Calculus

Problem 10801

Find the tangent line equation for $y=4 \sin x+\cos 2 x$ at $x=\frac{5 \pi}{2}$ . The equation is $y=$

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Problem 10802

How long to double a $\$ 1400$ deposit at $6 \%$ continuous compounding? Answer in years and days.

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Problem 10803

Find the following for the function $f(x)=x-4$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

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Problem 10804

Find the following for the function $f(x)=-x^{2}-7$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

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Problem 10805

Calculate the average rate of change for $f(x)=\frac{9}{2 x+4}$ on $[3,4]$ . Average rate of change $=$

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Problem 10806

Find the function $f$ with derivative $f'(x) = -2x \sqrt{8 - x^2}$ that passes through the point $(2, 3)$ .

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Problem 10807

Find the acceleration and position functions for a particle with velocity $v(t)=5/\sqrt{t}$ and $x(1)=12$ .

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Problem 10808

Given the cost function $C(x)=67600+500 x+x^{2}$ , find: a) Cost at $x=1900$ b) Average cost at $x=1900$ c) Marginal cost at $x=1900$ d) Production level minimizing average cost e) Minimal average cost

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Problem 10809

Find the average rate of change of $f(x)=7 x^{2}-4$ from $x=5$ to $x=a$ . Express your answer in terms of $a$ .

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Problem 10810

Find the average rate of change of $f(x)=x^{2}+11x$ on $[4,4+h]$ . Provide your answer as an expression in terms of $h$ .

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Problem 10811

Consider the curve $y + x^{2} = 6xy$ . Find $\frac{dy}{dx}$ and the equation of horizontal tangent lines.

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Problem 10812

Determine where the function $f(x)=\frac{20 x}{x^{2}+25}$ is increasing.

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Problem 10813

Find the second derivative of $f(t)=\ln(1-e^{t})$ .

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Problem 10814

Find the derivative of $f(x)=18 x^{2}-(36 x+9) \ln (x)$ .

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Problem 10815

Find the derivative of $f(x)=\frac{5 x^{2}}{x^{3}+1}$ and evaluate it at $x=2$ . Choices: $-13$ , 5.25, $-20 / 27$ , 15.

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Problem 10816

Find the cost, average cost, marginal cost, and optimal production level for $C(x)=40000+400x+x^{2}$ at $x=1500$ .

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Problem 10817

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

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Problem 10818

Determine where the function $f(x)=\ln(2x^{2}+3)$ is concave up.

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Problem 10819

Find the average rate of change of $f(x)=4 x^{2}-7$ from $x=2$ to $x=b$ . Express your answer in terms of $b$ .

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Problem 10820

Determine where the function $f(x)=\ln(2x^{2}+3)$ is concave down.

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Problem 10821

Determine the concavity of the function $f(x)=\ln(2x^{2}+3)$ .

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Problem 10822

Find the second derivative of $f(x)=18x^2 - (36x+9) \ln(x)$ .

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Problem 10823

Determine the inflection points of the function $f(x)=\ln(2x^{2}+3)$ .

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Problem 10824

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-1}-x\right)$

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Problem 10825

Find the amount of carbon-14 left after 7174 years using the model $A=16 e^{-0.000121 t}$ . Round to the nearest gram.

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Problem 10826

If $f$ is concave up on $(-\infty, 0)$ and $(0, \infty)$ with a local max at $x=0$ , what can you say about $\mathrm{f}^{\prime}(0)$ ?

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Problem 10827

Given the cost function $C(x)=128 \sqrt{x}+\frac{x^{2}}{216000}$ , find: a) Cost at $x=1550$ b) Average cost at $x=1550$ c) Marginal cost at $x=1550$ d) Production level for minimal average cost e) Minimal average cost.

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Problem 10828

Given the cost function $C(x)=250 \sqrt{x}+\frac{x^{2}}{166375}$ , find: 1) Cost at $x=1450$ 2) Average cost at $x=1450$ 3) Marginal cost at $x=1450$ 4) Production level minimizing average cost 5) Minimal average cost.

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Problem 10829

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

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Problem 10830

Determine the relative max and min of $f(x)=x^{4}-32 x^{2}-8$ .

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Problem 10831

Find the derivative of $f(x) = (4x+2)e^{-x}$ using the product and chain rules.

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Problem 10832

Deposit \ $2000 at 6% interest compounded continuously. Find the amount in 15 years using$ A=P e^{\tau t}$.

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Problem 10833

Find the limit: $\lim _{x \rightarrow+\infty}\left(e^{-x}+2 \cos 3 x\right)$ .

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Problem 10834

Find the derivative of $2 e^{-x} - 4 x e^{-x}$ .

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Problem 10835

Calculate the average rate of change of $g(x)=-2 x^{2}-5$ from the points $(-1,-7)$ to $(2,-13)$ .

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Problem 10836

Find the average rate of change of $f(x)$ from $x=3$ to $x=5$ , where $f(3)=0$ and $f(5)=0$ .

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Problem 10837

Find the second derivative of $f(x)=4 x^{2}-3 x+7$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

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Problem 10838

Approximate $\sqrt{17}$ using linear approximation of $f(x)=\sqrt{x}$ at $x=16$ . Provide the answer to 3 decimal places.

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Problem 10839

Find the critical points of $f(x)=(x+5)^{4}$ and analyze them using the first derivative test. Use the second derivative test if applicable.

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Problem 10840

A scale in an elevator stretches by $0.04 \mathrm{~m}$ with a $5 \mathrm{~kg}$ block. What is the max speed of the elevator?

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Problem 10841

Explore squirrel population growth in Calgary:
(a) Show $y=c e^{r t}$ satisfies $\frac{d y}{d t}=b y-d y$ for a specific $r$ . (b) Find $\lim_{t \rightarrow \infty} f(t)$ for $r<0$ , $r=0$ , $r>0$ . (c) Discuss if $r>0$ long-term behavior is realistic.

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Problem 10842

Find the first and second derivatives of $f(x)=\frac{1}{2} x^{2}-9 x+9 \ln (x)$ .

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Problem 10843

Consider the logistic model $\frac{d y}{d t}=r y\left(1-\frac{y}{n}\right)$ for squirrel population.
(a) Show $y=f(t)=\frac{n}{1+k e^{-r t}}$ is a solution. Find $k$ in terms of initial population $c=f(0)$ .
(b) If $r>0$ , what is $\lim _{t \rightarrow \infty} f(t)$ ? Does this match your intuition?

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Problem 10844

Find the derivatives: (a) $f^{\prime}(x)$ for $f(x)=\sin(\cos(x))$ , (b) $(f \circ f)^{\prime}(x)$ , (c) $(f \circ f \circ f)^{\prime}(x)$ .

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Problem 10845

Calculate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}+y^{3}}{x^{2}+y^{2}}$

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Problem 10846

Find $g'(2)$ for $g(x)=3 f(x)+x^{5} f(x)$ and $h'(2)$ for $h(x)=(9 x+8) f(x)$ given $f(2)=5$ and $f'(2)=-2$ .

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Problem 10847

Find the derivative of $f(x) = x - 9 + \frac{9}{x}$ .

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Problem 10848

Find the derivatives: (a) $f'(x)$ for $f(x)=\sin(\cos(x))$ , (b) $(f \circ f)'(x)$ , (c) $(f \circ f \circ f)'(x)$ .

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Problem 10849

How long to double an investment at a continuous compounding rate of $6.5\%$ ?

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Problem 10850

Find constants $c$ and $d$ such that the derivative $f'(x)$ is continuous for the piecewise function $f(x)=\{c x^{2}$ if $x \leq 3$ , $5-d x$ if $x>3\}$ .

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Problem 10851

Skizzieren Sie den Graphen von $f(x)=x-2+e^{-x}$ und untersuchen Sie lokale Extrema.

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Problem 10852

Untersuchen Sie die Geburten- und Sterbezahlen über 50 Jahre. Bestimmen Sie Maxima, Bevölkerungszahlen nach 5, 15, 30, 50 Jahren und Wachstumsraten.

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Problem 10853

Bestimme die Höhe des Felsplateaus, die Aufprallgeschwindigkeit des Steins und die Zeit bis zum Schallereignis bei $340 \mathrm{~m/s}$ .

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Problem 10854

A falcon descends from 3,000 ft at $250 \mathrm{ft/sec}$ . Find the rate of change and initial value.

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Problem 10855

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (x y)=x+y$ .

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Problem 10856

Find the maximum size $A$ of the logistic growth model $A=\frac{c}{1+a e^{-b t}}$ and the max values for parameters a, b, and c.

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Problem 10857

Find $x$ such that $\frac{d y}{d x}=17$ for the relation $\sqrt{x+y}=3 x$ when $y=8$ .

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Problem 10858

Find the derivative $\frac{d y}{d x}$ for the hyperbola defined by the equation $x^{2}-y^{2}=16$ .

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Problem 10859

Trouvez la dérivée de $f(x)=\frac{1}{6 x^{2}+5}$ .

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Problem 10860

Calcule la dérivée $f^{\prime}(x)$ si $f(x)=\frac{3}{x^{2}}-2^{\pi+1}+\sqrt{x}+3 x-x^{2 / 5}$ .

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Problem 10861

Encuentra $\frac{d k}{d t}$ para $k(t)=\frac{-3}{\sqrt[3]{3 t^{5}-2 t^{2}+3}}$ .

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Problem 10862

Find $\frac{d y}{d x}$ for the function $y=(x^{2}+1)^{5}(3 x-2)^{3}$ .

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Problem 10863

Calculez $h^{\prime}(x)$ pour $h(x)=\sqrt[3]{\frac{3 x^{2}-2}{6-7 x^{3}}}$ .

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Problem 10864

Find the derivative of $y=\sin ^{3} \frac{x}{3}+\cos ^{3} \frac{x}{3}$ .

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Problem 10865

Find the average lifespan $\mu$ of a neon tube given the function $f(t)=0.0002 e^{-0.0002 t}$ for $t \geq 0$ .

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Problem 10866

Find the average lifespan $\mu$ of a neon tube using $f(t)=0.0002 e^{-0.0002 t}$ for $t \geq 0$ .

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Problem 10867

Étudiez la population de bonobos avec $P(t)=\frac{6000+15 t^{2}}{2+0,01 t^{3}}$ .
a) Taille initiale $P(0)$ ? b) Taille à long terme $P(\infty)$ ? c) Taux de variation moyen entre $t=5$ et $t=10$ . Interprétation ?

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Problem 10868

For the function $f(x)=\frac{x+6}{x+2}$ , find (a) critical numbers, (b) intervals where $f$ is increasing, and (c) where $f$ is decreasing.

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Problem 10869

Find the absolute extrema of $f(x)=3 x^{2}-24 x$ on the interval $[0,7]$ . Provide as $(x, f(x))$ .

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Problem 10870

Find the absolute extrema of $f(x)=8 x^{2}+32 x+1$ on $[-3,2]$ . Provide your answer as an ordered pair $(x, f(x))$ .

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Problem 10871

Find the intervals where the profit function, given by $P(x) = R(x) - C(x)$ , is increasing. Use $C(x)=0.19 x^{2}-0.0001 x^{3}$ and $R(x)=0.700 x^{2}-0.0002 x^{3}$ .

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Problem 10872

Find the point on the graph of $f(x)=\sqrt{-x-4}$ where the tangent line passes through the origin.

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Problem 10873

Find the derivative using the product rule for $y=(3 x^{4}-x+1)(-x^{5}+6)$ . What is $y'$ ?

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Problem 10874

Find the derivative using the product rule for $y=(7 x^{4}-x+6)(-x^{5}+3)$ . What is $y^{\prime}=$ ?

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Problem 10875

Évaluez $f^{\prime}\left(\frac{\pi}{2}\right)$ pour $f(x)=\frac{x^{4}+3 \cos (x)}{2 \sin (x)}$ . Donnez la valeur exacte.

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Problem 10876

Trouvez l'équation de la tangente à $y=-6 x \sin (x)$ au point $(\pi / 2,-3 \pi)$ sous la forme $y=m x+b$ .

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Problem 10877

Find the point on the graph of $f(x)=\ln (-x)$ where the tangent line passes through the origin.

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Problem 10878

Trouver la dérivée de $f(x)=\frac{1-x e^{x}}{2+e^{x}}$ . Quelle est $f^{\prime}(x)$ ?

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Problem 10879

Find the point on the graph of $f(x)=\ln (-x)$ where the tangent line passes through the origin.

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Problem 10880

Find the derivative using the quotient rule for the function $y=\frac{2 x-3}{x^{2}-1}$ . What is $\frac{d y}{d x}=?$

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Problem 10881

Soit $f(x)=\frac{5}{11+2 x}$ . Calculez $f^{\prime}(x)$ avec $f^{\prime}(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .
a) Trouvez $\frac{f(x+h)-f(x)}{h}$ .
b) Simplifiez pour $A$ et $B$ dans $\frac{f(x+h)-f(x)}{h}=\frac{A}{B}$ . Donnez $[A, B]$ .

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Problem 10882

Find the derivative of the function using the quotient rule: $f(t)=\frac{2 t^{2}-3 t}{7 t+4}$ . What is $f^{\prime}(t)=?$

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Problem 10883

Find $x$ that minimizes the average cost function given $C(x)=417600+1.9x+29x^{2}$ for $1 \leq x \leq 122$ .

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Problem 10884

Calculate the integral: $\int \sec ^{3} 5 x \tan ^{3} 5 x \, dx$

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Problem 10885

Calculate the integral: $\int 2 \arcsin x \, dx$

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Problem 10886

Find the value of $x$ that minimizes the average cost function for $C(x)=489888+1.7 x+42 x^{2}$ , $1 \leq x \leq 285$ .

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Problem 10887

Calculate the indefinite integral: $\int 2\left(\sin ^{3} x\right) \sqrt[4]{\cos x} \, dx$

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Problem 10888

Find the indefinite integral: $\int 2\left(\sin ^{3} x\right) \sqrt[4]{\cos x} \, dx$

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Problem 10889

Calculate the indefinite integral: $\int \frac{\sqrt{4 x^{2}-25}}{x} \, dx$

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Problem 10890

Find the average rate of change of $f(x)=6-2x$ from $x=3$ to $x=4$ .

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Problem 10891

Calculate the integral: $\int \frac{4 x^{2}+2 x+6}{(x-1)(x^{2}+2)} dx$

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Problem 10892

Calculate the integral: $\int 2\left(\sin ^{3} x\right) \sqrt[4]{\cos x} \, dx$

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Problem 10893

What height would a pitch at $100.9 \mathrm{mi}/\mathrm{hr}$ or $45.0 \mathrm{m/s}$ reach if thrown straight up?

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Problem 10894

Identify the function $f$ in the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} x_{k}^{} \tan ^{2} x_{k}^{} \Delta x_{k} ;[2,3]$ and express it as a definite integral.

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Problem 10895

Rewrite the integral using $u$ : $\int_{2}^{3} \frac{7}{9 x^{2}+24 x+16} d x=\int_{10}^{\square}(\square) d u$

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Problem 10896

Find the particle's velocity at $t=4$ seconds given $s(t)=9 t^{2}+36 t$ .

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Problem 10897

Evaluate the integral $\int_{2}^{3} \frac{7}{9 x^{2}+24 x+16} d x$ using a variable change. Choose $u$ from options A-D.

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Problem 10898

Evaluate the integral $\int_{-\pi}^{0} \frac{\sin x}{6+\cos x} d x$ using a change of variables. Choose $u$ from options A-D and rewrite the integral.

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Problem 10899

Determine the indeterminate form and evaluate: $\lim _{x \rightarrow 0} \frac{3 x^{2}}{1-\cos (2 x)}$ using L'Hopital's Rule.

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Problem 10900

Find $f^{\prime \prime}(1)$ , $f^{\prime \prime}(5)$ , and $f^{\prime \prime}(7)$ for the function $f(x) = -8 x^{3}+6 x^{2}-9 x$ .

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Calculus

Problem 10801

Find the tangent line equation for y=4sin⁡x+cos⁡2xy=4 \sin x+\cos 2 xy=4sinx+cos2x at x=5π2x=\frac{5 \pi}{2}x=25π​. The equation is y=y=y=

Problem 10802

How long to double a $1400\$ 1400$1400 deposit at 6%6 \%6% continuous compounding? Answer in years and days.

Problem 10803

Find the following for the function f(x)=x−4f(x)=x-4f(x)=x−4: a. f(x)−f(a)x−a\frac{f(x)-f(a)}{x-a}x−af(x)−f(a)​; b. f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 10804

Find the following for the function f(x)=−x2−7f(x)=-x^{2}-7f(x)=−x2−7: a. f(x)−f(a)x−a\frac{f(x)-f(a)}{x-a}x−af(x)−f(a)​; b. f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​.

Problem 10805

Calculate the average rate of change for f(x)=92x+4f(x)=\frac{9}{2 x+4}f(x)=2x+49​ on [3,4][3,4][3,4]. Average rate of change ===

Problem 10806

Find the function fff with derivative f′(x)=−2x8−x2f'(x) = -2x \sqrt{8 - x^2}f′(x)=−2x8−x2​ that passes through the point (2,3)(2, 3)(2,3).

Problem 10807

Find the acceleration and position functions for a particle with velocity v(t)=5/tv(t)=5/\sqrt{t}v(t)=5/t​ and x(1)=12x(1)=12x(1)=12.

Problem 10808

Given the cost function C(x)=67600+500x+x2C(x)=67600+500 x+x^{2}C(x)=67600+500x+x2, find: a) Cost at x=1900x=1900x=1900 b) Average cost at x=1900x=1900x=1900 c) Marginal cost at x=1900x=1900x=1900 d) Production level minimizing average cost e) Minimal average cost

Problem 10809

Find the average rate of change of f(x)=7x2−4f(x)=7 x^{2}-4f(x)=7x2−4 from x=5x=5x=5 to x=ax=ax=a. Express your answer in terms of aaa.

Problem 10810

Find the average rate of change of f(x)=x2+11xf(x)=x^{2}+11xf(x)=x2+11x on [4,4+h][4,4+h][4,4+h]. Provide your answer as an expression in terms of hhh.

Problem 10811

Consider the curve y+x2=6xyy + x^{2} = 6xyy+x2=6xy. Find dydx\frac{dy}{dx}dxdy​ and the equation of horizontal tangent lines.

Problem 10812

Determine where the function f(x)=20xx2+25f(x)=\frac{20 x}{x^{2}+25}f(x)=x2+2520x​ is increasing.

Problem 10813

Find the second derivative of f(t)=ln⁡(1−et)f(t)=\ln(1-e^{t})f(t)=ln(1−et).

Problem 10814

Find the derivative of f(x)=18x2−(36x+9)ln⁡(x)f(x)=18 x^{2}-(36 x+9) \ln (x)f(x)=18x2−(36x+9)ln(x).

Problem 10815

Find the derivative of f(x)=5x2x3+1f(x)=\frac{5 x^{2}}{x^{3}+1}f(x)=x3+15x2​ and evaluate it at x=2x=2x=2. Choices: −13-13−13, 5.25, −20/27-20 / 27−20/27, 15.

Problem 10816

Find the cost, average cost, marginal cost, and optimal production level for C(x)=40000+400x+x2C(x)=40000+400x+x^{2}C(x)=40000+400x+x2 at x=1500x=1500x=1500.

Problem 10817

Find the critical points of the function f(x)=x2/3(52−x)f(x)=x^{2/3}\left(\frac{5}{2}-x\right)f(x)=x2/3(25​−x).

Problem 10818

Determine where the function f(x)=ln⁡(2x2+3)f(x)=\ln(2x^{2}+3)f(x)=ln(2x2+3) is concave up.

Problem 10819

Find the average rate of change of f(x)=4x2−7f(x)=4 x^{2}-7f(x)=4x2−7 from x=2x=2x=2 to x=bx=bx=b. Express your answer in terms of bbb.

Problem 10820

Determine where the function f(x)=ln⁡(2x2+3)f(x)=\ln(2x^{2}+3)f(x)=ln(2x2+3) is concave down.

Problem 10821

Determine the concavity of the function f(x)=ln⁡(2x2+3)f(x)=\ln(2x^{2}+3)f(x)=ln(2x2+3).

Problem 10822

Find the second derivative of f(x)=18x2−(36x+9)ln⁡(x)f(x)=18x^2 - (36x+9) \ln(x)f(x)=18x2−(36x+9)ln(x).

Problem 10823

Determine the inflection points of the function f(x)=ln⁡(2x2+3)f(x)=\ln(2x^{2}+3)f(x)=ln(2x2+3).

Problem 10824

Calculate the limit: lim⁡x→∞(x2−1−x)\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-1}-x\right)limx→∞​(x2−1​−x)

Problem 10825

Find the amount of carbon-14 left after 7174 years using the model A=16e−0.000121tA=16 e^{-0.000121 t}A=16e−0.000121t. Round to the nearest gram.

Problem 10826

If fff is concave up on (−∞,0)(-\infty, 0)(−∞,0) and (0,∞)(0, \infty)(0,∞) with a local max at x=0x=0x=0, what can you say about f′(0)\mathrm{f}^{\prime}(0)f′(0)?

Problem 10827

Given the cost function C(x)=128x+x2216000C(x)=128 \sqrt{x}+\frac{x^{2}}{216000}C(x)=128x​+216000x2​, find: a) Cost at x=1550x=1550x=1550 b) Average cost at x=1550x=1550x=1550 c) Marginal cost at x=1550x=1550x=1550 d) Production level for minimal average cost e) Minimal average cost.

Problem 10828

Given the cost function C(x)=250x+x2166375C(x)=250 \sqrt{x}+\frac{x^{2}}{166375}C(x)=250x​+166375x2​, find: 1) Cost at x=1450x=1450x=1450 2) Average cost at x=1450x=1450x=1450 3) Marginal cost at x=1450x=1450x=1450 4) Production level minimizing average cost 5) Minimal average cost.

Problem 10829

Find the limit: lim⁡x→+∞1−ex1+2ex\lim _{x \rightarrow+\infty} \frac{1-e^{x}}{1+2 e^{x}}limx→+∞​1+2ex1−ex​.

Problem 10830

Determine the relative max and min of f(x)=x4−32x2−8f(x)=x^{4}-32 x^{2}-8f(x)=x4−32x2−8.

Problem 10831

Find the derivative of f(x)=(4x+2)e−xf(x) = (4x+2)e^{-x}f(x)=(4x+2)e−x using the product and chain rules.

Problem 10832

Deposit \2000at62000 at 6% interest compounded continuously. Find the amount in 15 years using 2000at6A=P e^{\tau t}$.

Problem 10833

Find the limit: lim⁡x→+∞(e−x+2cos⁡3x)\lim _{x \rightarrow+\infty}\left(e^{-x}+2 \cos 3 x\right)limx→+∞​(e−x+2cos3x).

Problem 10834

Find the derivative of 2e−x−4xe−x2 e^{-x} - 4 x e^{-x}2e−x−4xe−x.

Problem 10835

Calculate the average rate of change of g(x)=−2x2−5g(x)=-2 x^{2}-5g(x)=−2x2−5 from the points (−1,−7)(-1,-7)(−1,−7) to (2,−13)(2,-13)(2,−13).

Problem 10836

Find the average rate of change of f(x)f(x)f(x) from x=3x=3x=3 to x=5x=5x=5, where f(3)=0f(3)=0f(3)=0 and f(5)=0f(5)=0f(5)=0.

Problem 10837

Find the second derivative of f(x)=4x2−3x+7f(x)=4 x^{2}-3 x+7f(x)=4x2−3x+7, then calculate f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(7)f^{\prime \prime}(7)f′′(7).

Problem 10838

Approximate 17\sqrt{17}17​ using linear approximation of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=16x=16x=16. Provide the answer to 3 decimal places.

Problem 10839

Find the critical points of f(x)=(x+5)4f(x)=(x+5)^{4}f(x)=(x+5)4 and analyze them using the first derivative test. Use the second derivative test if applicable.

Problem 10840

Find the tangent line equation for $y=4 \sin x+\cos 2 x$ at $x=\frac{5 \pi}{2}$ . The equation is $y=$

How long to double a $\$ 1400$ deposit at $6 \%$ continuous compounding? Answer in years and days.

Find the following for the function $f(x)=x-4$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

Find the following for the function $f(x)=-x^{2}-7$ : a. $\frac{f(x)-f(a)}{x-a}$ ; b. $\frac{f(x+h)-f(x)}{h}$ .

Calculate the average rate of change for $f(x)=\frac{9}{2 x+4}$ on $[3,4]$ . Average rate of change $=$

Find the function $f$ with derivative $f'(x) = -2x \sqrt{8 - x^2}$ that passes through the point $(2, 3)$ .

Find the acceleration and position functions for a particle with velocity $v(t)=5/\sqrt{t}$ and $x(1)=12$ .

Given the cost function $C(x)=67600+500 x+x^{2}$ , find: a) Cost at $x=1900$ b) Average cost at $x=1900$ c) Marginal cost at $x=1900$ d) Production level minimizing average cost e) Minimal average cost

Find the average rate of change of $f(x)=7 x^{2}-4$ from $x=5$ to $x=a$ . Express your answer in terms of $a$ .

Find the average rate of change of $f(x)=x^{2}+11x$ on $[4,4+h]$ . Provide your answer as an expression in terms of $h$ .

Consider the curve $y + x^{2} = 6xy$ . Find $\frac{dy}{dx}$ and the equation of horizontal tangent lines.

Determine where the function $f(x)=\frac{20 x}{x^{2}+25}$ is increasing.

Find the second derivative of $f(t)=\ln(1-e^{t})$ .

Find the derivative of $f(x)=18 x^{2}-(36 x+9) \ln (x)$ .

Find the derivative of $f(x)=\frac{5 x^{2}}{x^{3}+1}$ and evaluate it at $x=2$ . Choices: $-13$ , 5.25, $-20 / 27$ , 15.

Find the cost, average cost, marginal cost, and optimal production level for $C(x)=40000+400x+x^{2}$ at $x=1500$ .

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

Determine where the function $f(x)=\ln(2x^{2}+3)$ is concave up.

Find the average rate of change of $f(x)=4 x^{2}-7$ from $x=2$ to $x=b$ . Express your answer in terms of $b$ .

Determine where the function $f(x)=\ln(2x^{2}+3)$ is concave down.

Determine the concavity of the function $f(x)=\ln(2x^{2}+3)$ .

Find the second derivative of $f(x)=18x^2 - (36x+9) \ln(x)$ .

Determine the inflection points of the function $f(x)=\ln(2x^{2}+3)$ .

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}-1}-x\right)$

Find the amount of carbon-14 left after 7174 years using the model $A=16 e^{-0.000121 t}$ . Round to the nearest gram.

If $f$ is concave up on $(-\infty, 0)$ and $(0, \infty)$ with a local max at $x=0$ , what can you say about $\mathrm{f}^{\prime}(0)$ ?

Given the cost function $C(x)=128 \sqrt{x}+\frac{x^{2}}{216000}$ , find: a) Cost at $x=1550$ b) Average cost at $x=1550$ c) Marginal cost at $x=1550$ d) Production level for minimal average cost e) Minimal average cost.

Given the cost function $C(x)=250 \sqrt{x}+\frac{x^{2}}{166375}$ , find: 1) Cost at $x=1450$ 2) Average cost at $x=1450$ 3) Marginal cost at $x=1450$ 4) Production level minimizing average cost 5) Minimal average cost.

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

Determine the relative max and min of $f(x)=x^{4}-32 x^{2}-8$ .

Find the derivative of $f(x) = (4x+2)e^{-x}$ using the product and chain rules.

Deposit \ $2000 at 6% interest compounded continuously. Find the amount in 15 years using$ A=P e^{\tau t}$.

Find the limit: $\lim _{x \rightarrow+\infty}\left(e^{-x}+2 \cos 3 x\right)$ .

Find the derivative of $2 e^{-x} - 4 x e^{-x}$ .

Calculate the average rate of change of $g(x)=-2 x^{2}-5$ from the points $(-1,-7)$ to $(2,-13)$ .

Find the average rate of change of $f(x)$ from $x=3$ to $x=5$ , where $f(3)=0$ and $f(5)=0$ .

Find the second derivative of $f(x)=4 x^{2}-3 x+7$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

Approximate $\sqrt{17}$ using linear approximation of $f(x)=\sqrt{x}$ at $x=16$ . Provide the answer to 3 decimal places.

Find the critical points of $f(x)=(x+5)^{4}$ and analyze them using the first derivative test. Use the second derivative test if applicable.

A scale in an elevator stretches by $0.04 \mathrm{~m}$ with a $5 \mathrm{~kg}$ block. What is the max speed of the elevator?

Find the first and second derivatives of $f(x)=\frac{1}{2} x^{2}-9 x+9 \ln (x)$ .

Find the derivatives: (a) $f^{\prime}(x)$ for $f(x)=\sin(\cos(x))$ , (b) $(f \circ f)^{\prime}(x)$ , (c) $(f \circ f \circ f)^{\prime}(x)$ .

Calculate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}+y^{3}}{x^{2}+y^{2}}$

Find $g'(2)$ for $g(x)=3 f(x)+x^{5} f(x)$ and $h'(2)$ for $h(x)=(9 x+8) f(x)$ given $f(2)=5$ and $f'(2)=-2$ .

Find the derivative of $f(x) = x - 9 + \frac{9}{x}$ .

Find the derivatives: (a) $f'(x)$ for $f(x)=\sin(\cos(x))$ , (b) $(f \circ f)'(x)$ , (c) $(f \circ f \circ f)'(x)$ .

How long to double an investment at a continuous compounding rate of $6.5\%$ ?

Find constants $c$ and $d$ such that the derivative $f'(x)$ is continuous for the piecewise function $f(x)=\{c x^{2}$ if $x \leq 3$ , $5-d x$ if $x>3\}$ .

Skizzieren Sie den Graphen von $f(x)=x-2+e^{-x}$ und untersuchen Sie lokale Extrema.

Bestimme die Höhe des Felsplateaus, die Aufprallgeschwindigkeit des Steins und die Zeit bis zum Schallereignis bei $340 \mathrm{~m/s}$ .

A falcon descends from 3,000 ft at $250 \mathrm{ft/sec}$ . Find the rate of change and initial value.

Find the derivative $\frac{d y}{d x}$ for the equation $\tan (x y)=x+y$ .

Find the maximum size $A$ of the logistic growth model $A=\frac{c}{1+a e^{-b t}}$ and the max values for parameters a, b, and c.

Find $x$ such that $\frac{d y}{d x}=17$ for the relation $\sqrt{x+y}=3 x$ when $y=8$ .

Find the derivative $\frac{d y}{d x}$ for the hyperbola defined by the equation $x^{2}-y^{2}=16$ .

Trouvez la dérivée de $f(x)=\frac{1}{6 x^{2}+5}$ .

Calcule la dérivée $f^{\prime}(x)$ si $f(x)=\frac{3}{x^{2}}-2^{\pi+1}+\sqrt{x}+3 x-x^{2 / 5}$ .

Encuentra $\frac{d k}{d t}$ para $k(t)=\frac{-3}{\sqrt[3]{3 t^{5}-2 t^{2}+3}}$ .

Find $\frac{d y}{d x}$ for the function $y=(x^{2}+1)^{5}(3 x-2)^{3}$ .

Calculez $h^{\prime}(x)$ pour $h(x)=\sqrt[3]{\frac{3 x^{2}-2}{6-7 x^{3}}}$ .

Find the derivative of $y=\sin ^{3} \frac{x}{3}+\cos ^{3} \frac{x}{3}$ .

Find the average lifespan $\mu$ of a neon tube given the function $f(t)=0.0002 e^{-0.0002 t}$ for $t \geq 0$ .

Find the average lifespan $\mu$ of a neon tube using $f(t)=0.0002 e^{-0.0002 t}$ for $t \geq 0$ .

Étudiez la population de bonobos avec $P(t)=\frac{6000+15 t^{2}}{2+0,01 t^{3}}$ .
a) Taille initiale $P(0)$ ? b) Taille à long terme $P(\infty)$ ? c) Taux de variation moyen entre $t=5$ et $t=10$ . Interprétation ?

For the function $f(x)=\frac{x+6}{x+2}$ , find (a) critical numbers, (b) intervals where $f$ is increasing, and (c) where $f$ is decreasing.

Find the absolute extrema of $f(x)=3 x^{2}-24 x$ on the interval $[0,7]$ . Provide as $(x, f(x))$ .

Find the absolute extrema of $f(x)=8 x^{2}+32 x+1$ on $[-3,2]$ . Provide your answer as an ordered pair $(x, f(x))$ .

Find the intervals where the profit function, given by $P(x) = R(x) - C(x)$ , is increasing. Use $C(x)=0.19 x^{2}-0.0001 x^{3}$ and $R(x)=0.700 x^{2}-0.0002 x^{3}$ .

Find the point on the graph of $f(x)=\sqrt{-x-4}$ where the tangent line passes through the origin.

Find the derivative using the product rule for $y=(3 x^{4}-x+1)(-x^{5}+6)$ . What is $y'$ ?

Find the derivative using the product rule for $y=(7 x^{4}-x+6)(-x^{5}+3)$ . What is $y^{\prime}=$ ?