Calculus

Problem 10601

Find the average change in revenue from selling 1,000 to 1,050 car seats using $R(x)=64x-0.020x^2$ . Also, find $R'(x)$ and the revenue at 1,000 seats.

See Solution

Problem 10602

An island with 20,000 people grows at $5\%$ per year. Find the population in 5 years using the exponential growth formula.

See Solution

Problem 10603

Solve the equation $L \frac{d i}{d t}+\frac{1}{C} \int_{0}^{t} i d t=0$ .

See Solution

Problem 10604

Find $f'(9)$ for the function $f(x)=6 x \sqrt{x}+\frac{3}{x^{2} \sqrt{x}}$ .

See Solution

Problem 10605

Find the end behavior of $f(x)=2^{-x}-1$ as $x \to \infty$ .

See Solution

Problem 10606

Approximate the volume of a bird's egg shell using differentials. Shell radii: inside $7$ mm, outside $7.5$ mm. Use $V(r)=\frac{4}{3} \pi r^{3}$ . Volume in $\mathrm{mm}^{3}$ , round to nearest hundredth.

See Solution

Problem 10607

Approximate the volume of a bird's egg shell using differentials. Given inner radius 8 mm and outer radius 8.3 mm, use $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ .

See Solution

Problem 10608

Graph the function and estimate its relative extrema for $f(x)=-3 x^{6}-8 x^{5}+108 x^{4}+200 x^{3}-700 x^{2}-1000 x+1,000$ .

See Solution

Problem 10609

Approximate the volume of a bird's egg shell with inner radius 7 mm and outer radius 7.5 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume: $\square \mathrm{mm}^{3}$ .

See Solution

Problem 10610

Find the marginal cost function for $C(x)=174+0.2 x$ . What is $C^{\prime}(x)$ ?

See Solution

Problem 10611

Find the marginal profit function given $C(x)=163+0.8 x$ and $R(x)=9 x-0.03 x^{2}$ . What is $P^{\prime}(x)$ ?

See Solution

Problem 10612

Find the marginal cost function for $C(x) = 170 + 3.2x - 0.03x^2$ . What is $C'(x)$ ?

See Solution

Problem 10613

Find the marginal revenue function for $R(x)=5x-0.05x^{2}$ . Calculate $R^{\prime}(x)$ .

See Solution

Problem 10614

Find the cost of producing the 41st food processor using $C(x)=2500+90x-0.2x^{2}$ : exact and marginal cost.

See Solution

Problem 10615

Évaluer les limites suivantes : a) $\lim _{x \rightarrow 3} \frac{x^{2}-5 x+6}{x^{2}-2 x-3}$ , b) $\lim _{x \rightarrow 2} \frac{3-\sqrt{x^{2}+5}}{x-2}$

See Solution

Problem 10616

Given the curve $f(x)=(2x-5)^{2}(x+3)$ , find $k$ for $f(x)-k$ at origin, $c$ for min at origin, and $B$ 's $x$ if $A$ is at $x=3$ .

See Solution

Problem 10617

Find the average cost per unit for $C(x)=30,000+900x$ at $x=80$ , and estimate it for $x=81$ . What is the marginal cost?

See Solution

Problem 10618

A $5.00 \mu \mathrm{C}$ charge at the origin and a $-3.00 \mu \mathrm{C}$ charge at $(3.00,0) \mathrm{m}$ . Find potential at $(0,4.00) \mathrm{m}$ and work to bring $4.00 \mu \mathrm{C}$ there.

See Solution

Problem 10619

Given the velocity field $\mathbf{F}(x, y)=\left(\cos \left(\theta_{0}\right)\right) \mathbf{i}+\left(\sin \left(\theta_{0}\right)\right) \mathbf{j}$ , find $f(z)$ , $g(z)=\overline{f(z)}$ , and verify if $g^{\prime}(z)$ is continuous in domain $D$ .

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Problem 10620

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-1$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 10621

Find the derivative of $y=x^{9}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 10622

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-3 x^{8}$ .

See Solution

Problem 10623

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{1}{x^{4}}$ . What is $\frac{d y}{d x}=$ ?

See Solution

Problem 10624

Find the derivative $f^{\prime}(x)$ for the function $f(x)=2 x^{3}$ .

See Solution

Problem 10625

Find the derivative of $\frac{x^{8}}{32}$ with respect to $x$ .

See Solution

Problem 10626

Find the derivative of the function -5x + 2 with respect to x. What is $\frac{d}{d x}(-5 x+2)$ ?

See Solution

Problem 10627

Find the derivative $y^{\prime}$ for the function $y=5 x^{-4}+5 x^{-1}$ .

See Solution

Problem 10628

Find the derivative of $x^{11}$ with respect to $x$ : $\frac{d}{d x} x^{11} =$

See Solution

Problem 10629

Find the derivative $h^{\prime}(t)$ for the function $h(t)=13.4-0.8 t+1.9 t^{2}$ .

See Solution

Problem 10630

Find $h^{\prime}(5)$ if $h(x)=2 f(x)+4 g(x)+6$ , given $f^{\prime}(5)=3$ and $g^{\prime}(5)=6$ .

See Solution

Problem 10631

Find the derivative of $y=x^{-10}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 10632

Find the derivative $g^{\prime}(x)$ of the function $g(x)=x^{-1/6}$ .

See Solution

Problem 10633

Find the derivative $f^{\prime}(t)$ for the function $f(t)=-9 t^{2}+3 t+6$ .

See Solution

Problem 10634

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{2 x^{4}}$ .

See Solution

Problem 10635

Compute circulation and net flux for the flow and contour $C$ : 25. $f(z)=\frac{1}{z}$ , $C$ : $|z|=1$ ; 26. $f(z)=2z$ , $C$ : $|z|=1$ ; 27. $f(z)=\frac{1}{\overline{z-1}}$ , $C$ : $|z-1|=2$ .

See Solution

Problem 10636

Differentiate $\frac{5 x^{2}}{8}-\frac{5}{9 x^{2}}$ with respect to $x$ .

See Solution

Problem 10637

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{5}{4 w^{6}}+2 \sqrt[3]{w}$ .

See Solution

Problem 10638

Find the derivative $y^{\prime}$ of the function $y=\frac{2}{7 x^{2}}$ . What is $y^{\prime}=$ ?

See Solution

Problem 10639

Find the derivative $G^{\prime}(w)$ for $G(w)=\frac{8}{5 w^{4}}+2 \sqrt[3]{w}$ .

See Solution

Problem 10640

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(8x-7)^{2}$ .

See Solution

Problem 10641

Find the derivative $y^{\prime}$ of the function $y=\frac{7}{5 x^{2}}$ . What is $y^{\prime}=?$

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Problem 10642

Find the absolute extreme values of $f(x)=x^{2}-10$ on the interval $[-2,4]$ . Where is the maximum?

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Problem 10643

For $f(x)=x^{4}-4 x^{3}+9$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line equation at $x=-2$ , (D) where tangent is horizontal.

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Problem 10644

Find the derivative $y^{\prime}$ for the function $y=\frac{x^{7}-5 x^{6}+9}{x^{5}}$ .

See Solution

Problem 10645

Find the velocity function $v=f'(x)$ for $f(x)=156x-12x^2$ , then determine $v(0)$ , $v(2)$ , and when $v=0$ .

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Problem 10646

Differentiate the function $f(x)=6-2x+8e^{x}$ . Find $f^{\prime}(x)=$ .

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Problem 10647

Differentiate the function $f(x)=3-3x+6e^{x}$ . Find $f^{\prime}(x)$ .

See Solution

Problem 10648

Find the absolute extrema of $f(x)=2 x^{3}-39 x^{2}+240 x$ on the interval $[4,9]$ .

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Problem 10649

Find the absolute extrema of $f(x)=2 x^{3}-39 x^{2}+216 x$ on $[3,10]$ . What are the max and min values?

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Problem 10650

Find the derivative $f^{\prime}(x)$ for the function $f(x)=2 e^{x}+5 x-3 \ln x$ .

See Solution

Problem 10651

Find the absolute extreme values of $f(x)=2 x^{3}-39 x^{2}+216 x$ on the interval $[3,10]$ .

See Solution

Problem 10652

Is the series $\sum_{k=1}^{\infty} 4 i\left(\frac{1}{3}\right)^{k-1}$ convergent or divergent? If convergent, find the sum.

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Problem 10653

Evaluate the integral $\pi \int_{0}^{2} \frac{0}{1}$ .

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Problem 10654

Find the derivative of $f(x)=\ln x^{14}$ with respect to $x$ .

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Problem 10655

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-15 \ln x + 4 x^{2} - 11$ .

See Solution

Problem 10656

Find the tangent line equation for $f(x)=17 e^{x}+11 x$ at $x=0$ . What is $y=$ ?

See Solution

Problem 10657

Rewrite $f(x)=24x+\ln(24x)$ using logarithm properties and find $f^{\prime}(x)$ .

See Solution

Problem 10658

Find the tangent line equation for $f(x)=2-3 \ln x$ at $x=1$ . What is $y=$ ?

See Solution

Problem 10659

Determine the max and min values of $g(x)=\sqrt{9-x^{2}}$ on the interval $[-3, 2]$ .

See Solution

Problem 10660

Rewrite $f(x)=2 \ln \left(\frac{13}{x}\right)$ using logarithm properties, then find $f^{\prime}(x)$ .

See Solution

Problem 10661

Find the rate of change of blood pressure $P(x)=12.7(1+\ln x)$ at $x=30$ and $x=40$ pounds.

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Problem 10662

Find the rate of change of blood pressure $P(x)=12.7(1+\ln x)$ at $x=30$ pounds. Answer in $\mathrm{mm}/$ pound.

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Problem 10663

Determine if the series $\sum_{k=0}^{\infty} 2\left(\frac{4}{1+4 i}\right)^{k}$ converges or diverges. If it converges, find the sum.

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Problem 10664

Given the cost function $C(x)=16900+800 x+x^{2}$ , find: a) $C(1200)$ , b) average cost at 1200, c) marginal cost at 1200, d) production level for minimal average cost, e) minimal average cost.

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Problem 10665

Rewrite $f(x)=7x+\ln(7x)$ using logarithm properties and find $f'(x)$ .

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Problem 10666

Find $c$ such that $\frac{f(3)-f(-1)}{3-(-1)}=f^{\prime}(c)$ for $f(x)=5x^2-2x-3$ on $[-1,3]$ .

See Solution

Problem 10667

Find values of $c$ satisfying $\frac{f(3)-f(-1)}{3-(-1)}=f^{\prime}(c)$ for $f(x)=3x^{2}+5x-2$ on $[-1,3]$ .

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Problem 10668

Is the series $\sum_{k=0}^{\infty} 5\left(\frac{6}{1+6 i}\right)^{k}$ convergent or divergent? If convergent, find the sum.

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Problem 10669

Find $c$ such that $\frac{f(10)-f(6)}{10-6}=f^{\prime}(c)$ for $f(x)=\ln(x-5)$ on $[6,10]$ .

See Solution

Problem 10670

Given $g(3)=2$ and $\lim_{x \rightarrow 3}[6 f(x)+f(x) g(x)]=32$ , find $f(3)$ .

See Solution

Problem 10671

Find values of $c$ for $f(x)=3x^2-5x-2$ in $[-1,2]$ satisfying $\frac{f(2)-f(-1)}{2-(-1)}=f'(c)$ .

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Problem 10672

Given continuous functions $f$ and $g$ with $g(3)=2$ and $\lim_{x \rightarrow 3}[6 f(x)+f(x) g(x)]=32$ , find $f(3)$ .

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Problem 10673

Find the average velocity of the ball using $h(t)=-\frac{1}{2}t^2+5t$ over the first 3 seconds.

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Problem 10674

Find $d y$ for $y=2 x^{2}+2 x+2$ at $x=1$ , $d x=0.3$ and $d x=0.6$ .

See Solution

Problem 10675

Find $c$ such that $\frac{f(11)-f(6)}{11-6}=f'(c)$ for $f(x)=\sqrt{x-6}$ on the interval [6,11].

See Solution

Problem 10676

Find the learning rate after 100 hours using the model $N(t)=6+7 \ln t$ . Answer in words/minute per hour.

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Problem 10677

A car's position is given by $s(t)=\frac{1}{80} t^{3}$ for $0 \leq t \leq 7$ . Find: (a) average velocity from $t=0$ to $t=7$ , (b) instantaneous velocity for $t \in(0,7)$ , (c) when is instantaneous velocity equal to average velocity?

See Solution

Problem 10678

Is there a root for $\cos x = x$ in the interval $(0, 1)$ by the Intermediate Value Theorem? Answer " $y$ " or " $n$ ".

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Problem 10679

A car's position is given by $s(t)=\frac{1}{100} t^{3}$ for $0 \leq t \leq 11$ . Find average velocity (a), instantaneous velocity (b), and when they equal (c).

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Problem 10680

Find the rate of learning after 80 hours using the model $N(t)=2+\ln t$ , where $t \geq 1$ .

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Problem 10681

Calculate the slope of the secant line for $f(x)=x^2-5$ from $x=0$ to $x=3$ .

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Problem 10682

Find the derivative of the function $f(x)=5 \sin (3 x) \arcsin (x)$ . What is $f^{\prime}(x)$ ?

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Problem 10683

Find the slope of the secant line between $x=0$ and $x=h$ for any number $h>0$ .

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Problem 10684

An investment of \$12,000 earns 6.5% interest compounded continuously. Find the rate of change after 2 years: \$\square.

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Problem 10685

Does the sequence $\{a_n\}_{n=1}^{\infty} = \{10^n - 1\}_{n=1}^{\infty}$ converge or diverge?

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Problem 10686

An investment of \$9,000 earns 8% interest compounded continuously. Find rates of change after 3 years and at \$12,500.

See Solution

Problem 10687

Evaluate the integral: $\int(5 x^{3}-6 x^{2}+4 x-3) \, dx$

See Solution

Problem 10688

Determine if the sequence $a_{n}=\frac{(-1)^{n}}{n^{2}+1}$ converges or diverges, and explain your reasoning.

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Problem 10689

An investment of \$9,000 earns 8% interest compounded continuously. Find rates of change after 2 years and at \$12,500.

See Solution

Problem 10690

Analyze the sequence $a_{n}=\frac{(-1)^{n}}{n^{2}+1}$ . Does it converge or diverge? If it converges, state its limit.

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Problem 10691

Find the tangent line approximation for the zero of $f$ at $x=-2$ given $f(-2)=1$ and $f'(-2)=5$ . Choose A, B, C, or D.

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Problem 10692

Find the $y$ -intercept of the tangent line to $f(x)=x^{2}+\frac{1}{x}$ at $x=2$ . A) $\frac{15}{4}$ B) -3 C) 3 D) $\frac{9}{2}$

See Solution

Problem 10693

Find the limit: $\lim _{x \rightarrow \infty} \frac{-3 x^{2}+2 x+1}{x^{3}+13 x^{2}+x}$ .

See Solution

Problem 10694

Determine the circle of convergence for the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k 3^{k}}(z-2-i)^{k}$ .

See Solution

Problem 10695

Is the function $h(x)$ continuous at $x=1$ ? Analyze the piecewise definition to explain your reasoning.

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Problem 10696

The value that $\left(1+\frac{1}{n}\right)^{n}$ approaches as $n$ increases is approximately equal to the irrational number e.

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Problem 10697

An investment of \$9,000 earns 8% interest compounded continuously. Find the rate of change after 2 years and at \$12,500.

See Solution

Problem 10698

Given revenue $R(q)=-q^{3}+310 q^{2}$ and cost $C(q)=480+19 q$ , find the marginal profit $M P(q)$ and max profit quantity (in hundreds).

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Problem 10699

Find the 1st and 2nd derivatives of $f(x)=e^{b x}(e^{b x}+x^{2} e^{-b x})$ and calculate $f^{\prime \prime}(0)$ .

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Problem 10700

Given the profit function $P(x)=-3.75 x^{2}+1600 x-5000$ , find:
a) Profit for 55 units b) Average profit per unit for 55 units c) Rate of profit change at 55 units d) Average profit change from 55 to 110 units e) Units sold when profit stops increasing.

See Solution

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Calculus

Problem 10601

Find the average change in revenue from selling 1,000 to 1,050 car seats using R(x)=64x−0.020x2R(x)=64x-0.020x^2R(x)=64x−0.020x2. Also, find R′(x)R'(x)R′(x) and the revenue at 1,000 seats.

Problem 10602

An island with 20,000 people grows at 5%5\%5% per year. Find the population in 5 years using the exponential growth formula.

Problem 10603

Solve the equation Ldidt+1C∫0tidt=0L \frac{d i}{d t}+\frac{1}{C} \int_{0}^{t} i d t=0Ldtdi​+C1​∫0t​idt=0.

Problem 10604

Find f′(9)f'(9)f′(9) for the function f(x)=6xx+3x2xf(x)=6 x \sqrt{x}+\frac{3}{x^{2} \sqrt{x}}f(x)=6xx​+x2x​3​.

Problem 10605

Find the end behavior of f(x)=2−x−1f(x)=2^{-x}-1f(x)=2−x−1 as x→∞x \to \inftyx→∞.

Problem 10606

Approximate the volume of a bird's egg shell using differentials. Shell radii: inside 777 mm, outside 7.57.57.5 mm. Use V(r)=43πr3V(r)=\frac{4}{3} \pi r^{3}V(r)=34​πr3. Volume in mm3\mathrm{mm}^{3}mm3, round to nearest hundredth.

Problem 10607

Approximate the volume of a bird's egg shell using differentials. Given inner radius 8 mm and outer radius 8.3 mm, use V(r)=43πr3V(r)=\frac{4}{3} \pi r^{3}V(r)=34​πr3. Volume is approximately □mm3\square \mathrm{mm}^{3}□mm3.

Problem 10608

Graph the function and estimate its relative extrema for f(x)=−3x6−8x5+108x4+200x3−700x2−1000x+1,000 f(x)=-3 x^{6}-8 x^{5}+108 x^{4}+200 x^{3}-700 x^{2}-1000 x+1,000 f(x)=−3x6−8x5+108x4+200x3−700x2−1000x+1,000.

Problem 10609

Approximate the volume of a bird's egg shell with inner radius 7 mm and outer radius 7.5 mm using V(r)=43πr3V(r)=\frac{4}{3} \pi r^{3}V(r)=34​πr3. Volume: □mm3\square \mathrm{mm}^{3}□mm3.

Problem 10610

Find the marginal cost function for C(x)=174+0.2xC(x)=174+0.2 xC(x)=174+0.2x. What is C′(x)C^{\prime}(x)C′(x)?

Problem 10611

Find the marginal profit function given C(x)=163+0.8xC(x)=163+0.8 xC(x)=163+0.8x and R(x)=9x−0.03x2R(x)=9 x-0.03 x^{2}R(x)=9x−0.03x2. What is P′(x)P^{\prime}(x)P′(x)?

Problem 10612

Find the marginal cost function for C(x)=170+3.2x−0.03x2 C(x) = 170 + 3.2x - 0.03x^2 C(x)=170+3.2x−0.03x2. What is C′(x) C'(x) C′(x)?

Problem 10613

Find the marginal revenue function for R(x)=5x−0.05x2R(x)=5x-0.05x^{2}R(x)=5x−0.05x2. Calculate R′(x)R^{\prime}(x)R′(x).

Problem 10614

Find the cost of producing the 41st food processor using C(x)=2500+90x−0.2x2C(x)=2500+90x-0.2x^{2}C(x)=2500+90x−0.2x2: exact and marginal cost.

Problem 10615

Évaluer les limites suivantes : a) lim⁡x→3x2−5x+6x2−2x−3\lim _{x \rightarrow 3} \frac{x^{2}-5 x+6}{x^{2}-2 x-3}limx→3​x2−2x−3x2−5x+6​, b) lim⁡x→23−x2+5x−2\lim _{x \rightarrow 2} \frac{3-\sqrt{x^{2}+5}}{x-2}limx→2​x−23−x2+5​​

Problem 10616

Given the curve f(x)=(2x−5)2(x+3)f(x)=(2x-5)^{2}(x+3)f(x)=(2x−5)2(x+3), find kkk for f(x)−kf(x)-kf(x)−k at origin, ccc for min at origin, and BBB's xxx if AAA is at x=3x=3x=3.

Problem 10617

Find the average cost per unit for C(x)=30,000+900xC(x)=30,000+900xC(x)=30,000+900x at x=80x=80x=80, and estimate it for x=81x=81x=81. What is the marginal cost?

Problem 10618

A 5.00μC5.00 \mu \mathrm{C}5.00μC charge at the origin and a −3.00μC-3.00 \mu \mathrm{C}−3.00μC charge at (3.00,0)m(3.00,0) \mathrm{m}(3.00,0)m. Find potential at (0,4.00)m(0,4.00) \mathrm{m}(0,4.00)m and work to bring 4.00μC4.00 \mu \mathrm{C}4.00μC there.

Problem 10619

Problem 10620

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=−1f(x)=-1f(x)=−1. What is f′(x)f^{\prime}(x)f′(x)?

Problem 10621

Find the derivative of y=x9y=x^{9}y=x9. What is dydx\frac{d y}{d x}dxdy​?

Problem 10622

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=−3x8f(x)=-3 x^{8}f(x)=−3x8.

Problem 10623

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=1x4y=\frac{1}{x^{4}}y=x41​. What is dydx=\frac{d y}{d x}=dxdy​=?

Problem 10624

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=2x3f(x)=2 x^{3}f(x)=2x3.

Problem 10625

Find the derivative of x832\frac{x^{8}}{32}32x8​ with respect to xxx.

Problem 10626

Find the derivative of the function -5x + 2 with respect to x. What is ddx(−5x+2)\frac{d}{d x}(-5 x+2)dxd​(−5x+2)?

Problem 10627

Find the derivative y′y^{\prime}y′ for the function y=5x−4+5x−1y=5 x^{-4}+5 x^{-1}y=5x−4+5x−1.

Problem 10628

Find the derivative of x11x^{11}x11 with respect to xxx: ddxx11=\frac{d}{d x} x^{11} = dxd​x11=

Problem 10629

Find the derivative h′(t)h^{\prime}(t)h′(t) for the function h(t)=13.4−0.8t+1.9t2h(t)=13.4-0.8 t+1.9 t^{2}h(t)=13.4−0.8t+1.9t2.

Problem 10630

Find h′(5)h^{\prime}(5)h′(5) if h(x)=2f(x)+4g(x)+6h(x)=2 f(x)+4 g(x)+6h(x)=2f(x)+4g(x)+6, given f′(5)=3f^{\prime}(5)=3f′(5)=3 and g′(5)=6g^{\prime}(5)=6g′(5)=6.

Problem 10631

Find the derivative of y=x−10y=x^{-10}y=x−10. What is dydx\frac{d y}{d x}dxdy​?

Problem 10632

Find the derivative g′(x)g^{\prime}(x)g′(x) of the function g(x)=x−1/6g(x)=x^{-1/6}g(x)=x−1/6.

Problem 10633

Find the derivative f′(t)f^{\prime}(t)f′(t) for the function f(t)=−9t2+3t+6f(t)=-9 t^{2}+3 t+6f(t)=−9t2+3t+6.

Problem 10634

Find the derivative y′y^{\prime}y′ for the function y=92x4y=\frac{9}{2 x^{4}}y=2x49​.

Problem 10635

Compute circulation and net flux for the flow and contour CCC: 25. f(z)=1zf(z)=\frac{1}{z}f(z)=z1​, CCC: ∣z∣=1|z|=1∣z∣=1; 26. f(z)=2zf(z)=2zf(z)=2z, CCC: ∣z∣=1|z|=1∣z∣=1; 27. f(z)=1z−1‾f(z)=\frac{1}{\overline{z-1}}f(z)=z−1​1​, CCC: ∣z−1∣=2|z-1|=2∣z−1∣=2.

Problem 10636

Differentiate 5x28−59x2\frac{5 x^{2}}{8}-\frac{5}{9 x^{2}}85x2​−9x25​ with respect to xxx.

Problem 10637

Find the derivative G′(w)G^{\prime}(w)G′(w) for the function G(w)=54w6+2w3G(w)=\frac{5}{4 w^{6}}+2 \sqrt[3]{w}G(w)=4w65​+23w​.

Problem 10638

Find the derivative y′y^{\prime}y′ of the function y=27x2y=\frac{2}{7 x^{2}}y=7x22​. What is y′=y^{\prime}=y′=?

Problem 10639

Find the derivative G′(w)G^{\prime}(w)G′(w) for G(w)=85w4+2w3G(w)=\frac{8}{5 w^{4}}+2 \sqrt[3]{w}G(w)=5w48​+23w​.

Problem 10640

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(8x−7)2f(x)=(8x-7)^{2}f(x)=(8x−7)2.

Find the average change in revenue from selling 1,000 to 1,050 car seats using $R(x)=64x-0.020x^2$ . Also, find $R'(x)$ and the revenue at 1,000 seats.

An island with 20,000 people grows at $5\%$ per year. Find the population in 5 years using the exponential growth formula.

Solve the equation $L \frac{d i}{d t}+\frac{1}{C} \int_{0}^{t} i d t=0$ .

Find $f'(9)$ for the function $f(x)=6 x \sqrt{x}+\frac{3}{x^{2} \sqrt{x}}$ .

Find the end behavior of $f(x)=2^{-x}-1$ as $x \to \infty$ .

Approximate the volume of a bird's egg shell using differentials. Shell radii: inside $7$ mm, outside $7.5$ mm. Use $V(r)=\frac{4}{3} \pi r^{3}$ . Volume in $\mathrm{mm}^{3}$ , round to nearest hundredth.

Approximate the volume of a bird's egg shell using differentials. Given inner radius 8 mm and outer radius 8.3 mm, use $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ .

Graph the function and estimate its relative extrema for $f(x)=-3 x^{6}-8 x^{5}+108 x^{4}+200 x^{3}-700 x^{2}-1000 x+1,000$ .

Approximate the volume of a bird's egg shell with inner radius 7 mm and outer radius 7.5 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume: $\square \mathrm{mm}^{3}$ .

Find the marginal cost function for $C(x)=174+0.2 x$ . What is $C^{\prime}(x)$ ?

Find the marginal profit function given $C(x)=163+0.8 x$ and $R(x)=9 x-0.03 x^{2}$ . What is $P^{\prime}(x)$ ?

Find the marginal cost function for $C(x) = 170 + 3.2x - 0.03x^2$ . What is $C'(x)$ ?

Find the marginal revenue function for $R(x)=5x-0.05x^{2}$ . Calculate $R^{\prime}(x)$ .

Find the cost of producing the 41st food processor using $C(x)=2500+90x-0.2x^{2}$ : exact and marginal cost.

Évaluer les limites suivantes : a) $\lim _{x \rightarrow 3} \frac{x^{2}-5 x+6}{x^{2}-2 x-3}$ , b) $\lim _{x \rightarrow 2} \frac{3-\sqrt{x^{2}+5}}{x-2}$

Given the curve $f(x)=(2x-5)^{2}(x+3)$ , find $k$ for $f(x)-k$ at origin, $c$ for min at origin, and $B$ 's $x$ if $A$ is at $x=3$ .

Find the average cost per unit for $C(x)=30,000+900x$ at $x=80$ , and estimate it for $x=81$ . What is the marginal cost?

A $5.00 \mu \mathrm{C}$ charge at the origin and a $-3.00 \mu \mathrm{C}$ charge at $(3.00,0) \mathrm{m}$ . Find potential at $(0,4.00) \mathrm{m}$ and work to bring $4.00 \mu \mathrm{C}$ there.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-1$ . What is $f^{\prime}(x)$ ?

Find the derivative of $y=x^{9}$ . What is $\frac{d y}{d x}$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-3 x^{8}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{1}{x^{4}}$ . What is $\frac{d y}{d x}=$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=2 x^{3}$ .

Find the derivative of $\frac{x^{8}}{32}$ with respect to $x$ .

Find the derivative of the function -5x + 2 with respect to x. What is $\frac{d}{d x}(-5 x+2)$ ?

Find the derivative $y^{\prime}$ for the function $y=5 x^{-4}+5 x^{-1}$ .

Find the derivative of $x^{11}$ with respect to $x$ : $\frac{d}{d x} x^{11} =$

Find the derivative $h^{\prime}(t)$ for the function $h(t)=13.4-0.8 t+1.9 t^{2}$ .

Find $h^{\prime}(5)$ if $h(x)=2 f(x)+4 g(x)+6$ , given $f^{\prime}(5)=3$ and $g^{\prime}(5)=6$ .

Find the derivative of $y=x^{-10}$ . What is $\frac{d y}{d x}$ ?

Find the derivative $g^{\prime}(x)$ of the function $g(x)=x^{-1/6}$ .

Find the derivative $f^{\prime}(t)$ for the function $f(t)=-9 t^{2}+3 t+6$ .

Find the derivative $y^{\prime}$ for the function $y=\frac{9}{2 x^{4}}$ .

Compute circulation and net flux for the flow and contour $C$ : 25. $f(z)=\frac{1}{z}$ , $C$ : $|z|=1$ ; 26. $f(z)=2z$ , $C$ : $|z|=1$ ; 27. $f(z)=\frac{1}{\overline{z-1}}$ , $C$ : $|z-1|=2$ .

Differentiate $\frac{5 x^{2}}{8}-\frac{5}{9 x^{2}}$ with respect to $x$ .

Find the derivative $G^{\prime}(w)$ for the function $G(w)=\frac{5}{4 w^{6}}+2 \sqrt[3]{w}$ .

Find the derivative $y^{\prime}$ of the function $y=\frac{2}{7 x^{2}}$ . What is $y^{\prime}=$ ?

Find the derivative $G^{\prime}(w)$ for $G(w)=\frac{8}{5 w^{4}}+2 \sqrt[3]{w}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(8x-7)^{2}$ .

Find the derivative $y^{\prime}$ of the function $y=\frac{7}{5 x^{2}}$ . What is $y^{\prime}=?$

Find the absolute extreme values of $f(x)=x^{2}-10$ on the interval $[-2,4]$ . Where is the maximum?

For $f(x)=x^{4}-4 x^{3}+9$ , find: (A) $f^{\prime}(x)$ , (B) slope at $x=-2$ , (C) tangent line equation at $x=-2$ , (D) where tangent is horizontal.

Find the derivative $y^{\prime}$ for the function $y=\frac{x^{7}-5 x^{6}+9}{x^{5}}$ .

Find the velocity function $v=f'(x)$ for $f(x)=156x-12x^2$ , then determine $v(0)$ , $v(2)$ , and when $v=0$ .

Differentiate the function $f(x)=6-2x+8e^{x}$ . Find $f^{\prime}(x)=$ .

Differentiate the function $f(x)=3-3x+6e^{x}$ . Find $f^{\prime}(x)$ .

Find the absolute extrema of $f(x)=2 x^{3}-39 x^{2}+240 x$ on the interval $[4,9]$ .

Find the absolute extrema of $f(x)=2 x^{3}-39 x^{2}+216 x$ on $[3,10]$ . What are the max and min values?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=2 e^{x}+5 x-3 \ln x$ .

Find the absolute extreme values of $f(x)=2 x^{3}-39 x^{2}+216 x$ on the interval $[3,10]$ .

Is the series $\sum_{k=1}^{\infty} 4 i\left(\frac{1}{3}\right)^{k-1}$ convergent or divergent? If convergent, find the sum.

Evaluate the integral $\pi \int_{0}^{2} \frac{0}{1}$ .

Find the derivative of $f(x)=\ln x^{14}$ with respect to $x$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-15 \ln x + 4 x^{2} - 11$ .

Find the tangent line equation for $f(x)=17 e^{x}+11 x$ at $x=0$ . What is $y=$ ?

Rewrite $f(x)=24x+\ln(24x)$ using logarithm properties and find $f^{\prime}(x)$ .

Find the tangent line equation for $f(x)=2-3 \ln x$ at $x=1$ . What is $y=$ ?

Determine the max and min values of $g(x)=\sqrt{9-x^{2}}$ on the interval $[-3, 2]$ .

Rewrite $f(x)=2 \ln \left(\frac{13}{x}\right)$ using logarithm properties, then find $f^{\prime}(x)$ .

Find the rate of change of blood pressure $P(x)=12.7(1+\ln x)$ at $x=30$ and $x=40$ pounds.

Find the rate of change of blood pressure $P(x)=12.7(1+\ln x)$ at $x=30$ pounds. Answer in $\mathrm{mm}/$ pound.

Determine if the series $\sum_{k=0}^{\infty} 2\left(\frac{4}{1+4 i}\right)^{k}$ converges or diverges. If it converges, find the sum.

Given the cost function $C(x)=16900+800 x+x^{2}$ , find: a) $C(1200)$ , b) average cost at 1200, c) marginal cost at 1200, d) production level for minimal average cost, e) minimal average cost.

Rewrite $f(x)=7x+\ln(7x)$ using logarithm properties and find $f'(x)$ .

Find $c$ such that $\frac{f(3)-f(-1)}{3-(-1)}=f^{\prime}(c)$ for $f(x)=5x^2-2x-3$ on $[-1,3]$ .

Find values of $c$ satisfying $\frac{f(3)-f(-1)}{3-(-1)}=f^{\prime}(c)$ for $f(x)=3x^{2}+5x-2$ on $[-1,3]$ .

Is the series $\sum_{k=0}^{\infty} 5\left(\frac{6}{1+6 i}\right)^{k}$ convergent or divergent? If convergent, find the sum.

Find $c$ such that $\frac{f(10)-f(6)}{10-6}=f^{\prime}(c)$ for $f(x)=\ln(x-5)$ on $[6,10]$ .

Given $g(3)=2$ and $\lim_{x \rightarrow 3}[6 f(x)+f(x) g(x)]=32$ , find $f(3)$ .

Find values of $c$ for $f(x)=3x^2-5x-2$ in $[-1,2]$ satisfying $\frac{f(2)-f(-1)}{2-(-1)}=f'(c)$ .

Given continuous functions $f$ and $g$ with $g(3)=2$ and $\lim_{x \rightarrow 3}[6 f(x)+f(x) g(x)]=32$ , find $f(3)$ .

Find the average velocity of the ball using $h(t)=-\frac{1}{2}t^2+5t$ over the first 3 seconds.

Find $d y$ for $y=2 x^{2}+2 x+2$ at $x=1$ , $d x=0.3$ and $d x=0.6$ .