\pi,-4 \pi)$ . Determine $m$ and $b$ for $y=m x+b$ .

See Solution

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Calculus

Problem 10501

Find f′(x)f^{\prime}(x)f′(x) for f(x)=x2−4xf(x)=x^{2}-4xf(x)=x2−4x, then find slopes at x=−2,0,4x=-2,0,4x=−2,0,4 and sketch tangent lines.

Problem 10502

Find the instantaneous velocity function v=f′(x)v=f^{\prime}(x)v=f′(x) for y=f(x)=5x2−4xy=f(x)=5 x^{2}-4 xy=f(x)=5x2−4x and evaluate at x=1,3,5x=1,3,5x=1,3,5 sec.

Problem 10503

Find the average velocity of y=f(x)=x2+xy=f(x)=x^{2}+xy=f(x)=x2+x for xxx from 2 to 5, then from 2 to 2+h2+h2+h, and the instantaneous velocity at x=2x=2x=2.

Problem 10504

An object moves along the yyy axis with y=f(x)=x2+xy=f(x)=x^{2}+xy=f(x)=x2+x. Find: (A) average velocity from x=4x=4x=4 to 666; (B) from 444 to 4+h4+h4+h; (C) instantaneous velocity at x=4x=4x=4.

Problem 10505

Find the cylinder dimensions for maximum volume that fits inside a sphere of radius RRR.

Problem 10506

Find the limit: lim⁡x→e+(ln⁡x)1x−e\lim _{x \rightarrow e^{+}}(\ln x)^{\frac{1}{x-e}}limx→e+​(lnx)x−e1​.

Problem 10507

Find f′(2)f^{\prime}(2)f′(2) given f(x)=g(h(x))f(x)=g(h(x))f(x)=g(h(x)), g′(3)=12g^{\prime}(3)=\frac{1}{2}g′(3)=21​, h(2)=3h(2)=3h(2)=3, and h′(2)=10h^{\prime}(2)=10h′(2)=10.

Problem 10508

Differentiate the equation x4=300+4y2x^{4}=300+4 y^{2}x4=300+4y2 implicitly and find y′y'y′. Choose the correct option from a to f.

Problem 10509

Find the yyy-intercept of the tangent line to y=3.14+4xy=\frac{3.1}{\sqrt{4+4 x}}y=4+4x​3.1​ at the point (2,0.8949)(2,0.8949)(2,0.8949).

Problem 10510

Find the derivative of g(x)=35xg(x)=3^{5 x}g(x)=35x. What is it? a. ln⁡(3)35x\ln (3) 3^{5 x}ln(3)35x b. 3ln⁡(5)3x3 \ln (5) 3^{x}3ln(5)3x c. ln⁡(3)3x\ln (3) 3^{x}ln(3)3x d. 5ln⁡(3)35x5 \ln (3) 3^{5 x}5ln(3)35x e. 5ln⁡(3)3x5 \ln (3) 3^{x}5ln(3)3x f. 34x3^{4 x}34x

Problem 10511

Find the instantaneous velocity function v=f′(x)v=f^{\prime}(x)v=f′(x) for y=f(x)=5x2−4xy=f(x)=5 x^{2}-4 xy=f(x)=5x2−4x and evaluate it at x=1,3,5x=1,3,5x=1,3,5.

Problem 10512

Find the derivative f′(x)f'(x)f′(x) for f(x)=πsin⁡(x)exx4f(x)=\pi \sin (x) e^{x} x^{4}f(x)=πsin(x)exx4. Choose the correct option.

Problem 10513

Differentiate sin⁡3(5cos⁡x)\sin^{3}(5 \cos x)sin3(5cosx) with respect to xxx.

Problem 10514

Find the velocity function v=f′(x)v=f^{\prime}(x)v=f′(x) for y=f(x)=5x2−4xy=f(x)=5 x^{2}-4 xy=f(x)=5x2−4x and calculate it at x=1,3,5x=1, 3, 5x=1,3,5 seconds.

Problem 10515

Determine the true statement about inflection points for f′′(x)=(x−5)6(x−4)9(x+5)3f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}f′′(x)=(x−5)6(x−4)9(x+5)3.

Problem 10516

Find the absolute maximum of f(x)=x33−x22−2x+136f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{13}{6}f(x)=3x3​−2x2​−2x+613​ on [−2,2][-2,2][−2,2].

Problem 10517

Identify conclusions about f(x)f(x)f(x) given critical points at x=8x=8x=8, x=10x=10x=10 and its behavior in intervals.

Problem 10518

Find the minimum value of f(7)f(7)f(7) given f(4)=9f(4)=9f(4)=9 and f′(x)≥3f^{\prime}(x) \geq 3f′(x)≥3 for all xxx.

Problem 10519

Find the minimum value of f(γ)f(\gamma)f(γ) given that f(4)=9f(4)=9f(4)=9 and f′(x)≥3f^{\prime}(x) \geq 3f′(x)≥3 for all xxx.

Problem 10520

Problem 10521

For f′′(x)=(x−5)6(x−4)9(x+5)3f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}f′′(x)=(x−5)6(x−4)9(x+5)3, which statements about inflection points are true?

Problem 10522

Problem 10523

Calculate the area of the surface formed by rotating y=13x3/2y=\frac{1}{3} x^{3 / 2}y=31​x3/2, for 0≤x≤120 \leq x \leq 120≤x≤12, around the yyy-axis.

Problem 10524

Find the marginal cost at a production level of 60 items for the functions C(q)=q3−10q2+53q+5000C(q)=q^{3}-10 q^{2}+53 q+5000C(q)=q3−10q2+53q+5000 and R(q)=−3q2+2500qR(q)=-3 q^{2}+2500 qR(q)=−3q2+2500q.

Problem 10525

Find the velocity of a projectile at t=2.6t = 2.6t=2.6 seconds given the height function f(t)=−16t2+301t+5f(t) = -16 t^{2} + 301 t + 5f(t)=−16t2+301t+5.

Problem 10526

Cost function: C(q)=100q+98C(q)=100q+98C(q)=100q+98, Revenue function: R(q)=100q+51qln⁡qR(q)=100q+\frac{51q}{\ln q}R(q)=100q+lnq51q​. Find marginal cost, profit function, and profit for 9 units sold when 8 units are sold.

Problem 10527

Find the oblique asymptote for f(x)=2x3−2x2+x+1−2x2−1f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}f(x)=−2x2−12x3−2x2+x+1​. Answer: y=y= y=

Problem 10528

Find the limit using I'Hopital's Rule: lim⁡x→0+∫0xtcos⁡t dtx2\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t \, dt}{x^{2}}x→0+lim​x2∫0x​t​costdt​

Problem 10529

Evaluate the limit of f(x)=3x2−2x2+2f(x) = \frac{3x^2 - 2}{x^2 + 2}f(x)=x2+23x2−2​ as x→−∞x \to -\inftyx→−∞. What is the limit?

Problem 10530

Find the oblique asymptote for f(x)=2x3−2x2+x+1−2x2−1f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}f(x)=−2x2−12x3−2x2+x+1​. Answer: y=y=y=

Problem 10531

Evaluate f(x)=3x2−2x2+2f(x)=\frac{3 x^{2}-2}{x^{2}+2}f(x)=x2+23x2−2​ as x→−∞x \to -\inftyx→−∞. What is the limit? lim⁡x→−∞f(x)=\lim _{x \rightarrow-\infty} f(x) =x→−∞lim​f(x)=

Problem 10532

Find lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) for f(x)=2x+4x2+1f(x)=2x+\sqrt{4x^{2}+1}f(x)=2x+4x2+1​.

Problem 10533

Evaluate lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) for f(x)=−(x−1)(x+1)2(2x2+1)f(x)=-(x-1)(x+1)^{2}(2x^{2}+1)f(x)=−(x−1)(x+1)2(2x2+1).

Problem 10534

Evaluate lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x) for f(x)=−4+x4−2x3−5x+x2+1f(x)=\frac{-4+x^{4}-2 x^{3}}{-5 x+x^{2}+1}f(x)=−5x+x2+1−4+x4−2x3​.

Problem 10535

Find the derivative f′(x)f^{\prime}(x)f′(x) and evaluate it at c=0c=0c=0 for the function f(x)=(x4+3x)(3x4+3x−3)f(x)=(x^{4}+3x)(3x^{4}+3x-3)f(x)=(x4+3x)(3x4+3x−3).

Problem 10536

Evaluate lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow -\infty} f(x)limx→−∞​f(x) for f(x)=6cos⁡(1−4x−8)f(x)=6 \cos \left(\frac{1}{-4 x-8}\right)f(x)=6cos(−4x−81​).

Problem 10537

Find the derivative f′(x)f^{\prime}(x)f′(x) for f(x)=xcos⁡(x)f(x)=x \cos (x)f(x)=xcos(x) and evaluate it at c=π4c=\frac{\pi}{4}c=4π​.

Problem 10538

Find the rate of change of the ant population given by P(t)=(t+120)ln⁡(t+1)P(t)=(t+120) \ln (t+1)P(t)=(t+120)ln(t+1) on days 4 and 7.

Problem 10539

Find the derivatives f′(x)f^{\prime}(x)f′(x) and f′(c)f^{\prime}(c)f′(c) for f(x)=xcos⁡(x)f(x)=x \cos(x)f(x)=xcos(x) where c=π4c=\frac{\pi}{4}c=4π​.

Problem 10540

Evaluate the function f(x)=−(3)x+3x2+1−2f(x)=-(\sqrt{3}) x+\sqrt{3 x^{2}+1}-2f(x)=−(3​)x+3x2+1​−2 as x→∞x \rightarrow \inftyx→∞ and find its limit.

Problem 10541

Find $f^{\prime}(x)$ for $f(x)=x^{2}-4x$ , then find slopes at $x=-2,0,4$ and sketch tangent lines.

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and evaluate at $x=1,3,5$ sec.

Find the average velocity of $y=f(x)=x^{2}+x$ for $x$ from 2 to 5, then from 2 to $2+h$ , and the instantaneous velocity at $x=2$ .

An object moves along the $y$ axis with $y=f(x)=x^{2}+x$ . Find: (A) average velocity from $x=4$ to $6$ ; (B) from $4$ to $4+h$ ; (C) instantaneous velocity at $x=4$ .

Find the cylinder dimensions for maximum volume that fits inside a sphere of radius $R$ .

Find the limit: $\lim _{x \rightarrow e^{+}}(\ln x)^{\frac{1}{x-e}}$ .

Find $f^{\prime}(2)$ given $f(x)=g(h(x))$ , $g^{\prime}(3)=\frac{1}{2}$ , $h(2)=3$ , and $h^{\prime}(2)=10$ .

Differentiate the equation $x^{4}=300+4 y^{2}$ implicitly and find $y'$ . Choose the correct option from a to f.

Find the $y$ -intercept of the tangent line to $y=\frac{3.1}{\sqrt{4+4 x}}$ at the point $(2,0.8949)$ .

Find the derivative of $g(x)=3^{5 x}$ . What is it? a. $\ln (3) 3^{5 x}$ b. $3 \ln (5) 3^{x}$ c. $\ln (3) 3^{x}$ d. $5 \ln (3) 3^{5 x}$ e. $5 \ln (3) 3^{x}$ f. $3^{4 x}$

Find the instantaneous velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and evaluate it at $x=1,3,5$ .

Find the derivative $f'(x)$ for $f(x)=\pi \sin (x) e^{x} x^{4}$ . Choose the correct option.

Differentiate $\sin^{3}(5 \cos x)$ with respect to $x$ .

Find the velocity function $v=f^{\prime}(x)$ for $y=f(x)=5 x^{2}-4 x$ and calculate it at $x=1, 3, 5$ seconds.

Determine the true statement about inflection points for $f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}$ .

Find the absolute maximum of $f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{13}{6}$ on $[-2,2]$ .

Identify conclusions about $f(x)$ given critical points at $x=8$ , $x=10$ and its behavior in intervals.

Find the minimum value of $f(7)$ given $f(4)=9$ and $f^{\prime}(x) \geq 3$ for all $x$ .

Find the minimum value of $f(\gamma)$ given that $f(4)=9$ and $f^{\prime}(x) \geq 3$ for all $x$ .

For $f^{\prime \prime}(x)=(x-5)^{6}(x-4)^{9}(x+5)^{3}$ , which statements about inflection points are true?

Calculate the area of the surface formed by rotating $y=\frac{1}{3} x^{3 / 2}$ , for $0 \leq x \leq 12$ , around the $y$ -axis.

Find the marginal cost at a production level of 60 items for the functions $C(q)=q^{3}-10 q^{2}+53 q+5000$ and $R(q)=-3 q^{2}+2500 q$ .

Find the velocity of a projectile at $t = 2.6$ seconds given the height function $f(t) = -16 t^{2} + 301 t + 5$ .

Cost function: $C(q)=100q+98$ , Revenue function: $R(q)=100q+\frac{51q}{\ln q}$ . Find marginal cost, profit function, and profit for 9 units sold when 8 units are sold.

Find the oblique asymptote for $f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}$ . Answer: $y=$

Find the limit using I'Hopital's Rule: $\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x} \sqrt{t} \cos t \, dt}{x^{2}}$

Evaluate the limit of $f(x) = \frac{3x^2 - 2}{x^2 + 2}$ as $x \to -\infty$ . What is the limit?

Find the oblique asymptote for $f(x)=\frac{2 x^{3}-2 x^{2}+x+1}{-2 x^{2}-1}$ . Answer: $y=$

Evaluate $f(x)=\frac{3 x^{2}-2}{x^{2}+2}$ as $x \to -\infty$ . What is the limit? $\lim _{x \rightarrow-\infty} f(x) =$

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=2x+\sqrt{4x^{2}+1}$ .

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-(x-1)(x+1)^{2}(2x^{2}+1)$ .

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{-4+x^{4}-2 x^{3}}{-5 x+x^{2}+1}$ .

Find the derivative $f^{\prime}(x)$ and evaluate it at $c=0$ for the function $f(x)=(x^{4}+3x)(3x^{4}+3x-3)$ .

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=6 \cos \left(\frac{1}{-4 x-8}\right)$ .

Find the derivative $f^{\prime}(x)$ for $f(x)=x \cos (x)$ and evaluate it at $c=\frac{\pi}{4}$ .

Find the rate of change of the ant population given by $P(t)=(t+120) \ln (t+1)$ on days 4 and 7.

Find the derivatives $f^{\prime}(x)$ and $f^{\prime}(c)$ for $f(x)=x \cos(x)$ where $c=\frac{\pi}{4}$ .

Evaluate the function $f(x)=-(\sqrt{3}) x+\sqrt{3 x^{2}+1}-2$ as $x \rightarrow \infty$ and find its limit.

Find the integral: $y=\int \frac{6}{\sqrt{3 x+1}} \, dx$

Find the limit: $\lim _{x \rightarrow \infty} \frac{10 x^{3}+x^{2}-5}{8-4 x-4 x^{3}}$ . Choose the answer: $\infty$ , 0, $\frac{-5}{2}$ , $-\infty$ .

Find the rate of change of the current $A(t)=5 \sin(120 \pi t)$ at $t=2 \mathrm{~s}$ .

Use Newton's method to find a root of $f(x)=7 x^{2}-\sqrt{x}-2$ starting with $x_{0}=1$ . Show your work in a table.

Differentiate the function $F(X, Y)=(2 X^{2}-2 X+4)(-Y+6)$ with respect to $X$ : $\frac{\partial F}{\partial X}=$

Find the derivative of $f(x)=\frac{3}{x^{2}}$ using the limit definition: $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Find the first and second derivatives of the function $f(x)=\sec(x)$ .

Find the integral $\int \cos (\ln (x)) d x$ using integration by parts.

Find the first and second derivatives of the function $f(x)=\sec(x)$ . First derivative: $f'(x)=\sec(x) \cdot \tan(x)$ .

Find the first and second derivatives of the function $f(x)=\sec(x)$ . First: $f'(x)=\sec(x) \cdot \tan(x)$ ; Second: $f''(x)=2 \sec^2(x) - \tan(x)$ .

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=-5 x+\sqrt{25 x^{2}+3 x}$ .

Find the volume of soup in a hemispherical bowl with radius $5 \mathrm{~cm}$ filled to a height of $4 \mathrm{~cm}$ .

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x) = 2x + \sqrt{4x^2 + 1}$ .

Find the maximum revenue for the function $R(x)=x(400-0.4x)$ .

Find the integral for the area between $y=x^{2}+5$ and the $x$ -axis from $x=-2$ to $x=5$ .

Find the derivative of $y=\sqrt{\frac{x-1}{x^{8}+1}}$ using logarithmic differentiation. What is $y^{\prime}(x)=?$

A culture starts with 500 bacteria and grows as $P(t)=500\left(5+\frac{3 t}{44+t^{2}}\right)$ . Find $P^{\prime}(t)$ and $P^{\prime}(3)$ (rounded to two decimal places).

Find the volume $V$ of a sphere with radius $r$ in terms of $r$ , then determine $r$ when diameter is $60 \mathrm{~mm}$ . Also, find the rate of volume increase when radius grows at $3 \mathrm{~mm/s}$ and diameter is $60 \mathrm{~mm}$ .

Find the period $T$ of a dwarf planet with a semimajor axis of $454 \mathrm{AU}$ .

How fast (in $\mathrm{m}^{2} / \mathrm{s}$ ) is the area of an oil spill increasing when the radius is $39 \mathrm{~m}$ ?

Find the producer surplus for the supply function $p(x)=3+0.01 x^{2}$ when $X=10$ .

Find the correct limits of integration for the region $R$ bounded by $y=(x+5)^{2}-2$ and $y=2$ when rotated about $x=7$ .

Find the derivative $\frac{dy}{dx}$ for the parametric equations $x=t^{3}+1$ and $y=t^{2}-t$ .

Find the correct limits of integration for the region $R$ bounded by $y=(x+5)^{2}-2$ and $y=2$ when rotated about $x=7$ .

Find the linear approximation $L(x)$ of $g(x)=\sqrt[5]{1+x}$ at $a=0$ . Use it to approximate $\sqrt[5]{0.95}$ and $\sqrt[5]{1.1}$ .

Find the derivative of the function $y=e^{4 x}$ .

Find the differential of the function $y=e^{4 x}$ . What is $d y$ ?

Find the limit: $\lim _{x \rightarrow \infty} g(x)=\frac{2-x}{|x|}$ .

A circular disk has a radius of $18 \mathrm{~cm}$ and a measurement error of $0.2 \mathrm{~cm}$ . Find the max area error (a) and relative & percentage errors (b).

Evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=-5 x^{4}-3 x+2 \cos(3 x^{3})$ .

Find $\Delta y$ and $d y$ for $y=\sqrt{x-3}$ at $x=4$ and $\Delta x=0.8$ . Round to three decimal places.

Find the second derivative of $g(x)=x^{10}-10 x^{5}+16$ .