Calculus

Problem 32501

Find the average rate of change (ARC) for $y=16t^{2}$ over the interval $[1, 1.1]$ . Use $\frac{\Delta y}{\Delta t}$ .

See Solution

Problem 32502

Identify which functions are valid in quantum mechanics: $x^{2}$ or $e^{-x^{2}}$ . Normalize the valid ones and plot them.

See Solution

Problem 32503

Find the secant line's equation using $y - y_{1} = m(x - x_{1})$ and slope $m$ . Also, find IRC for $y = 16t^2$ at $t_0 = 1$ .

See Solution

Problem 32504

Find the limit: $\lim _{x \rightarrow 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3}$ .

See Solution

Problem 32505

Evaluate the limit: $\lim _{t \rightarrow-2}(6 t-5 a t+2 a)=$

See Solution

Problem 32506

Find the average rate of change of $R(\theta)=\sqrt{3\theta+1}$ over the interval $[0,8]$ . Simplify $\frac{\Delta R}{\Delta \theta}=\square$ .

See Solution

Problem 32507

Find the slope of the curve $y=x^{3}+2$ at $P(-2,-6)$ and the equation of the tangent line at that point.

See Solution

Problem 32508

Find the slope of the curve $y=x^{3}+2$ at point $P(2,10)$ and the equation of the tangent line there.

See Solution

Problem 32509

Find the slope of the curve $y=-3-5x^{2}$ at point $P(-2,-23)$ and the equation of the tangent line at $P$ .

See Solution

Problem 32510

Find the slope of the curve $y=x^{3}+2$ at $P(-2,-6)$ using secant lines, then find the tangent line equation at $P$ .

See Solution

Problem 32511

Solve the differential equation $\frac{d y}{d x}=\frac{3 x^{2}}{e^{2 y}}$ with the condition $f(0)=\frac{1}{2}$ .

See Solution

Problem 32512

Find the slope and equation of the tangent line at point $P(2,2)$ for the curve $y=\frac{4}{x}$ .

See Solution

Problem 32513

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{n+a}{n+b}\right)^{n^{2}}$ converges or diverges for $b<a$ .

See Solution

Problem 32514

Find values of $a$ where limits $\lim _{x \rightarrow a} f(x)$ exist for these piecewise functions.

See Solution

Problem 32515

Estimate $f^{\prime}(7)$ , evaluate $\int_{2}^{13}(2-f^{\prime}(x))dx$ , and approximate $\int_{2}^{13}f(x)dx$ . Show all work. Also, prove $f(7) \leq 4$ using the tangent at $x=5$ .

See Solution

Problem 32516

Find the limits of $f(x)=\frac{x^{2}+1}{x^{2}-10 x+25}$ as $x$ approaches 5 from the left, right, and both.

See Solution

Problem 32517

Find values of $a$ and $b$ for which $\lim _{x \rightarrow 2} f(x)$ and $\lim _{x \rightarrow 3} f(x)$ exist.

See Solution

Problem 32518

Calculate the integral $I=\int 2 x(\ln (x))^{2} d x$ .

See Solution

Problem 32519

What is the limit of $f(x)$ as $x$ approaches 2 if $f(2)=-3$ and $f(x)$ is continuous at $x=2$ ? Options: $-3$ , 2, $-2$ , 3

See Solution

Problem 32520

Calculate the integral $I=\int_{0}^{1} 3 e^{7 \sqrt{x}} dx$ .

See Solution

Problem 32521

Calculate the integral $I=\int_{0}^{\pi} e^{x} \cos x \, dx$ .

See Solution

Problem 32522

Evaluate the integral: $\int \frac{u}{27-3 u^{2}} d u$

See Solution

Problem 32523

Calculate the integral $I=\int_{e}^{8} \frac{\ln (x)}{x^{2}} d x$ .

See Solution

Problem 32524

How long will it take to double \ $2,000 invested at a$ 5.75\%$ annual interest rate, compounded continuously? Round up.

See Solution

Problem 32525

Calculate the integral $I=\int_{0}^{\pi / 4}(4 x-5) \cos 2 x \, dx$ .

See Solution

Problem 32526

Find the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{11}$ on $[2,6]$ using 8 rectangles. What is the sum?

See Solution

Problem 32527

Explain why the function $f(x)$ is discontinuous at $x=1$ :
$f(x)=\left\{\begin{array}{ll} \frac{1}{x-1} & \text { if } x \neq 1 \\ 2 & \text { if } x=1 \end{array}\right.$
Options: (a) $f(1)$ does not exist. (b) $\lim _{x \rightarrow 1} f(x)$ does not exist. (c) Both (a) and (b). (d) They exist but are not equal.

See Solution

Problem 32528

Differentiate $6xy + y + 185 = 0$ to find $y'$ and evaluate it at $(6, -5)$ . Simplify your answer.

See Solution

Problem 32529

Find $y^{\prime}$ using implicit differentiation for $9xy + y + 210 = 0$ and evaluate at $(-4, 6)$ .

See Solution

Problem 32530

Find $y^{\prime}$ using implicit differentiation for $x^{2} y - 3 x^{2} - 4 = 0$ and evaluate it at $(2,4)$ .

See Solution

Problem 32531

Find $y^{\prime}$ using implicit differentiation for $-32 e^{y}=x^{5}+y^{3}$ and evaluate at $(-2,0)$ .

See Solution

Problem 32532

Find $y^{\prime}$ using implicit differentiation for $x^{3}+y^{5}=\ln y$ and evaluate it at $(-1,1)$ .

See Solution

Problem 32533

Identify acceptable quantum mechanics functions: $x^{2}$ and $e^{-x^{2}}$ . Plot and normalize if acceptable.

See Solution

Problem 32534

Differentiate $x^{3}-8 y^{5}=\ln y$ implicitly to find $y^{\prime}$ , then evaluate at $(2,1)$ .

See Solution

Problem 32535

Find $x^{\prime}$ for $x(t)$ from $x^{4}+t^{4} x+t^{3}-9=0$ and evaluate at $(-2,1)$ . Simplify your answer.

See Solution

Problem 32536

Analyze an object's motion based on graphs and answer:
1. Change in velocity?
2. Gaining or losing velocity?
3. Distance traveled in 8s?
4. Acceleration?
5. Position at $10 \mathrm{~s}$ ?
6. Why negative acceleration if gaining velocity?

See Solution

Problem 32537

Provide examples of functions $f(x)$ and $g(x)$ to show these statements are false: (a) $\lim_{x \to 0} f(x)+g(x)$ exists $\Rightarrow$ both $\lim_{x \to 0} f(x)$ and $\lim_{x \to 0} g(x)$ exist. (b) $\lim_{x \to 0} f(x)g(x)$ exists $\Rightarrow$ both $\lim_{x \to 0} f(x)$ and $\lim_{x \to a} g(x)$ exist.

See Solution

Problem 32538

Find the limit: $\lim _{x \rightarrow-5} \frac{x+5}{x^{3}+125}$ . If it doesn't exist, write DNE.

See Solution

Problem 32539

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{2}{t^{1/3}}-\frac{9}{t^{3/7}}$ .

See Solution

Problem 32540

Find $y^{\prime}$ and the slope of the tangent line to $(2x - 3y)^5 = 3y^2 - 2$ at point $(2,1)$ .

See Solution

Problem 32541

Find the limits and function value for the piecewise function $f(x)$ given below:
$f(x)=\left\{\begin{array}{ll} x-6, & x \leq-1 \\ x^{2}+3, & -1<x \leq 4 \\ 23-x, & x>4 \end{array}\right.$
a) $\lim _{x \rightarrow-1^{-}} f(x)$ b) $\lim _{x \rightarrow-1^{+}} f(x)$ c) $\lim _{x \rightarrow-1} f(x)$ d) $\lim _{x \rightarrow 4^{-}} f(x)$ e) $\lim _{x \rightarrow 4^{+}} f(x)$ f) $\lim _{x \rightarrow 4} f(x)$ g) $f(-1)$

See Solution

Problem 32542

Find $y^{\prime}$ and the slope of the tangent line to $(3x - 5y)^5 = 2y^2 - 9$ at point $(3,2)$ .

See Solution

Problem 32543

A ball drops for $3 \mathrm{~s}$ . Find the roof height in meters, ignoring air resistance.

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Problem 32544

Evaluate these limits: (a) $\lim _{x \rightarrow 5} \frac{x^{2}-6 x+5}{x-5}$ , (b) $\lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h}$ , (c) $\lim _{x \rightarrow 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3}$ , (d) $\lim _{t \rightarrow-0} \frac{\sqrt{1+t}-\sqrt{1-t}}{t}$ , (e) $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)$ .

See Solution

Problem 32545

Find the slope of secant line $PQ$ for $P(-3,0)$ and $Q(x, \sin(3 \pi x))$ at specified $x$ values. Then estimate the tangent slope and find the tangent line's equation.

See Solution

Problem 32546

Find the slope of secant line $PQ$ for $P(-3,0)$ and $Q(x, \sin(3\pi x))$ at given $x$ values. Then, guess the tangent slope and find the tangent line equation at $P$ .

See Solution

Problem 32547

Find $y^{\prime}$ and the slope at $(3,3)$ for the equation $\sqrt{72+y^{2}}-x^{3}+18=0$ .

See Solution

Problem 32548

Find $y^{\prime}$ and the slope at (3,3) for $\sqrt{40+y^{2}}-x^{3}+20=0$ .

See Solution

Problem 32549

Find the following limits and value for the piecewise function $f(x)$ :
a) $\lim _{x \rightarrow 0^{-}} f(x)$ b) $\lim _{x \rightarrow 0^{+}} f(x)$ c) $\lim _{x \rightarrow 0} f(x)$ d) $\lim _{x \rightarrow \pi^{-}} f(x)$ e) $\lim _{x \rightarrow \pi^{+}} f(x)$ f) $\lim _{x \rightarrow \pi} f(x)$ g) $f(\pi)$

See Solution

Problem 32550

Find the derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $y=5 \ln x+4 \log _{5} x$ . Type an exact answer.

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Problem 32551

Find the limits and value for the piecewise function $f(x)$ at $x=-1$ and classify the discontinuity type.

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Problem 32552

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0^{+}} 11 x \ln x=\square$ . Enter INF, -INF, or DNE.

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Problem 32553

Evaluate the limit using continuity: $\lim _{x \rightarrow \frac{3 \pi}{2}} \sin (x+\sin (x))$ .

See Solution

Problem 32554

Find $y^{\prime}$ and the slope at the point (1,1) for the equation $13 \ln (x y)=y^{2}-1$ .

See Solution

Problem 32555

Calculate the limit: $\lim _{x \rightarrow 1} \frac{5-5 x}{1-\sqrt{x}}$ .

See Solution

Problem 32556

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x+\tan (4 x)}{3 x-\tan (4 x)}$ . Enter -Inf, Inf, or DNE for your answer.

See Solution

Problem 32557

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 1} \frac{x^{2}+4 x-5}{\ln x}=\square$

See Solution

Problem 32558

Find $y^{\prime}$ and the slope at the point (1,1) for the equation $9 \ln (x y)=y^{2}-1$ .

See Solution

Problem 32559

Evaluate the limit using continuity: $\lim _{x \rightarrow \pi} \cos (x+\cos (x))$ .

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Problem 32560

Find the smallest interval where $f(x)$ has a root using the Intermediate Value Theorem from the values of $f$ given.

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Problem 32561

Find the initial population of arctic flounder modeled by $P(t)=\frac{12 t+240}{0.7 t^{2}+4}$ and calculate $P^{\prime}(8)$ .

See Solution

Problem 32562

Find the limits: $\lim_{x \rightarrow 8^{-}} f(x)$ , $\lim_{x \rightarrow 8^{+}} f(x)$ , $\lim_{x \rightarrow -8^{-}} f(x)$ , and $\lim_{x \rightarrow -8^{+}} f(x)$ for the piecewise function $f(x)$ .

See Solution

Problem 32563

Given the piecewise function
$f(x)=\left\{\begin{array}{ll} -x-3, & \text { if } x<5 \\ x^{2}-16, & \text { if } x>5 \\ -18, & \text { if } x=5 \end{array}\right.$
find: a) $\lim _{x \rightarrow 5^{-}} f(x)$ b) $\lim _{x \rightarrow 5^{+}} f(x)$ c) $f(5)$ d) classify the discontinuity at $x=5$ .

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Problem 32564

Find the limits of the following indeterminate forms (0/0). Enter "D" if the limit does not exist:
$\lim _{x \rightarrow 0} \frac{\sin x}{x}=\square, \quad \lim _{x \rightarrow 0} \frac{e^{x^{x}}-1-x}{x^{2}}=\square,$ $\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{\sin (x)}=\square, \quad \lim _{x \rightarrow-2} \frac{x^{2}+5 x+6}{x-2}=\square,$ $\lim _{x \rightarrow 2} \frac{x^{2}+5 x+6}{x-2}=\square.$

See Solution

Problem 32565

Evaluate the limit: $\lim _{t \rightarrow 0} \frac{t^{3} \cos (3 t)}{4 t-3}$ . If it doesn't exist, enter DNE.

See Solution

Problem 32566

Find the rate of change of price $p$ with respect to demand $x$ for $x=\sqrt[3]{10,000-p^{4}}$ by implicit differentiation.

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Problem 32567

Calculate the limit using L'Hospital's Rule if needed: $\lim _{x \rightarrow 9} \frac{x-9}{x^{2}-15 x+54}=\square$

See Solution

Problem 32568

Find the limit as $x$ approaches negative infinity: $\lim _{x \rightarrow-\infty} \frac{3 x^{2}+1}{x^{2}+5}=\square$

See Solution

Problem 32569

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

See Solution

Problem 32570

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{x^{2}-1}{x+5}.$

See Solution

Problem 32571

Find the limit: $\lim _{x \rightarrow 9} \frac{x-9}{\sqrt{x}-3}$ . If it doesn't exist, write DNE.

See Solution

Problem 32572

Given the piecewise function $f(x)$ , find the limits as $x$ approaches 2 from the right and left, and values of $m$ for equality of these limits.

See Solution

Problem 32573

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{-7 x-6}{-2 x-6}=\square$

See Solution

Problem 32574

Find $g^{\prime}(t)$ using the Chain Rule for $g(t)=f(x(t), y(t))$ , where $f(x, y)=x^{2} y-\sin y$ , $x(t)=\sqrt{t^{2}+1}$ , $y(t)=e^{t}$ .

See Solution

Problem 32575

Find the limit: $\lim _{x \rightarrow \infty} \frac{7 x^{2}-5 x-3}{x-4 x^{2}-7}=\square$

See Solution

Problem 32576

Evaluate the limit: $\lim _{x \rightarrow 0}\left(x^{3}-\cos \left(\frac{3 x}{1000}\right)\right)$ . If it doesn't exist, enter DNE.

See Solution

Problem 32577

Given the function $f(x)$ , find the limits as $x$ approaches 2 from the right and left, and determine values of $m$ for continuity.

See Solution

Problem 32578

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{2 x^{2}-5 x-3}{3 x^{3}-3 x-7}=\square$

See Solution

Problem 32579

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{2 x^{4}+6}{11 x^{2}+3}$ . Limit =

See Solution

Problem 32580

Find the initial population of arctic flounder modeled by $P(t)=\frac{12 t+240}{0.7 t^{2}+4}$ and calculate $P^{\prime}(8)$ .

See Solution

Problem 32581

Evaluate the limit as $x$ approaches negative infinity:
$\lim _{x \rightarrow-\infty} \frac{12 x^{7}+3 x}{2 x^{3}-4 x^{2}}$

See Solution

Problem 32582

Evaluate the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{3-8 x^{4}}{5+8 x^{4}}$ . Limit =

See Solution

Problem 32583

A stone is thrown down at $1.84 \, \mathrm{m/s}$ from a $51.0 \, \mathrm{m}$ cliff.
(a) When do both stones hit the water?
(b) What speed must the second stone have to hit simultaneously?
(c) What is the speed of each stone upon impact?

See Solution

Problem 32584

Find the slope of the secant line from $P\left(\frac{5}{2}, 0\right)$ to $Q(x, \cos(3\pi x))$ for $x = 0, 2.4, 2.49, 2.499, 3, 2.6, 2.51, 2.501$ . Then, estimate the tangent slope at $P$ and find the tangent line equation.

See Solution

Problem 32585

Determine if the series $\sum_{n=1}^{\infty} \frac{6}{(2 n-1)(2 n+1)}$ converges or diverges and find its sum.

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Problem 32586

Evaluate the series $\sum_{n=1}^{\infty} \ln \frac{n}{n+1}$ . Which statement is true about its convergence?

See Solution

Problem 32587

Differentiate $7 x^{3}-2 y-13=0$ implicitly to find $y^{\prime}$ , then solve for $y$ and differentiate directly.

See Solution

Problem 32588

Determine if the series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}$ converges or diverges using the Integral Test.

See Solution

Problem 32589

Find the derivative of $f(x)=x^{6} \cos x$ .

See Solution

Problem 32590

Determine the convergence of the series $\sum_{n=2}^{\infty} \frac{\ln n^{2}}{n}$ . Choose: a. Diverges by Integral Test b. Diverges by nth term test c. Converges by Integral Test d. Converges to $(\ln 2)^{2}$ by Integral Test.

See Solution

Problem 32591

Determine if the series $\sum_{n=1}^{\infty}(\ln 2)^{n}$ is convergent or divergent and find its sum.

See Solution

Problem 32592

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{3}-64}{x-4}$ .

See Solution

Problem 32593

Find the limit as $t$ approaches 0 for the expression $\frac{\sin 3 t}{2 t}$ .

See Solution

Problem 32594

Find the limit as $x$ approaches 0 for the expression $\frac{\sin x}{8 x}$ .

See Solution

Problem 32595

Find the derivative of the function $f(x)=\frac{4}{4x+4}$ .

See Solution

Problem 32596

Find the limit: $\lim _{x \rightarrow 8} \frac{\sqrt{x+1}-3}{x-8}$ .

See Solution

Problem 32597

Find the limits: (a) $\lim _{x \rightarrow \infty} \frac{5+4 x}{8-7 x}=$ and (b) $\lim _{x \rightarrow-\infty} \frac{5+4 x}{8-7 x}=$ .

See Solution

Problem 32598

Find the value of the integral $\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$ using the substitution $u=\sqrt{x}$ .

See Solution

Problem 32599

Find the limit as $x$ approaches 0 for $\frac{x^{2}+5x}{x}$ . Create a simpler function $g(x)$ that matches it everywhere except one point. Use a graphing tool to verify.

See Solution

Problem 32600

Find the derivative of the function $f(x)=(3x-5x^{3})(6+\sqrt{x})$ . What is $f^{\prime}(x)$ ?

See Solution

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Calculus

Problem 32501

Find the average rate of change (ARC) for y=16t2y=16t^{2}y=16t2 over the interval [1,1.1][1, 1.1][1,1.1]. Use ΔyΔt\frac{\Delta y}{\Delta t}ΔtΔy​.

Problem 32502

Identify which functions are valid in quantum mechanics: x2x^{2}x2 or e−x2e^{-x^{2}}e−x2. Normalize the valid ones and plot them.

Problem 32503

Find the secant line's equation using y−y1=m(x−x1)y - y_{1} = m(x - x_{1})y−y1​=m(x−x1​) and slope mmm. Also, find IRC for y=16t2y = 16t^2y=16t2 at t0=1t_0 = 1t0​=1.

Problem 32504

Find the limit: lim⁡x→31x−13x−3\lim _{x \rightarrow 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3}limx→3​x−3x1​−31​​.

Problem 32505

Evaluate the limit: lim⁡t→−2(6t−5at+2a)=\lim _{t \rightarrow-2}(6 t-5 a t+2 a)=limt→−2​(6t−5at+2a)=

Problem 32506

Find the average rate of change of R(θ)=3θ+1R(\theta)=\sqrt{3\theta+1}R(θ)=3θ+1​ over the interval [0,8][0,8][0,8]. Simplify ΔRΔθ=□\frac{\Delta R}{\Delta \theta}=\squareΔθΔR​=□.

Problem 32507

Find the slope of the curve y=x3+2y=x^{3}+2y=x3+2 at P(−2,−6)P(-2,-6)P(−2,−6) and the equation of the tangent line at that point.

Problem 32508

Find the slope of the curve y=x3+2y=x^{3}+2y=x3+2 at point P(2,10)P(2,10)P(2,10) and the equation of the tangent line there.

Problem 32509

Find the slope of the curve y=−3−5x2y=-3-5x^{2}y=−3−5x2 at point P(−2,−23)P(-2,-23)P(−2,−23) and the equation of the tangent line at PPP.

Problem 32510

Find the slope of the curve y=x3+2y=x^{3}+2y=x3+2 at P(−2,−6)P(-2,-6)P(−2,−6) using secant lines, then find the tangent line equation at PPP.

Problem 32511

Solve the differential equation dydx=3x2e2y\frac{d y}{d x}=\frac{3 x^{2}}{e^{2 y}}dxdy​=e2y3x2​ with the condition f(0)=12f(0)=\frac{1}{2}f(0)=21​.

Problem 32512

Find the slope and equation of the tangent line at point P(2,2)P(2,2)P(2,2) for the curve y=4xy=\frac{4}{x}y=x4​.

Problem 32513

Determine if the series ∑n=1∞(n+an+b)n2\sum_{n=1}^{\infty}\left(\frac{n+a}{n+b}\right)^{n^{2}}∑n=1∞​(n+bn+a​)n2 converges or diverges for b<ab<ab<a.

Problem 32514

Find values of aaa where limits lim⁡x→af(x)\lim _{x \rightarrow a} f(x)limx→a​f(x) exist for these piecewise functions.

Problem 32515

Estimate f′(7)f^{\prime}(7)f′(7), evaluate ∫213(2−f′(x))dx\int_{2}^{13}(2-f^{\prime}(x))dx∫213​(2−f′(x))dx, and approximate ∫213f(x)dx\int_{2}^{13}f(x)dx∫213​f(x)dx. Show all work. Also, prove f(7)≤4f(7) \leq 4f(7)≤4 using the tangent at x=5x=5x=5.

Problem 32516

Find the limits of f(x)=x2+1x2−10x+25f(x)=\frac{x^{2}+1}{x^{2}-10 x+25}f(x)=x2−10x+25x2+1​ as xxx approaches 5 from the left, right, and both.

Problem 32517

Find values of aaa and bbb for which lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) and lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x)limx→3​f(x) exist.

Problem 32518

Calculate the integral I=∫2x(ln⁡(x))2dxI=\int 2 x(\ln (x))^{2} d xI=∫2x(ln(x))2dx.

Problem 32519

What is the limit of f(x)f(x)f(x) as xxx approaches 2 if f(2)=−3f(2)=-3f(2)=−3 and f(x)f(x)f(x) is continuous at x=2x=2x=2? Options: −3-3−3, 2, −2-2−2, 3

Problem 32520

Calculate the integral I=∫013e7xdxI=\int_{0}^{1} 3 e^{7 \sqrt{x}} dxI=∫01​3e7x​dx.

Problem 32521

Calculate the integral I=∫0πexcos⁡x dxI=\int_{0}^{\pi} e^{x} \cos x \, dxI=∫0π​excosxdx.

Problem 32522

Evaluate the integral: ∫u27−3u2du\int \frac{u}{27-3 u^{2}} d u∫27−3u2u​du

Problem 32523

Calculate the integral I=∫e8ln⁡(x)x2dxI=\int_{e}^{8} \frac{\ln (x)}{x^{2}} d xI=∫e8​x2ln(x)​dx.

Problem 32524

How long will it take to double \2,000investedata 2,000 invested at a 2,000investedata5.75\%$ annual interest rate, compounded continuously? Round up.

Problem 32525

Calculate the integral I=∫0π/4(4x−5)cos⁡2x dxI=\int_{0}^{\pi / 4}(4 x-5) \cos 2 x \, dxI=∫0π/4​(4x−5)cos2xdx.

Problem 32526

Find the left endpoint Riemann sum for f(x)=x211f(x)=\frac{x^{2}}{11}f(x)=11x2​ on [2,6][2,6][2,6] using 8 rectangles. What is the sum?

Problem 32527

Problem 32528

Differentiate 6xy+y+185=06xy + y + 185 = 06xy+y+185=0 to find y′y'y′ and evaluate it at (6,−5)(6, -5)(6,−5). Simplify your answer.

Problem 32529

Find y′y^{\prime}y′ using implicit differentiation for 9xy+y+210=09xy + y + 210 = 09xy+y+210=0 and evaluate at (−4,6)(-4, 6)(−4,6).

Problem 32530

Find y′y^{\prime}y′ using implicit differentiation for x2y−3x2−4=0x^{2} y - 3 x^{2} - 4 = 0x2y−3x2−4=0 and evaluate it at (2,4)(2,4)(2,4).

Problem 32531

Find y′y^{\prime}y′ using implicit differentiation for −32ey=x5+y3-32 e^{y}=x^{5}+y^{3}−32ey=x5+y3 and evaluate at (−2,0)(-2,0)(−2,0).

Problem 32532

Find y′y^{\prime}y′ using implicit differentiation for x3+y5=ln⁡yx^{3}+y^{5}=\ln yx3+y5=lny and evaluate it at (−1,1)(-1,1)(−1,1).

Problem 32533

Identify acceptable quantum mechanics functions: x2x^{2}x2 and e−x2e^{-x^{2}}e−x2. Plot and normalize if acceptable.

Problem 32534

Differentiate x3−8y5=ln⁡yx^{3}-8 y^{5}=\ln yx3−8y5=lny implicitly to find y′y^{\prime}y′, then evaluate at (2,1)(2,1)(2,1).

Problem 32535

Find x′x^{\prime}x′ for x(t)x(t)x(t) from x4+t4x+t3−9=0x^{4}+t^{4} x+t^{3}-9=0x4+t4x+t3−9=0 and evaluate at (−2,1)(-2,1)(−2,1). Simplify your answer.

Problem 32536

Analyze an object's motion based on graphs and answer: 1. Change in velocity? 2. Gaining or losing velocity? 3. Distance traveled in 8s? 4. Acceleration? 5. Position at 10 s10 \mathrm{~s}10 s? 6. Why negative acceleration if gaining velocity?

Problem 32537

Problem 32538

Find the limit: lim⁡x→−5x+5x3+125\lim _{x \rightarrow-5} \frac{x+5}{x^{3}+125}x→−5lim​x3+125x+5​. If it doesn't exist, write DNE.

Problem 32539

Find the derivative h′(t)h^{\prime}(t)h′(t) for the function h(t)=2t1/3−9t3/7h(t)=\frac{2}{t^{1/3}}-\frac{9}{t^{3/7}}h(t)=t1/32​−t3/79​.

Problem 32540

Find y′y^{\prime}y′ and the slope of the tangent line to (2x−3y)5=3y2−2(2x - 3y)^5 = 3y^2 - 2(2x−3y)5=3y2−2 at point (2,1)(2,1)(2,1).

Problem 32541

Find the average rate of change (ARC) for $y=16t^{2}$ over the interval $[1, 1.1]$ . Use $\frac{\Delta y}{\Delta t}$ .

Identify which functions are valid in quantum mechanics: $x^{2}$ or $e^{-x^{2}}$ . Normalize the valid ones and plot them.

Find the secant line's equation using $y - y_{1} = m(x - x_{1})$ and slope $m$ . Also, find IRC for $y = 16t^2$ at $t_0 = 1$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3}$ .

Evaluate the limit: $\lim _{t \rightarrow-2}(6 t-5 a t+2 a)=$

Find the average rate of change of $R(\theta)=\sqrt{3\theta+1}$ over the interval $[0,8]$ . Simplify $\frac{\Delta R}{\Delta \theta}=\square$ .

Find the slope of the curve $y=x^{3}+2$ at $P(-2,-6)$ and the equation of the tangent line at that point.

Find the slope of the curve $y=x^{3}+2$ at point $P(2,10)$ and the equation of the tangent line there.

Find the slope of the curve $y=-3-5x^{2}$ at point $P(-2,-23)$ and the equation of the tangent line at $P$ .

Find the slope of the curve $y=x^{3}+2$ at $P(-2,-6)$ using secant lines, then find the tangent line equation at $P$ .

Solve the differential equation $\frac{d y}{d x}=\frac{3 x^{2}}{e^{2 y}}$ with the condition $f(0)=\frac{1}{2}$ .

Find the slope and equation of the tangent line at point $P(2,2)$ for the curve $y=\frac{4}{x}$ .

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{n+a}{n+b}\right)^{n^{2}}$ converges or diverges for $b<a$ .

Find values of $a$ where limits $\lim _{x \rightarrow a} f(x)$ exist for these piecewise functions.

Estimate $f^{\prime}(7)$ , evaluate $\int_{2}^{13}(2-f^{\prime}(x))dx$ , and approximate $\int_{2}^{13}f(x)dx$ . Show all work. Also, prove $f(7) \leq 4$ using the tangent at $x=5$ .

Find the limits of $f(x)=\frac{x^{2}+1}{x^{2}-10 x+25}$ as $x$ approaches 5 from the left, right, and both.

Find values of $a$ and $b$ for which $\lim _{x \rightarrow 2} f(x)$ and $\lim _{x \rightarrow 3} f(x)$ exist.

Calculate the integral $I=\int 2 x(\ln (x))^{2} d x$ .

What is the limit of $f(x)$ as $x$ approaches 2 if $f(2)=-3$ and $f(x)$ is continuous at $x=2$ ? Options: $-3$ , 2, $-2$ , 3

Calculate the integral $I=\int_{0}^{1} 3 e^{7 \sqrt{x}} dx$ .

Calculate the integral $I=\int_{0}^{\pi} e^{x} \cos x \, dx$ .

Evaluate the integral: $\int \frac{u}{27-3 u^{2}} d u$

Calculate the integral $I=\int_{e}^{8} \frac{\ln (x)}{x^{2}} d x$ .

How long will it take to double \ $2,000 invested at a$ 5.75\%$ annual interest rate, compounded continuously? Round up.

Calculate the integral $I=\int_{0}^{\pi / 4}(4 x-5) \cos 2 x \, dx$ .

Find the left endpoint Riemann sum for $f(x)=\frac{x^{2}}{11}$ on $[2,6]$ using 8 rectangles. What is the sum?

Differentiate $6xy + y + 185 = 0$ to find $y'$ and evaluate it at $(6, -5)$ . Simplify your answer.

Find $y^{\prime}$ using implicit differentiation for $9xy + y + 210 = 0$ and evaluate at $(-4, 6)$ .

Find $y^{\prime}$ using implicit differentiation for $x^{2} y - 3 x^{2} - 4 = 0$ and evaluate it at $(2,4)$ .

Find $y^{\prime}$ using implicit differentiation for $-32 e^{y}=x^{5}+y^{3}$ and evaluate at $(-2,0)$ .

Find $y^{\prime}$ using implicit differentiation for $x^{3}+y^{5}=\ln y$ and evaluate it at $(-1,1)$ .

Identify acceptable quantum mechanics functions: $x^{2}$ and $e^{-x^{2}}$ . Plot and normalize if acceptable.

Differentiate $x^{3}-8 y^{5}=\ln y$ implicitly to find $y^{\prime}$ , then evaluate at $(2,1)$ .

Find $x^{\prime}$ for $x(t)$ from $x^{4}+t^{4} x+t^{3}-9=0$ and evaluate at $(-2,1)$ . Simplify your answer.

Analyze an object's motion based on graphs and answer:
1. Change in velocity?
2. Gaining or losing velocity?
3. Distance traveled in 8s?
4. Acceleration?
5. Position at $10 \mathrm{~s}$ ?
6. Why negative acceleration if gaining velocity?

Find the limit: $\lim _{x \rightarrow-5} \frac{x+5}{x^{3}+125}$ . If it doesn't exist, write DNE.

Find the derivative $h^{\prime}(t)$ for the function $h(t)=\frac{2}{t^{1/3}}-\frac{9}{t^{3/7}}$ .

Find $y^{\prime}$ and the slope of the tangent line to $(2x - 3y)^5 = 3y^2 - 2$ at point $(2,1)$ .

Find $y^{\prime}$ and the slope of the tangent line to $(3x - 5y)^5 = 2y^2 - 9$ at point $(3,2)$ .

A ball drops for $3 \mathrm{~s}$ . Find the roof height in meters, ignoring air resistance.

Find the slope of secant line $PQ$ for $P(-3,0)$ and $Q(x, \sin(3 \pi x))$ at specified $x$ values. Then estimate the tangent slope and find the tangent line's equation.

Find the slope of secant line $PQ$ for $P(-3,0)$ and $Q(x, \sin(3\pi x))$ at given $x$ values. Then, guess the tangent slope and find the tangent line equation at $P$ .

Find $y^{\prime}$ and the slope at $(3,3)$ for the equation $\sqrt{72+y^{2}}-x^{3}+18=0$ .

Find $y^{\prime}$ and the slope at (3,3) for $\sqrt{40+y^{2}}-x^{3}+20=0$ .

Find the derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ for $y=5 \ln x+4 \log _{5} x$ . Type an exact answer.

Find the limits and value for the piecewise function $f(x)$ at $x=-1$ and classify the discontinuity type.

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 0^{+}} 11 x \ln x=\square$ . Enter INF, -INF, or DNE.

Evaluate the limit using continuity: $\lim _{x \rightarrow \frac{3 \pi}{2}} \sin (x+\sin (x))$ .

Find $y^{\prime}$ and the slope at the point (1,1) for the equation $13 \ln (x y)=y^{2}-1$ .

Calculate the limit: $\lim _{x \rightarrow 1} \frac{5-5 x}{1-\sqrt{x}}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{2 x+\tan (4 x)}{3 x-\tan (4 x)}$ . Enter -Inf, Inf, or DNE for your answer.

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow 1} \frac{x^{2}+4 x-5}{\ln x}=\square$

Find $y^{\prime}$ and the slope at the point (1,1) for the equation $9 \ln (x y)=y^{2}-1$ .

Evaluate the limit using continuity: $\lim _{x \rightarrow \pi} \cos (x+\cos (x))$ .

Find the smallest interval where $f(x)$ has a root using the Intermediate Value Theorem from the values of $f$ given.

Find the initial population of arctic flounder modeled by $P(t)=\frac{12 t+240}{0.7 t^{2}+4}$ and calculate $P^{\prime}(8)$ .

Evaluate the limit: $\lim _{t \rightarrow 0} \frac{t^{3} \cos (3 t)}{4 t-3}$ . If it doesn't exist, enter DNE.

Find the rate of change of price $p$ with respect to demand $x$ for $x=\sqrt[3]{10,000-p^{4}}$ by implicit differentiation.

Calculate the limit using L'Hospital's Rule if needed: $\lim _{x \rightarrow 9} \frac{x-9}{x^{2}-15 x+54}=\square$

Find the limit as $x$ approaches negative infinity: $\lim _{x \rightarrow-\infty} \frac{3 x^{2}+1}{x^{2}+5}=\square$

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

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{x^{2}-1}{x+5}.$

Find the limit: $\lim _{x \rightarrow 9} \frac{x-9}{\sqrt{x}-3}$ . If it doesn't exist, write DNE.

Given the piecewise function $f(x)$ , find the limits as $x$ approaches 2 from the right and left, and values of $m$ for equality of these limits.

Evaluate the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \infty} \frac{-7 x-6}{-2 x-6}=\square$

Find $g^{\prime}(t)$ using the Chain Rule for $g(t)=f(x(t), y(t))$ , where $f(x, y)=x^{2} y-\sin y$ , $x(t)=\sqrt{t^{2}+1}$ , $y(t)=e^{t}$ .