Calculus

Problem 22701

Calculez la dérivée de $f(x)=(4 x+1)^{3}-\frac{2}{\sqrt[4]{x}}+\frac{x^{2}+1}{x^{2}-1}$ .

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Problem 22702

Find the horizontal asymptote of $C(t)=\frac{1}{2t^{2}+1}$ and describe the drug concentration as $t$ increases.

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Problem 22703

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

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Problem 22704

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find: (a) decay rate, (b) amount after 30 yrs, (c) time for 100g, (d) half-life.

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Problem 22705

Find the limit: $\lim _{x \rightarrow \infty}(15 / x)^{14 x}=\square$

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Problem 22706

Evaluate the integral from 1 to e of $\frac{\ln x}{x} \, dx$ .

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Problem 22707

Find the interval where the functions $f(x)=x^{2}+2x+2$ and $g(x)=-x^{2}+2x+10$ have the same average rate of change.

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Problem 22708

Find $x$ values that satisfy the Mean Value Theorem for $f(x)=x^{3}-x$ on the interval $[1,2]$ .

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Problem 22709

Estimate $\sqrt{9.8}$ using linear approximation at $a=9$ .

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Problem 22710

Find the average rate of change of the function $f(x)=2 x^{2}+5 x-6$ on the interval $-4 \leq x \leq 0$ .

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Problem 22711

Find the max and min of $f(t)=\frac{t^{2}}{t-8}$ on $-4 \leq t \leq -\frac{1}{4}$ .

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Problem 22712

Find the future value of an investment of \$10,000 for 13 years with continuous compounding at: (a) 1\%, (b) 3\%, (c) 5.5\%.

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Problem 22713

Find the velocity of a particle with position $s(t)=-4 \cos t-\frac{t^{2}}{2}+10$ when its acceleration is zero.

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Problem 22714

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow+\infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Enter $-I$ for $-\infty$ , $I$ for $\infty$ , or DNE if it doesn't exist.

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Problem 22715

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 9 x}{x}$ using l'Hôpital's Rule and limit laws. Rewrite it as $\lim _{x \rightarrow 0}(\square)$ .

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Problem 22716

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Does it exist?

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Problem 22717

Find the limit as $n$ approaches infinity of the sum $\sum_{k=1}^{n}\left(\frac{6}{n}\right)\left(4+\frac{6 k}{n}\right)^{2}$ .

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Problem 22718

Evaluate $\lim _{x \rightarrow \infty} \frac{4 x^{2}+3 x}{4 x^{3}+x+3}$ using l'Hôpital's Rule and limit laws. Rewrite the limit as $\lim _{x \rightarrow \infty}(\square)$ .

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Problem 22719

Which inequality shows that the change in people inside a building is increasing at time $t$ given rates $f(t)$ and $g(t)$ ?

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Problem 22720

Evaluate $\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}$ using l'Hôpital's Rule and limit laws.

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Problem 22721

Find the limit using l'Hôpital's rule: $$\lim _{x \rightarrow \infty} \frac{3 x^{3}-2 x}{7 x^{3}+3}=\square($ Type an integer or a fraction.)

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Problem 22722

A softball weighing $88.9 \mathrm{~g}$ is dropped from $2.00 \mathrm{~m}$ . Find its velocity upon hitting the floor.

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Problem 22723

Find the limit using l'Hôpital's rule: $\lim _{x \rightarrow \infty} \frac{\ln (x+2)}{\log _{5} x} = \square$

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Problem 22724

Find the limit using l'Hôpital's Rule: $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{2-2 \sin \theta}{7+7 \cos 2 \theta}$ . What is the answer?

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Problem 22725

Evaluate $\lim _{t \rightarrow 0} \frac{-3 \sin \left(9 t^{6}\right)}{5 t}$ . What is the exact answer?

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Problem 22726

Find the limit using l'Hôpital's Rule: $$\lim _{x \rightarrow 0} \frac{6 x^{2}}{\cos (x)-1}=\square($ Type an exact answer.)

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Problem 22727

Finden Sie Hoch- und Tiefpunkte der Funktionen und skizzieren Sie deren Graphen: a) $f(x)=x^{4}-4 x^{2}+3$ b) $f(x)=-x^{3}+3 x^{2}-4$ c) $f(x)=0,5 x^{3}+x^{2}-3,5 x$ d) $f(x)=\frac{1}{9} x^{3}-3 x+1$ e) $f(x)=x^{4}-4 x^{2}$ f) $f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}+\frac{1}{4}$

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Problem 22728

A 225 kg boulder falls from 25 m. Calculate its impact velocity using energy principles. (4 marks)

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Problem 22729

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+4}}{\sqrt{x+2}}$ and simplify your answer.

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Problem 22730

Find $\mathrm{c}$ for continuity of $f(x)$ at $x=0$ :
$f(x)=\begin{cases} \frac{15 x-5 \sin (3 x)}{4 x^{3}}, & x \neq 0 \\ c, & x=0 \end{cases}$
What is $\mathrm{c}$ ?

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Problem 22731

Find the time $m$ for the cooling liquid to reach 90°F using $f(m)=112 e^{-0.5 m}+64$ . Choose from 34, 5960.50, 5966.40, 5968.25 minutes.

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Problem 22732

Prove that $\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}=e^{a}$ for $a \neq 0$ . Identify the indeterminate form. A. $\infty^{0}$ B. $1^{\infty}$ C. $0^{0}$ D. $\infty^{\infty}$ E. $0 \cdot \infty$

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Problem 22733

Start with 16 blackberry plants growing at $75\%$ monthly. Estimate plants after 4 months with a limit of 100.

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Problem 22734

Find the tangent line equation to $y=f(x)$ at $x=0$ where $f(x)=-2 x e^{-x}$ .

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Problem 22735

Find the tangent line equation to $y=f(x)$ at $x=0$ for $f(x)=-2 x e^{-x}$ .

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Problem 22736

Find relative extrema of $f(x)=2 x^{3}-3 x^{2}-4$ using the Second Derivative Test and justify your answers.

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Problem 22737

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x} \sin x \tan x}{x^{2}}$ .

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Problem 22738

How much to deposit today to reach \$1500 in 10 years at a continuous compounding rate of 6.5%?

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Problem 22739

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{3 x^{3}+4 x}{5-2 x^{4}}$

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Problem 22740

At 2.5 hours after noon, find your distance from home given $v(t)=-5 t^{4}+41 t^{3}-142 t^{2}+190 t$ and start 150 miles away.

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Problem 22741

Find the height of a projectile after 3 seconds using $v(t)=-15.5 t+147$ and 6 rectangles. Round to nearest tenth.

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Problem 22742

Find the absolute maximum of $f(x)=-\sin(2x)$ on $[0, 2\pi]$ . The derivative is $f'(x)=-2\cos(2x)$ .

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Problem 22743

Find the derivative of the function: $\frac{d}{d x}(\sec x \sin (3 x))$ .

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Problem 22744

A 500 g isotope decays as $A(t)=500 e^{-0.026 t}$ . Find: (a) remaining mass after 10 years, (b) time to halve the sample.

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Problem 22745

Find the derivatives: 27. $\frac{d}{d x}\left(\sqrt{\frac{2 x+5}{7 x-9}}\right)$ , 28. $\frac{d}{d x}\left(\frac{x}{\sqrt{1-(\ln x)^{2}}}\right)$ .

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Problem 22746

Find the limit as $u$ approaches $a$ : $\lim _{u \rightarrow a}\left(\frac{2 \sqrt{u}+\sqrt{a}}{4 u-a}\right)$ .

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Problem 22747

Find the intervals where the graph of $f(x)=x^{2} e^{x}$ is decreasing. Choose from the options given.

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Problem 22748

Find the derivative of $\frac{x}{\sqrt{1-(\ln x)^{2}}}$ with respect to $x$ .

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Problem 22749

Find the limit as $x$ approaches 0 for the expression $\frac{\sin (2 x)}{x}$ .

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Problem 22750

What are the units for $f^{\prime}(A)$ if $t=f(A)$ is time in minutes and $A$ is catalyst in milliliters? (A) minutes per milliliter (B) milliliters per minute (C) minutes per milliliter per milliliter (D) milliliters per minute per minute

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Problem 22751

Find $\lim _{x \rightarrow 5} M(x)$ for the piecewise function $M(x) = \{-4 \cos (\pi x), x<3; 0, x=3; \frac{x^{2}-25}{x^{2}-6 x+5}, x \geq 3\}$ .

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Problem 22752

Determine if the integral $\int_{c^{a}}^{\infty} \frac{\ln w-1}{w^{1 / a}-2} d w$ is convergent or divergent using the Comparison Theorem.

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Problem 22753

What are the units for $f^{\prime}(A)$ if $t=f(A)$ models time in minutes for catalyst amount $A$ in milliliters? (A) minutes per milliliter (B) milliliters per minute (C) minutes per milliliter per milliliter (D) milliliters per minute per minute

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Problem 22754

Given the equations $x=4t+3$ and $y=4t+8+\frac{5}{2t}$ for $t \neq 0$ , find $\frac{dy}{dx}$ at $t=2$ and show $y=\frac{x^2+ax+b}{x-3}$ .

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Problem 22755

Find $\lim _{x \rightarrow 5} M(x)$ and the horizontal asymptote of $M(x)$ for the piecewise function defined.

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Problem 22756

Find and classify critical points of $z=(x^{2}-5x)(y^{2}-8y)$ . List local max, min, and saddle points as $(x,y)$ or DNE.

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Problem 22757

Find the length of the curve given by $y=\int_{\pi / 3 a}^{x} \sqrt{\cos a \theta} d \theta$ for $\frac{\pi}{3 a} \leq x \leq \frac{\pi}{2 a}$ .

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Problem 22758

Find $S_{7}$ in the series $64,16,4,\ldots$ and explain when $S_{\infty}$ can be determined for an infinite series.

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Problem 22759

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve given by $x=4 t+3$ , $y=4 t+8+\frac{5}{2 t}$ at $t=2$ .

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Problem 22760

Analyze the motion of an object with position $s(t)=t^{2}-10t$ for $0 \leq t \leq 12$ ; find initial/final velocity and acceleration.

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Problem 22761

Water is removed at a rate $R(t)$ (liters/hour) shown in the table. Estimate water removed in 8 hours using a left Riemann sum. Is it an overestimate or underestimate?

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Problem 22762

Find the marginal revenue equations for $R(x, y)=150 x+60 y-3 x^{2}-2 y^{2}-x y$ . Set $R_{x}=0$ and $R_{y}=0$ to maximize revenue. Find $x$ and $y$ .

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Problem 22763

Water is removed from a tank at a rate $R(t)$ (liters/hour). Given values of $R$ over 8 hours, estimate water removed using a left Riemann sum. Is it an overestimate or underestimate? Explain why $R(t) = 900$ liters/hour occurs for some $t$ in $(0,8)$ .

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Problem 22764

Evaluate the integral $\int_{e^{a}}^{\infty} \frac{\ln w-1}{w^{1 / a}-2} d w$ for $a > 2$ .

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Problem 22765

Use l'Hôpital's Rule to evaluate $\lim _{x \rightarrow \infty} \frac{\ln (1+a / x)}{1 / x}$ .

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Problem 22766

Evaluate the series $\sum_{k=1}^{\infty} \frac{\sqrt{k} \ln k}{k^{3}+1}$ .

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Problem 22767

A cube's edges expand at $3 \mathrm{~cm/s}$ . Find the volume change rate when each edge is $10 \mathrm{~cm}$ .

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Problem 22768

How much more will $\$ 10,000$ at $7 \%$ continuous compounding for 9 years earn than quarterly compounding? Round to cents.

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Problem 22769

Calculate the average rate of change of $g(x)=4x+8$ between $x=2$ and $x=4$ . Simplify your answer.

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Problem 22770

Find $x$ in $0 \leq x \leq 4$ for which the sequence $a_{n}=n(2 \sin \frac{\pi x}{2})^{n}$ converges.

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Problem 22771

Find values of $p$ for which the series $\sum_{n=1}^{\infty}\left(\frac{p}{n+1}-\frac{1}{n+3}\right)$ converges.

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Problem 22772

Find the curve length for $y=\int_{\pi / 3 a}^{x} \sqrt{\cos a \theta} \, d \theta$ with $\frac{\pi}{3 a} \leq x \leq \frac{\pi}{2 a}$ .

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Problem 22773

A 120g radioactive substance decays at 0.056/min. How much remains after 40 min? Round to the nearest tenth.

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Problem 22774

Compare the earnings of a \ $25,000 investment at$ 5\%$ for 3 years, compounded continuously vs. monthly. Calculate the difference.

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Problem 22775

Find the value of the series: $\sum_{k=1}^{\infty} \frac{5^{k}+k}{k !+3}$ .

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Problem 22776

Find the antiderivative of $\int \frac{\sec 3 \theta}{\cos 3 \theta - \sec 3 \theta} d \theta$ .

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Problem 22777

Given the parametric equations $x=1-\frac{1}{a} \cos a t$ and $y=t-\frac{1}{a} \sin a t$ (where $a>0$ ), find points where the tangent line is vertical and horizontal.

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Problem 22778

Determine if the integral $\int_{e^{a}}^{\infty} \frac{\ln w-1}{w^{1 / a}-2} d w$ is convergent or divergent using the Comparison Theorem, where $a>2$ .

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Problem 22779

Determine if the integral $\int \frac{p(x)}{q(x)} d x$ results in the given logarithmic form for polynomials $p(x)$ and monic $q(x)$ .

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Problem 22780

Differentiate to verify: $\int \sec ^{2}(2 x+3) d x=\frac{1}{2} \tan (2 x+3)+C$ . Choose A or B.

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Problem 22781

Model the drug concentration in blood after one pill.
(a) Why does $\frac{d c}{d t}=-k_{1} c$ with $c(0)=c_{0}$ ?
(b) If $c(0)=60$ and $c(1)=51.2$ , solve the equation.

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Problem 22782

Find points on the curve where the tangent line is vertical and horizontal: $x=1-\frac{1}{a} \cos(at)$ , $y=t-\frac{1}{a} \sin(at)$ , $a>0$ .

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Problem 22783

Find the tangent line to $h(x)=x+3 e^{2 x-4}$ at $x=2$ and use it to estimate $h(3)$ .

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Problem 22784

Verify the integral formula by differentiation:
$\int \sec^2(2x+3) dx = \frac{1}{2} \tan(2x+3) + C$
What is $f'(g(x))$ ? Options: A. $\frac{1}{2} \tan(2x+3)$ B. $\tan(x)$ C. $\sec^2 x$ D. $\frac{1}{2} \sec^2(2x+3)$

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Problem 22785

Verify the integral formula:
$\int \sec^{2}(2x+3) dx = \frac{1}{2} \tan(2x+3) + C$
by using the Chain Rule. Define $f$ and $g$ and find their derivatives.

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Problem 22786

If $p(x)$ and monic $q(x)$ with $\operatorname{deg}(p)<3$ have roots $r_1, r_2, r_3$ , is the integral always, sometimes, or never true?

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Problem 22787

Verify the integral formula: $\int \sec ^{2}(2 x+3) d x=\frac{1}{2} \tan (2 x+3)+C$ . Find $f^{\prime}(g(x))$ .

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Problem 22788

Differentiate to verify: $\int \frac{2}{2 x-11} d x=\ln |2 x-11|+C, x \neq \frac{11}{2}$ . Use differentiation rules.

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Problem 22789

Find the length of the curve given by $y=\int_{\pi / 3 a}^{x} \sqrt{\cos a \theta} d \theta$ , for $\frac{\pi}{3 a}<x<\frac{\pi}{2 a}$ , where $a>0$ .

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Problem 22790

Find values of $x$ in $[0, 4]$ for which the sequence $a_{n}=n(2 \sin \frac{\pi x}{2})^{n}$ converges.

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Problem 22791

Find the absolute max and min of $g(\theta)=9 \theta^{\frac{3}{5}}$ for $-1 \leq \theta \leq 243$ . Where do they occur?

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Problem 22792

Model drug concentration in blood after one pill. Define constants $k_{1}$ and $c_{0}$ . Find $c(t)$ from data. $c(t)=\square$

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Problem 22793

Find the slope of the graph $y=\ln x^{4}$ at $x=2$ using the equation $(y-y_{1})=m(x-x_{1})$ .

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Problem 22794

If $p(x)$ and $q(x)$ are polynomials with $\operatorname{deg}(p(x))<3$ and $q$ monic with roots $r_1, r_2, r_3$ , is the integral $\int \frac{p(x)}{q(x)} d x$ always, sometimes, or never equal to the given expression? Explain.

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Problem 22795

A company makes 2 MP3 models. Revenue is given by $R(x, y)=150 x+60 y-3 x^{2}-2 y^{2}-x y$ . Find $R_{x}(x, y)$ and $R_{y}(x, y)$ , then set them to 0 to maximize revenue. Find $x$ and $y$ .

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Problem 22796

Find points on the curve with vertical and horizontal tangents: $x=1, \, y=\frac{1}{a} \sin(at), \, t=\frac{1}{a} \cos(at)$ , $a>0$ .

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Problem 22797

Find the derivative of the function $17^{x^4 + 7}$ .

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Problem 22798

Find the derivative of $y$ given that $\frac{d}{dx} y = \frac{d}{dx} 17^{x^4 + 7}$ .

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Problem 22799

Find $x$ in the interval $0 < x \leq 4$ where the sequence $a_{n} = n(2 \sin \frac{\pi x}{2})^{n}$ converges.

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Problem 22800

Find $x$ values in the interval $0 \leq x < 4$ for which the sequence $a_{n}=n\left(2 \sin \frac{\pi x}{2}\right)^{n}$ converges.

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Calculus

Problem 22701

Calculez la dérivée de f(x)=(4x+1)3−2x4+x2+1x2−1f(x)=(4 x+1)^{3}-\frac{2}{\sqrt[4]{x}}+\frac{x^{2}+1}{x^{2}-1}f(x)=(4x+1)3−4x​2​+x2−1x2+1​.

Problem 22702

Find the horizontal asymptote of C(t)=12t2+1C(t)=\frac{1}{2t^{2}+1}C(t)=2t2+11​ and describe the drug concentration as ttt increases.

Problem 22703

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

Problem 22704

Strontium-90 decays as A(t)=A0e−0.0244tA(t)=A_{0} e^{-0.0244 t}A(t)=A0​e−0.0244t. Given 400g, find: (a) decay rate, (b) amount after 30 yrs, (c) time for 100g, (d) half-life.

Problem 22705

Find the limit: lim⁡x→∞(15/x)14x=□\lim _{x \rightarrow \infty}(15 / x)^{14 x}=\squarelimx→∞​(15/x)14x=□

Problem 22706

Evaluate the integral from 1 to e of ln⁡xx dx\frac{\ln x}{x} \, dxxlnx​dx.

Problem 22707

Find the interval where the functions f(x)=x2+2x+2f(x)=x^{2}+2x+2f(x)=x2+2x+2 and g(x)=−x2+2x+10g(x)=-x^{2}+2x+10g(x)=−x2+2x+10 have the same average rate of change.

Problem 22708

Find xxx values that satisfy the Mean Value Theorem for f(x)=x3−xf(x)=x^{3}-xf(x)=x3−x on the interval [1,2][1,2][1,2].

Problem 22709

Estimate 9.8\sqrt{9.8}9.8​ using linear approximation at a=9a=9a=9.

Problem 22710

Find the average rate of change of the function f(x)=2x2+5x−6f(x)=2 x^{2}+5 x-6f(x)=2x2+5x−6 on the interval −4≤x≤0-4 \leq x \leq 0−4≤x≤0.

Problem 22711

Find the max and min of f(t)=t2t−8f(t)=\frac{t^{2}}{t-8}f(t)=t−8t2​ on −4≤t≤−14-4 \leq t \leq -\frac{1}{4}−4≤t≤−41​.

Problem 22712

Find the future value of an investment of \$10,000 for 13 years with continuous compounding at: (a) 1\%, (b) 3\%, (c) 5.5\%.

Problem 22713

Find the velocity of a particle with position s(t)=−4cos⁡t−t22+10s(t)=-4 \cos t-\frac{t^{2}}{2}+10s(t)=−4cost−2t2​+10 when its acceleration is zero.

Problem 22714

Evaluate the limit as xxx approaches infinity: lim⁡x→+∞(18x18x+4)6x\lim _{x \rightarrow+\infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}limx→+∞​(18x+418x​)6x. Enter −I-I−I for −∞-\infty−∞, III for ∞\infty∞, or DNE if it doesn't exist.

Problem 22715

Evaluate lim⁡x→0sin⁡9xx\lim _{x \rightarrow 0} \frac{\sin 9 x}{x}limx→0​xsin9x​ using l'Hôpital's Rule and limit laws. Rewrite it as lim⁡x→0(□)\lim _{x \rightarrow 0}(\square)limx→0​(□).

Problem 22716

Find the limit: lim⁡x→∞(18x18x+4)6x\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}limx→∞​(18x+418x​)6x. Does it exist?

Problem 22717

Find the limit as nnn approaches infinity of the sum ∑k=1n(6n)(4+6kn)2\sum_{k=1}^{n}\left(\frac{6}{n}\right)\left(4+\frac{6 k}{n}\right)^{2}∑k=1n​(n6​)(4+n6k​)2.

Problem 22718

Evaluate lim⁡x→∞4x2+3x4x3+x+3\lim _{x \rightarrow \infty} \frac{4 x^{2}+3 x}{4 x^{3}+x+3}limx→∞​4x3+x+34x2+3x​ using l'Hôpital's Rule and limit laws. Rewrite the limit as lim⁡x→∞(□)\lim _{x \rightarrow \infty}(\square)limx→∞​(□).

Problem 22719

Which inequality shows that the change in people inside a building is increasing at time ttt given rates f(t)f(t)f(t) and g(t)g(t)g(t)?

Problem 22720

Evaluate lim⁡x→∞2x2−7x9x2+7\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}limx→∞​9x2+72x2−7x​ using l'Hôpital's Rule and limit laws.

Problem 22721

Find the limit using l'Hôpital's rule: $$\lim _{x \rightarrow \infty} \frac{3 x^{3}-2 x}{7 x^{3}+3}=\square($ Type an integer or a fraction.)

Problem 22722

A softball weighing 88.9 g88.9 \mathrm{~g}88.9 g is dropped from 2.00 m2.00 \mathrm{~m}2.00 m. Find its velocity upon hitting the floor.

Problem 22723

Find the limit using l'Hôpital's rule: lim⁡x→∞ln⁡(x+2)log⁡5x=□\lim _{x \rightarrow \infty} \frac{\ln (x+2)}{\log _{5} x} = \squarelimx→∞​log5​xln(x+2)​=□

Problem 22724

Find the limit using l'Hôpital's Rule: lim⁡θ→π22−2sin⁡θ7+7cos⁡2θ\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{2-2 \sin \theta}{7+7 \cos 2 \theta}θ→2π​lim​7+7cos2θ2−2sinθ​. What is the answer?

Problem 22725

Evaluate lim⁡t→0−3sin⁡(9t6)5t\lim _{t \rightarrow 0} \frac{-3 \sin \left(9 t^{6}\right)}{5 t}limt→0​5t−3sin(9t6)​. What is the exact answer?

Problem 22726

Find the limit using l'Hôpital's Rule: $$\lim _{x \rightarrow 0} \frac{6 x^{2}}{\cos (x)-1}=\square($ Type an exact answer.)

Problem 22727

Problem 22728

A 225 kg boulder falls from 25 m. Calculate its impact velocity using energy principles. (4 marks)

Problem 22729

Find the limit: lim⁡x→∞25x+4x+2\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+4}}{\sqrt{x+2}}x→∞lim​x+2​25x+4​​ and simplify your answer.

Problem 22730

Find c\mathrm{c}c for continuity of f(x)f(x)f(x) at x=0x=0x=0: f(x)={15x−5sin⁡(3x)4x3,x≠0c,x=0 f(x)=\begin{cases} \frac{15 x-5 \sin (3 x)}{4 x^{3}}, & x \neq 0 \\ c, & x=0 \end{cases} f(x)={4x315x−5sin(3x)​,c,​x=0x=0​ What is c\mathrm{c}c?

Problem 22731

Find the time mmm for the cooling liquid to reach 90°F using f(m)=112e−0.5m+64f(m)=112 e^{-0.5 m}+64f(m)=112e−0.5m+64. Choose from 34, 5960.50, 5966.40, 5968.25 minutes.

Problem 22732

Problem 22733

Start with 16 blackberry plants growing at 75%75\%75% monthly. Estimate plants after 4 months with a limit of 100.

Problem 22734

Find the tangent line equation to y=f(x)y=f(x)y=f(x) at x=0x=0x=0 where f(x)=−2xe−xf(x)=-2 x e^{-x}f(x)=−2xe−x.

Problem 22735

Find the tangent line equation to y=f(x)y=f(x)y=f(x) at x=0x=0x=0 for f(x)=−2xe−xf(x)=-2 x e^{-x}f(x)=−2xe−x.

Problem 22736

Find relative extrema of f(x)=2x3−3x2−4f(x)=2 x^{3}-3 x^{2}-4f(x)=2x3−3x2−4 using the Second Derivative Test and justify your answers.

Problem 22737

Find the limit: lim⁡x→0exsin⁡xtan⁡xx2\lim _{x \rightarrow 0} \frac{e^{x} \sin x \tan x}{x^{2}}limx→0​x2exsinxtanx​.

Problem 22738

How much to deposit today to reach \$1500 in 10 years at a continuous compounding rate of 6.5%?

Problem 22739

Calculate the limit: lim⁡x→∞3x3+4x5−2x4\lim _{x \rightarrow \infty} \frac{3 x^{3}+4 x}{5-2 x^{4}}limx→∞​5−2x43x3+4x​

Problem 22740

At 2.5 hours after noon, find your distance from home given v(t)=−5t4+41t3−142t2+190tv(t)=-5 t^{4}+41 t^{3}-142 t^{2}+190 tv(t)=−5t4+41t3−142t2+190t and start 150 miles away.

Problem 22741

Calculez la dérivée de $f(x)=(4 x+1)^{3}-\frac{2}{\sqrt[4]{x}}+\frac{x^{2}+1}{x^{2}-1}$ .

Find the horizontal asymptote of $C(t)=\frac{1}{2t^{2}+1}$ and describe the drug concentration as $t$ increases.

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

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find: (a) decay rate, (b) amount after 30 yrs, (c) time for 100g, (d) half-life.

Find the limit: $\lim _{x \rightarrow \infty}(15 / x)^{14 x}=\square$

Evaluate the integral from 1 to e of $\frac{\ln x}{x} \, dx$ .

Find the interval where the functions $f(x)=x^{2}+2x+2$ and $g(x)=-x^{2}+2x+10$ have the same average rate of change.

Find $x$ values that satisfy the Mean Value Theorem for $f(x)=x^{3}-x$ on the interval $[1,2]$ .

Estimate $\sqrt{9.8}$ using linear approximation at $a=9$ .

Find the average rate of change of the function $f(x)=2 x^{2}+5 x-6$ on the interval $-4 \leq x \leq 0$ .

Find the max and min of $f(t)=\frac{t^{2}}{t-8}$ on $-4 \leq t \leq -\frac{1}{4}$ .

Find the velocity of a particle with position $s(t)=-4 \cos t-\frac{t^{2}}{2}+10$ when its acceleration is zero.

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow+\infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Enter $-I$ for $-\infty$ , $I$ for $\infty$ , or DNE if it doesn't exist.

Evaluate $\lim _{x \rightarrow 0} \frac{\sin 9 x}{x}$ using l'Hôpital's Rule and limit laws. Rewrite it as $\lim _{x \rightarrow 0}(\square)$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\frac{18 x}{18 x+4}\right)^{6 x}$ . Does it exist?

Find the limit as $n$ approaches infinity of the sum $\sum_{k=1}^{n}\left(\frac{6}{n}\right)\left(4+\frac{6 k}{n}\right)^{2}$ .

Evaluate $\lim _{x \rightarrow \infty} \frac{4 x^{2}+3 x}{4 x^{3}+x+3}$ using l'Hôpital's Rule and limit laws. Rewrite the limit as $\lim _{x \rightarrow \infty}(\square)$ .

Which inequality shows that the change in people inside a building is increasing at time $t$ given rates $f(t)$ and $g(t)$ ?

Evaluate $\lim _{x \rightarrow \infty} \frac{2 x^{2}-7 x}{9 x^{2}+7}$ using l'Hôpital's Rule and limit laws.

A softball weighing $88.9 \mathrm{~g}$ is dropped from $2.00 \mathrm{~m}$ . Find its velocity upon hitting the floor.

Find the limit using l'Hôpital's rule: $\lim _{x \rightarrow \infty} \frac{\ln (x+2)}{\log _{5} x} = \square$

Find the limit using l'Hôpital's Rule: $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{2-2 \sin \theta}{7+7 \cos 2 \theta}$ . What is the answer?

Evaluate $\lim _{t \rightarrow 0} \frac{-3 \sin \left(9 t^{6}\right)}{5 t}$ . What is the exact answer?

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+4}}{\sqrt{x+2}}$ and simplify your answer.

Find $\mathrm{c}$ for continuity of $f(x)$ at $x=0$ :
$f(x)=\begin{cases} \frac{15 x-5 \sin (3 x)}{4 x^{3}}, & x \neq 0 \\ c, & x=0 \end{cases}$
What is $\mathrm{c}$ ?

Find the time $m$ for the cooling liquid to reach 90°F using $f(m)=112 e^{-0.5 m}+64$ . Choose from 34, 5960.50, 5966.40, 5968.25 minutes.

Start with 16 blackberry plants growing at $75\%$ monthly. Estimate plants after 4 months with a limit of 100.

Find the tangent line equation to $y=f(x)$ at $x=0$ where $f(x)=-2 x e^{-x}$ .

Find the tangent line equation to $y=f(x)$ at $x=0$ for $f(x)=-2 x e^{-x}$ .

Find relative extrema of $f(x)=2 x^{3}-3 x^{2}-4$ using the Second Derivative Test and justify your answers.

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{x} \sin x \tan x}{x^{2}}$ .

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{3 x^{3}+4 x}{5-2 x^{4}}$

At 2.5 hours after noon, find your distance from home given $v(t)=-5 t^{4}+41 t^{3}-142 t^{2}+190 t$ and start 150 miles away.

Find the height of a projectile after 3 seconds using $v(t)=-15.5 t+147$ and 6 rectangles. Round to nearest tenth.

Find the absolute maximum of $f(x)=-\sin(2x)$ on $[0, 2\pi]$ . The derivative is $f'(x)=-2\cos(2x)$ .

Find the derivative of the function: $\frac{d}{d x}(\sec x \sin (3 x))$ .

A 500 g isotope decays as $A(t)=500 e^{-0.026 t}$ . Find: (a) remaining mass after 10 years, (b) time to halve the sample.

Find the derivatives: 27. $\frac{d}{d x}\left(\sqrt{\frac{2 x+5}{7 x-9}}\right)$ , 28. $\frac{d}{d x}\left(\frac{x}{\sqrt{1-(\ln x)^{2}}}\right)$ .

Find the limit as $u$ approaches $a$ : $\lim _{u \rightarrow a}\left(\frac{2 \sqrt{u}+\sqrt{a}}{4 u-a}\right)$ .

Find the intervals where the graph of $f(x)=x^{2} e^{x}$ is decreasing. Choose from the options given.

Find the derivative of $\frac{x}{\sqrt{1-(\ln x)^{2}}}$ with respect to $x$ .

Find the limit as $x$ approaches 0 for the expression $\frac{\sin (2 x)}{x}$ .

What are the units for $f^{\prime}(A)$ if $t=f(A)$ is time in minutes and $A$ is catalyst in milliliters? (A) minutes per milliliter (B) milliliters per minute (C) minutes per milliliter per milliliter (D) milliliters per minute per minute

Find $\lim _{x \rightarrow 5} M(x)$ for the piecewise function $M(x) = \{-4 \cos (\pi x), x<3; 0, x=3; \frac{x^{2}-25}{x^{2}-6 x+5}, x \geq 3\}$ .

Determine if the integral $\int_{c^{a}}^{\infty} \frac{\ln w-1}{w^{1 / a}-2} d w$ is convergent or divergent using the Comparison Theorem.

What are the units for $f^{\prime}(A)$ if $t=f(A)$ models time in minutes for catalyst amount $A$ in milliliters? (A) minutes per milliliter (B) milliliters per minute (C) minutes per milliliter per milliliter (D) milliliters per minute per minute

Given the equations $x=4t+3$ and $y=4t+8+\frac{5}{2t}$ for $t \neq 0$ , find $\frac{dy}{dx}$ at $t=2$ and show $y=\frac{x^2+ax+b}{x-3}$ .

Find $\lim _{x \rightarrow 5} M(x)$ and the horizontal asymptote of $M(x)$ for the piecewise function defined.

Find and classify critical points of $z=(x^{2}-5x)(y^{2}-8y)$ . List local max, min, and saddle points as $(x,y)$ or DNE.

Find the length of the curve given by $y=\int_{\pi / 3 a}^{x} \sqrt{\cos a \theta} d \theta$ for $\frac{\pi}{3 a} \leq x \leq \frac{\pi}{2 a}$ .

Find $S_{7}$ in the series $64,16,4,\ldots$ and explain when $S_{\infty}$ can be determined for an infinite series.

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for the curve given by $x=4 t+3$ , $y=4 t+8+\frac{5}{2 t}$ at $t=2$ .

Analyze the motion of an object with position $s(t)=t^{2}-10t$ for $0 \leq t \leq 12$ ; find initial/final velocity and acceleration.

Water is removed at a rate $R(t)$ (liters/hour) shown in the table. Estimate water removed in 8 hours using a left Riemann sum. Is it an overestimate or underestimate?

Find the marginal revenue equations for $R(x, y)=150 x+60 y-3 x^{2}-2 y^{2}-x y$ . Set $R_{x}=0$ and $R_{y}=0$ to maximize revenue. Find $x$ and $y$ .

Water is removed from a tank at a rate $R(t)$ (liters/hour). Given values of $R$ over 8 hours, estimate water removed using a left Riemann sum. Is it an overestimate or underestimate? Explain why $R(t) = 900$ liters/hour occurs for some $t$ in $(0,8)$ .

Evaluate the integral $\int_{e^{a}}^{\infty} \frac{\ln w-1}{w^{1 / a}-2} d w$ for $a > 2$ .

Use l'Hôpital's Rule to evaluate $\lim _{x \rightarrow \infty} \frac{\ln (1+a / x)}{1 / x}$ .

Evaluate the series $\sum_{k=1}^{\infty} \frac{\sqrt{k} \ln k}{k^{3}+1}$ .

A cube's edges expand at $3 \mathrm{~cm/s}$ . Find the volume change rate when each edge is $10 \mathrm{~cm}$ .

How much more will $\$ 10,000$ at $7 \%$ continuous compounding for 9 years earn than quarterly compounding? Round to cents.

Calculate the average rate of change of $g(x)=4x+8$ between $x=2$ and $x=4$ . Simplify your answer.

Find $x$ in $0 \leq x \leq 4$ for which the sequence $a_{n}=n(2 \sin \frac{\pi x}{2})^{n}$ converges.

Find values of $p$ for which the series $\sum_{n=1}^{\infty}\left(\frac{p}{n+1}-\frac{1}{n+3}\right)$ converges.

Find the curve length for $y=\int_{\pi / 3 a}^{x} \sqrt{\cos a \theta} \, d \theta$ with $\frac{\pi}{3 a} \leq x \leq \frac{\pi}{2 a}$ .

Compare the earnings of a \ $25,000 investment at$ 5\%$ for 3 years, compounded continuously vs. monthly. Calculate the difference.