Calculus

Problem 19701

Find the interval of $x$ where $f(x) = e^{\sin x} - 2\cos(3x)$ is concave down for $0 \leq x \leq 4$ . Options:
(A) $0 \leq x \leq 0.283$
(B) $0.605 \leq x \leq 1.571$ and $2.536 \leq x \leq 3.693$
(C) $0.283 \leq x \leq 3.082$
(D) $1.108 \leq x \leq 2.166$ and $3.082 \leq x \leq 4$
(E) $0 \leq x \leq 1.108$ and $2.166 \leq x \leq 3.082$

See Solution

Problem 19702

Compare $\Delta y$ and $d y$ for $f(x)=x^{4}-x+6$ as $x$ changes from 1 to 1.05 and 1 to 1.01. Round to four decimals.

See Solution

Problem 19703

Approximate the cubic $f(y)=y(y-1)(3-y)$ near steady states. Find linearized equations for $y(0)=y_0$ close to $a$ . Plot solutions for $y(0)=a-0.25, a, a+0.25$ and compare with original equation's solutions.

See Solution

Problem 19704

Is the series $10 - 4 + 1.6 - 0.64 + \cdots$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 19705

Is the series $\sum_{k=1}^{\infty} k e^{-k^{2}}$ convergent or divergent?

See Solution

Problem 19706

Is the series $\sum_{n=1}^{\infty} \frac{6}{n^{2}+n^{3}}$ convergent?

See Solution

Problem 19707

Is the series $\sum_{n=1}^{\infty} \frac{7}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 19708

Determine if the sequence $a_{n}=\frac{8 n}{4 n+1}$ converges.

See Solution

Problem 19709

Is the series $\sum_{n=1}^{\infty} \frac{8n}{4n+1}$ convergent or divergent?

See Solution

Problem 19710

Evaluate the integral from 0 to 1: $\int_{0}^{1} \frac{r^{3}}{\sqrt{25+r^{2}}} d r$

See Solution

Problem 19711

Find $F^{\prime}(x)$ using the Second Fundamental Theorem of Calculus for $F(x)=\int_{1}^{x} \frac{1}{t^{4}} dt$ .

See Solution

Problem 19712

Evaluate the integral from 0 to 3 of $\frac{x e^{x}}{(1+x)^{2}}$ dx.

See Solution

Problem 19713

Find the limit using l'Hospital's Rule or a simpler method: $\lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25}$

See Solution

Problem 19714

Evaluate if the following limits are indeterminate forms given: $\lim_{x \to a} f(x) = 0$ , $\lim_{x \to a} g(x) = 0$ , $\lim_{x \to a} h(x) = 1$ , $\lim_{x \to a} p(x) = \infty$ , $\lim_{x \to a} q(x) = \infty$ . (a) $\lim_{x \to a}[f(x) p(x)]$ (b) $\lim_{x \to a}[h(x) p(x)]$ (c) $\lim_{x \to a}[p(x) q(x)]$

See Solution

Problem 19715

Find the antiderivative $F$ of $f$ with $F(1)=0$ . Check by comparing the graphs of $f$ and $F$ . $f(x)=5-3(1+x^{2})^{-1}$

See Solution

Problem 19716

Evaluate the integral: $\int 7 \sin (8 x) \cos (5 x) d x$ , using the constant $C$ .

See Solution

Problem 19717

Berechne die Arbeit, um 100 kg um 10 km anzuheben, und finde eine Formel für die Arbeit W in Joule für Masse m und Höhe x.

See Solution

Problem 19718

Given $f^{\prime \prime}$ is continuous, determine local extrema for $f$ at $x=-3$ and $x=5$ based on $f^{\prime}$ and $f^{\prime \prime}$ .

See Solution

Problem 19719

Find the function $f$ such that $f^{\prime \prime}(x)=8 x^{3}+5$ , with conditions $f(1)=9$ and $f^{\prime}(1)=4$ .

See Solution

Problem 19720

Evaluate the integral from 0 to π/2: $\int_{0}^{\pi / 2} \cos (9 t) \cos (18 t) d t$ .

See Solution

Problem 19721

A sky diver falls for 5 seconds, increasing velocity by $49 \mathrm{~m/s}$ . What is the acceleration rate?

See Solution

Problem 19722

Evaluate the integral: $\int_{0}^{3} \frac{x e^{x}}{(1+x)^{2}} d x$

See Solution

Problem 19723

Evaluate the integrals given $\int_{4}^{8} f(x) dx=15$ and $\int_{4}^{8} g(x) dx=4$ for: (a) $\int_{4}^{8}[f(x)+g(x)] dx$ (b) $\int_{4}^{8}[f(x)-g(x)] dx$ (c) $\int_{4}^{8}[2 f(x)-3 g(x)] dx$ (d) $\int_{4}^{8} 8 f(x) dx$

See Solution

Problem 19724

Find the average value of $f(x)=\frac{1}{\sqrt{x}}$ over $[9,16]$ and where $f(x)$ equals that average. $x=$

See Solution

Problem 19725

Find the derivative of the function $f(x)$ defined as $f(x)=\begin{cases}(x-1)^{3} & x \leq 1 \\ (x-1)^{2} & x>1\end{cases}$ at $x=1$ .

See Solution

Problem 19726

Berechne die erste und zweite Ableitung für die Funktionen: a) $f(x)=x^{3}+3 x^{2}-17 x+1$ , b) $f(x)=x(3 x^{2}-1)$ , c) $f(x)=\sqrt[4]{x}-2 x$ , d) $f(x)=\frac{2}{x}+1$ , e) $f(x)=\frac{1}{3 \sqrt{x}}+x$ , f) $f(x)=x^{\frac{1}{3}}+2 \sin (x)$ .

See Solution

Problem 19727

Find the rate of volume growth of a watermelon rind after 5 weeks, with radius growth at 2 cm/week and rind thickness as 0.1 $r$ .

See Solution

Problem 19728

Evaluate the integral $\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x$ using the substitution $x=4 \tan (\theta)$ .

See Solution

Problem 19729

Is the function $f(x) = |x-1|$ differentiable at the point $x=1$ ?

See Solution

Problem 19730

Find the water rise rate in an inverted cone (depth 60 ft, diameter 30 ft) filled at 17 ft³/min when water is 1 ft deep.

See Solution

Problem 19731

Evaluate the integral $\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x$ with the substitution $x=4 \tan (\theta)$ .

See Solution

Problem 19732

Find the integral of $\frac{1}{x(x-a)}$ with respect to $x$ .

See Solution

Problem 19733

Evaluate the integral: $\int \frac{x^{2}-x+18}{x^{3}+3 x} \, dx$ (include absolute values and use $C$ for integration constant).

See Solution

Problem 19734

Evaluate the integral from 0 to 1 of $\frac{x^{3}+3 x}{x^{4}+6 x^{2}+2} \, dx$ .

See Solution

Problem 19735

Approximate $\sqrt{154}$ using $\Delta y \approx f^{\prime}(x) \Delta x$ . What is the rounded value? $\sqrt{154} \approx \square$ (3 decimals)

See Solution

Problem 19736

Find the inflection point (x, y) for the function $f(x)=\frac{e^{x}}{2+e^{x}}$ .

See Solution

Problem 19737

Find the derivative of $y=\sqrt{\frac{x(x+2)}{(2 x+1)(3 x+2)}}$ using logarithmic differentiation.

See Solution

Problem 19738

Find the derivative of the function $f(x)=x^{3}+3^{x}$ .

See Solution

Problem 19739

Determine when the inflection point of $c(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}$ occurs, given $\beta>\alpha>0$ .

See Solution

Problem 19740

Find $d y$ and $\Delta y$ for $y=f(x)=x^{3}-9 x^{2}+7$ at $x=5$ and $\Delta x=-0.1$ . $d y=\square$

See Solution

Problem 19741

Find $y(9)$ given $y^{\prime}(x)=\frac{3}{\sqrt{x}}$ , $y(16)=35$ .

See Solution

Problem 19742

Find the derivative of $f(x)=x^{e^{x}}$ using logarithmic differentiation.

See Solution

Problem 19743

Find the derivative of $f(x)=x^{3^{x}}$ using logarithmic differentiation.

See Solution

Problem 19744

Differentiate $x^{2} y^{3}$ with respect to $x$ , where $y$ is a function of $x$ .

See Solution

Problem 19745

Find the derivative $f^{\prime}(x)$ if $f(x)=(-3 x+3)(-x+4)$ . Choose from: (a) $6 x+9$ , (b) $6 x-15$ , (c) $4 x+11$ , (d) $2 x^{2}+5 x-17$ .

See Solution

Problem 19746

Find the derivative of $f(x) = x^{3} R^{5}$ with respect to $x$ .

See Solution

Problem 19747

Find the derivative of the function $f(x)=\frac{x-1}{x+2}$ .

See Solution

Problem 19748

Substitute to turn the integrand into a rational function and evaluate the integral: $\int \frac{1}{x-7 \sqrt{x}+12} d x$ .

See Solution

Problem 19749

Evaluate the integral $\int_{1}^{49} \frac{3+2 \sqrt{t}}{\sqrt{t}} d t=\square($ Type an exact answer.)

See Solution

Problem 19750

Determine the length of the curve $y=\frac{1}{4} x^{2}-\frac{1}{2} \ln (x)$ for $1 \leq x \leq 6$ .

See Solution

Problem 19751

Find the initial amount of a radioactive isotope given by $A(t)=500 e^{-0.02839 t}$ and its half-life.

See Solution

Problem 19752

Which statements are true?
1. Constant positive velocity means distance = velocity × time.
2. Positive but decreasing velocity overestimates distance. Choose: a. Neither b. Both c. Just 1 d. Just 2

See Solution

Problem 19753

Given $A(t)=700 e^{-0.02839 t}$ , find the initial amount and the half-life of the isotope.

See Solution

Problem 19754

Find the area of the surface formed by rotating $y=\cos \left(\frac{1}{4} x\right)$ , $0 \leq x \leq 2\pi$ , around the $x$ -axis.

See Solution

Problem 19755

Given the function $f(x)=\frac{e^{x}}{6+e^{x}}$ , find $f^{\prime}(x)$ , intervals of increase/decrease, local min/max, $f^{\prime \prime}(x)$ , and concavity.

See Solution

Problem 19756

Calculate the curve length for $x=\frac{1}{3} \sqrt{y}(y-3)$ from $y=1$ to $y=9$ .

See Solution

Problem 19757

Given the function $f(x)=e^{-3.5 x^{2}}$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

See Solution

Problem 19758

Find the surface area when the curve $y=\frac{1}{3} x^{3/2}$ , for $0 \leq x \leq 12$ , is rotated about the $y$ -axis.

See Solution

Problem 19759

Find the area of the surface formed by rotating the curve $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=1$ around the $x$ -axis.

See Solution

Problem 19760

A turkey cools from $185^{\circ}$ F to $140^{\circ}$ F in 30 min.
(a) Find its temp after 45 min.
(b) When will it reach $100^{\circ}$ F?

See Solution

Problem 19761

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2 x^{3}+x \cos x-\sin x}{x^{2}}$ .

See Solution

Problem 19762

Evaluate the limit $\lim _{t \rightarrow 0}(11 \sin (12 t) \ln (12 t))=\square$ using L'Hôpital's Rule.

See Solution

Problem 19763

Given the function $f(x)=3 x^{2} \ln (x)$ for $x>0$ , find:
(A) Critical values. (B) Intervals where $f(x)$ is increasing. (C) Intervals where $f(x)$ is decreasing. (D) $x$ values of local maxima. (E) $x$ values of local minima. (F) Intervals where $f(x)$ is concave up. (G) Intervals where $f(x)$ is concave down. (H) $x$ values of inflection points.

See Solution

Problem 19764

Find the limit: $\lim _{x \rightarrow-\infty}\left(\frac{3 x-4}{\sqrt{x^{2}+2}}\right)$

See Solution

Problem 19765

Show algebraically that as $x$ approaches 0, the slope of the tangent line $h^{\prime}(x)=\frac{2 x^{3}+x \cdot \cos x-\sin x}{x^{2}}$ approaches 0.

See Solution

Problem 19766

Find the limit as $x$ approaches 0 from the right of $(\sqrt[4]{x}-1)$ . What is the value?

See Solution

Problem 19767

Find the limit: $\lim _{x \rightarrow 49} \frac{x-49}{\sqrt{x}-7}$ using algebraic methods.

See Solution

Problem 19768

Find the limit as $x$ approaches infinity for $\frac{5+2x}{3-x}$ . Options: a) 0, b) $0-2$ , c) $\frac{5}{3}$ , d) 0, Does not exist.

See Solution

Problem 19769

Find $\lim _{x \rightarrow 0^{-}}(\sqrt[4]{x}-3)$ . What is the limit? a) 00 b) 01 c) -1 d) Does not exist

See Solution

Problem 19770

Calculate limits for the speed $v$ of a dropped object: (a) $\lim_{t \to \infty} v$ , (b) $\lim_{m \to \infty} v$ using L'Hospital's rule.

See Solution

Problem 19771

Find the limit as $x$ approaches -1.5 from the right of $\sqrt{3+2x} + x$ . Options: a) 1.5 b) $0-1.5$ c) 00 d) 1

See Solution

Problem 19772

Find the bacteria population after 3 hours if it grows at $16\%$ per hour from an initial count of 129. Round to the nearest tenth.

See Solution

Problem 19773

As $x$ approaches 0.5, find the limit of $f(x)=\frac{1}{x^{2}}$ . Choices: a) 16 b) 04 c) 0.025 d) 0.25

See Solution

Problem 19774

Find the limit as $x$ approaches $-\infty$ for $\mathbf{f}(x) = \frac{x+4}{x^2+16}$ . Options: a) $\frac{1}{4}$ b) $0-\frac{1}{4}$ c) 0 d) Does not exist.

See Solution

Problem 19775

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\frac{7 x+5}{6 x^{2}-4 x+9}\right)$

See Solution

Problem 19776

Find the limit: $\lim _{x \rightarrow \infty}\left[\frac{6 x^{2}-5 x+7}{2 x^{2}+9}\right]$ algebraically.

See Solution

Problem 19777

Find the indefinite integral and verify by differentiation: $\int\left(\sqrt{x}+\frac{1}{6 \sqrt{x}}\right) d x$

See Solution

Problem 19778

Find the limit as $x$ approaches 3 from the right for $\frac{|x-3|}{x-3}$ . Options: a) 01 b) $0-1$ c) $0-2$ d) 00

See Solution

Problem 19779

Is the function $f(x)=\left\{\begin{array}{c}\frac{x^{2}-4}{x+2}, x \neq 1 \\ -1, x=1\end{array}\right.$ continuous at $x=1$ ?

See Solution

Problem 19780

Find $f(-1)$ given that $f^{\prime}(x)=8 x^{3}+12 x+2$ and $f(1)=-4$ .

See Solution

Problem 19781

The substitution $u=3 x$ changes the integral $\int e^{3 x} d x$ . What is the new integral?

See Solution

Problem 19782

Evaluate the integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ .

See Solution

Problem 19783

Calculate the integral: $\int_{\frac{1}{\sqrt{e}}}^{e^{2}} \frac{3 \sqrt{5+2 \ln x}}{x} d x$

See Solution

Problem 19784

A biologist models bacteria population $P(t)=\frac{t+2}{t^{2}+t+3}$ . Find rates of change at $t=0$ and $t=10$ .

See Solution

Problem 19785

Given the function $f(x)=x^{2/5}(x-5)$ , find critical numbers $A$ and $B$ . Determine if $f(x)$ is increasing or decreasing in the intervals $(-\infty, A]$ , $[A, B]$ , and $[B, \infty)$ . Also, find $C$ and $D$ where $f''(x)=0$ or undefined, and check concavity in $(-\infty, C)$ , $(C, D)$ , and $(D, \infty)$ .

See Solution

Problem 19786

Determine if $h(\theta)=-3 \cos \frac{\theta}{2}$ has local extrema at $\theta=0$ and $\theta=2\pi$ .

See Solution

Problem 19787

Find the derivative of $f(x)=\frac{6 x^{2}}{x^{2}-25}$ , its critical numbers, intervals of increase/decrease, and local extrema.

See Solution

Problem 19788

Evaluate the integral $\int \frac{\sec ^{2}(1 / x^{3})}{x^{4}} dx$ using the substitution $u=1/x^{3}$ .

See Solution

Problem 19789

Find the value for the blank in $\int_{0}^{6}(\square s$ given $\mathbf{F}(\mathbf{G}(r, s)) \cdot \mathbf{N}(r, s) = -e^{r}+r^{2}(1-s)+r s^{2}$ .

See Solution

Problem 19790

Calculate the area under the curve $xy = -1$ from $x = 1$ to $x = 2$ .

See Solution

Problem 19791

Rewrite the equation $du=-\frac{9}{x^{10}} dx$ to show it as $-1/9 du$ .

See Solution

Problem 19792

Evaluate the integral: $\int_{1/2}^{1} \frac{3x \, dx}{1 + 4x^2}$ .

See Solution

Problem 19793

Find the zero(s) where the second derivative $f^{\prime \prime}$ of the polynomial $f(x)=(x+4)^{2}(x-1)^{5}(x-2)$ flattens out. Express as ordered pair(s).

See Solution

Problem 19794

Find the derivative of $18s^2 + \frac{216(1-s)}{3} - e^6 + 1$ and evaluate $\int_{0}^{6} (18s^2 + \frac{216(1-s)}{3} - e^6 + 1) ds$ .

See Solution

Problem 19795

Find and classify all critical points of the function $f(x, y)=9 e^{-\left(x^{2}+y^{2}-2 x\right)}$ .

See Solution

Problem 19796

Find the antiderivative of $18s^2 + \frac{216(1-s)}{3} - e^{6} + 1$ from $0$ to $6$ and evaluate it:
$\int_{0}^{6}\left(18 s^{2}+\frac{216(1-s)}{3}-e^{6}+1\right) ds = \square_{X}$

See Solution

Problem 19797

Find the first three nonzero terms in the Maclaurin series for $y=e^{x} \ln(1+x)$ using power series operations.

See Solution

Problem 19798

Evaluate the series: $3+\frac{9}{2!}+\frac{27}{3!}+\frac{81}{4!}+\cdots$

See Solution

Problem 19799

Calculate total cost for $q=20$ using marginal cost function $0.12 q^{2}-1.60 q+6.70$ and fixed costs of \{8000\}. Total cost is \$\square.

See Solution

Problem 19800

Find y given: $y^{\prime \prime \prime}=30 x$ , $y^{\prime \prime}(0)=8$ , $y^{\prime}(0)=8$ , $y(0)=5$ . $y(x)=\square$

See Solution

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Calculus

Problem 19701

Problem 19702

Compare Δy\Delta yΔy and dyd ydy for f(x)=x4−x+6f(x)=x^{4}-x+6f(x)=x4−x+6 as xxx changes from 1 to 1.05 and 1 to 1.01. Round to four decimals.

Problem 19703

Problem 19704

Is the series 10−4+1.6−0.64+⋯10 - 4 + 1.6 - 0.64 + \cdots10−4+1.6−0.64+⋯ convergent or divergent? If convergent, find the sum.

Problem 19705

Is the series ∑k=1∞ke−k2\sum_{k=1}^{\infty} k e^{-k^{2}}k=1∑∞​ke−k2 convergent or divergent?

Problem 19706

Is the series ∑n=1∞6n2+n3\sum_{n=1}^{\infty} \frac{6}{n^{2}+n^{3}}n=1∑∞​n2+n36​ convergent?

Problem 19707

Is the series ∑n=1∞7πn\sum_{n=1}^{\infty} \frac{7}{\pi^{n}}n=1∑∞​πn7​ convergent or divergent? If convergent, find the sum.

Problem 19708

Determine if the sequence an=8n4n+1a_{n}=\frac{8 n}{4 n+1}an​=4n+18n​ converges.

Problem 19709

Is the series ∑n=1∞8n4n+1\sum_{n=1}^{\infty} \frac{8n}{4n+1}∑n=1∞​4n+18n​ convergent or divergent?

Problem 19710

Evaluate the integral from 0 to 1: ∫01r325+r2dr\int_{0}^{1} \frac{r^{3}}{\sqrt{25+r^{2}}} d r∫01​25+r2​r3​dr

Problem 19711

Find F′(x)F^{\prime}(x)F′(x) using the Second Fundamental Theorem of Calculus for F(x)=∫1x1t4dtF(x)=\int_{1}^{x} \frac{1}{t^{4}} dtF(x)=∫1x​t41​dt.

Problem 19712

Evaluate the integral from 0 to 3 of xex(1+x)2\frac{x e^{x}}{(1+x)^{2}}(1+x)2xex​ dx.

Problem 19713

Find the limit using l'Hospital's Rule or a simpler method: lim⁡x→5x−5x2−25\lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25}x→5lim​x2−25x−5​

Problem 19714

Problem 19715

Find the antiderivative FFF of fff with F(1)=0F(1)=0F(1)=0. Check by comparing the graphs of fff and FFF. f(x)=5−3(1+x2)−1f(x)=5-3(1+x^{2})^{-1}f(x)=5−3(1+x2)−1

Problem 19716

Evaluate the integral: ∫7sin⁡(8x)cos⁡(5x)dx\int 7 \sin (8 x) \cos (5 x) d x∫7sin(8x)cos(5x)dx, using the constant CCC.

Problem 19717

Berechne die Arbeit, um 100 kg um 10 km anzuheben, und finde eine Formel für die Arbeit W in Joule für Masse m und Höhe x.

Problem 19718

Given f′′f^{\prime \prime}f′′ is continuous, determine local extrema for fff at x=−3x=-3x=−3 and x=5x=5x=5 based on f′f^{\prime}f′ and f′′f^{\prime \prime}f′′.

Problem 19719

Find the function fff such that f′′(x)=8x3+5f^{\prime \prime}(x)=8 x^{3}+5f′′(x)=8x3+5, with conditions f(1)=9f(1)=9f(1)=9 and f′(1)=4f^{\prime}(1)=4f′(1)=4.

Problem 19720

Evaluate the integral from 0 to π/2: ∫0π/2cos⁡(9t)cos⁡(18t)dt\int_{0}^{\pi / 2} \cos (9 t) \cos (18 t) d t∫0π/2​cos(9t)cos(18t)dt.

Problem 19721

A sky diver falls for 5 seconds, increasing velocity by 49 m/s49 \mathrm{~m/s}49 m/s. What is the acceleration rate?

Problem 19722

Evaluate the integral: ∫03xex(1+x)2dx\int_{0}^{3} \frac{x e^{x}}{(1+x)^{2}} d x∫03​(1+x)2xex​dx

Problem 19723

Problem 19724

Find the average value of f(x)=1xf(x)=\frac{1}{\sqrt{x}}f(x)=x​1​ over [9,16][9,16][9,16] and where f(x)f(x)f(x) equals that average. x=x=x=

Problem 19725

Find the derivative of the function f(x)f(x)f(x) defined as f(x)={(x−1)3x≤1(x−1)2x>1f(x)=\begin{cases}(x-1)^{3} & x \leq 1 \\ (x-1)^{2} & x>1\end{cases}f(x)={(x−1)3(x−1)2​x≤1x>1​ at x=1x=1x=1.

Problem 19726

Problem 19727

Find the rate of volume growth of a watermelon rind after 5 weeks, with radius growth at 2 cm/week and rind thickness as 0.1rrr.

Problem 19728

Evaluate the integral ∫x316+x2dx\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x∫16+x2​x3​dx using the substitution x=4tan⁡(θ)x=4 \tan (\theta)x=4tan(θ).

Problem 19729

Is the function f(x)=∣x−1∣f(x) = |x-1|f(x)=∣x−1∣ differentiable at the point x=1x=1x=1?

Problem 19730

Find the water rise rate in an inverted cone (depth 60 ft, diameter 30 ft) filled at 17 ft³/min when water is 1 ft deep.

Problem 19731

Evaluate the integral ∫x316+x2dx\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x∫16+x2​x3​dx with the substitution x=4tan⁡(θ)x=4 \tan (\theta)x=4tan(θ).

Problem 19732

Find the integral of 1x(x−a)\frac{1}{x(x-a)}x(x−a)1​ with respect to xxx.

Problem 19733

Evaluate the integral: ∫x2−x+18x3+3x dx\int \frac{x^{2}-x+18}{x^{3}+3 x} \, dx∫x3+3xx2−x+18​dx (include absolute values and use CCC for integration constant).

Problem 19734

Evaluate the integral from 0 to 1 of x3+3xx4+6x2+2 dx\frac{x^{3}+3 x}{x^{4}+6 x^{2}+2} \, dxx4+6x2+2x3+3x​dx.

Problem 19735

Approximate 154\sqrt{154}154​ using Δy≈f′(x)Δx\Delta y \approx f^{\prime}(x) \Delta xΔy≈f′(x)Δx. What is the rounded value? 154≈□\sqrt{154} \approx \square154​≈□ (3 decimals)

Problem 19736

Find the inflection point (x, y) for the function f(x)=ex2+exf(x)=\frac{e^{x}}{2+e^{x}}f(x)=2+exex​.

Problem 19737

Find the derivative of y=x(x+2)(2x+1)(3x+2)y=\sqrt{\frac{x(x+2)}{(2 x+1)(3 x+2)}}y=(2x+1)(3x+2)x(x+2)​​ using logarithmic differentiation.

Problem 19738

Find the derivative of the function f(x)=x3+3xf(x)=x^{3}+3^{x}f(x)=x3+3x.

Problem 19739

Determine when the inflection point of c(t)=e−αt−e−βtβ−αc(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}c(t)=β−αe−αt−e−βt​ occurs, given β>α>0\beta>\alpha>0β>α>0.

Problem 19740

Find dyd ydy and Δy\Delta yΔy for y=f(x)=x3−9x2+7y=f(x)=x^{3}-9 x^{2}+7y=f(x)=x3−9x2+7 at x=5x=5x=5 and Δx=−0.1\Delta x=-0.1Δx=−0.1. dy=□d y=\squaredy=□

Problem 19741

Find y(9)y(9)y(9) given y′(x)=3xy^{\prime}(x)=\frac{3}{\sqrt{x}}y′(x)=x​3​, y(16)=35y(16)=35y(16)=35.

Problem 19742

Find the derivative of f(x)=xexf(x)=x^{e^{x}}f(x)=xex using logarithmic differentiation.

Compare $\Delta y$ and $d y$ for $f(x)=x^{4}-x+6$ as $x$ changes from 1 to 1.05 and 1 to 1.01. Round to four decimals.

Is the series $10 - 4 + 1.6 - 0.64 + \cdots$ convergent or divergent? If convergent, find the sum.

Is the series $\sum_{k=1}^{\infty} k e^{-k^{2}}$ convergent or divergent?

Is the series $\sum_{n=1}^{\infty} \frac{6}{n^{2}+n^{3}}$ convergent?

Is the series $\sum_{n=1}^{\infty} \frac{7}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

Determine if the sequence $a_{n}=\frac{8 n}{4 n+1}$ converges.

Is the series $\sum_{n=1}^{\infty} \frac{8n}{4n+1}$ convergent or divergent?

Evaluate the integral from 0 to 1: $\int_{0}^{1} \frac{r^{3}}{\sqrt{25+r^{2}}} d r$

Find $F^{\prime}(x)$ using the Second Fundamental Theorem of Calculus for $F(x)=\int_{1}^{x} \frac{1}{t^{4}} dt$ .

Evaluate the integral from 0 to 3 of $\frac{x e^{x}}{(1+x)^{2}}$ dx.

Find the limit using l'Hospital's Rule or a simpler method: $\lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25}$

Find the antiderivative $F$ of $f$ with $F(1)=0$ . Check by comparing the graphs of $f$ and $F$ . $f(x)=5-3(1+x^{2})^{-1}$

Evaluate the integral: $\int 7 \sin (8 x) \cos (5 x) d x$ , using the constant $C$ .

Given $f^{\prime \prime}$ is continuous, determine local extrema for $f$ at $x=-3$ and $x=5$ based on $f^{\prime}$ and $f^{\prime \prime}$ .

Find the function $f$ such that $f^{\prime \prime}(x)=8 x^{3}+5$ , with conditions $f(1)=9$ and $f^{\prime}(1)=4$ .

Evaluate the integral from 0 to π/2: $\int_{0}^{\pi / 2} \cos (9 t) \cos (18 t) d t$ .

A sky diver falls for 5 seconds, increasing velocity by $49 \mathrm{~m/s}$ . What is the acceleration rate?

Evaluate the integral: $\int_{0}^{3} \frac{x e^{x}}{(1+x)^{2}} d x$

Find the average value of $f(x)=\frac{1}{\sqrt{x}}$ over $[9,16]$ and where $f(x)$ equals that average. $x=$

Find the derivative of the function $f(x)$ defined as $f(x)=\begin{cases}(x-1)^{3} & x \leq 1 \\ (x-1)^{2} & x>1\end{cases}$ at $x=1$ .

Find the rate of volume growth of a watermelon rind after 5 weeks, with radius growth at 2 cm/week and rind thickness as 0.1 $r$ .

Evaluate the integral $\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x$ using the substitution $x=4 \tan (\theta)$ .

Is the function $f(x) = |x-1|$ differentiable at the point $x=1$ ?

Evaluate the integral $\int \frac{x^{3}}{\sqrt{16+x^{2}}} d x$ with the substitution $x=4 \tan (\theta)$ .

Find the integral of $\frac{1}{x(x-a)}$ with respect to $x$ .

Evaluate the integral: $\int \frac{x^{2}-x+18}{x^{3}+3 x} \, dx$ (include absolute values and use $C$ for integration constant).

Evaluate the integral from 0 to 1 of $\frac{x^{3}+3 x}{x^{4}+6 x^{2}+2} \, dx$ .

Approximate $\sqrt{154}$ using $\Delta y \approx f^{\prime}(x) \Delta x$ . What is the rounded value? $\sqrt{154} \approx \square$ (3 decimals)

Find the inflection point (x, y) for the function $f(x)=\frac{e^{x}}{2+e^{x}}$ .

Find the derivative of $y=\sqrt{\frac{x(x+2)}{(2 x+1)(3 x+2)}}$ using logarithmic differentiation.

Find the derivative of the function $f(x)=x^{3}+3^{x}$ .

Determine when the inflection point of $c(t)=\frac{e^{-\alpha t}-e^{-\beta t}}{\beta-\alpha}$ occurs, given $\beta>\alpha>0$ .

Find $d y$ and $\Delta y$ for $y=f(x)=x^{3}-9 x^{2}+7$ at $x=5$ and $\Delta x=-0.1$ . $d y=\square$

Find $y(9)$ given $y^{\prime}(x)=\frac{3}{\sqrt{x}}$ , $y(16)=35$ .

Find the derivative of $f(x)=x^{e^{x}}$ using logarithmic differentiation.

Find the derivative of $f(x)=x^{3^{x}}$ using logarithmic differentiation.

Differentiate $x^{2} y^{3}$ with respect to $x$ , where $y$ is a function of $x$ .

Find the derivative $f^{\prime}(x)$ if $f(x)=(-3 x+3)(-x+4)$ . Choose from: (a) $6 x+9$ , (b) $6 x-15$ , (c) $4 x+11$ , (d) $2 x^{2}+5 x-17$ .

Find the derivative of $f(x) = x^{3} R^{5}$ with respect to $x$ .

Find the derivative of the function $f(x)=\frac{x-1}{x+2}$ .

Substitute to turn the integrand into a rational function and evaluate the integral: $\int \frac{1}{x-7 \sqrt{x}+12} d x$ .

Evaluate the integral $\int_{1}^{49} \frac{3+2 \sqrt{t}}{\sqrt{t}} d t=\square($ Type an exact answer.)

Determine the length of the curve $y=\frac{1}{4} x^{2}-\frac{1}{2} \ln (x)$ for $1 \leq x \leq 6$ .

Find the initial amount of a radioactive isotope given by $A(t)=500 e^{-0.02839 t}$ and its half-life.

Which statements are true?
1. Constant positive velocity means distance = velocity × time.
2. Positive but decreasing velocity overestimates distance. Choose: a. Neither b. Both c. Just 1 d. Just 2

Given $A(t)=700 e^{-0.02839 t}$ , find the initial amount and the half-life of the isotope.

Find the area of the surface formed by rotating $y=\cos \left(\frac{1}{4} x\right)$ , $0 \leq x \leq 2\pi$ , around the $x$ -axis.

Given the function $f(x)=\frac{e^{x}}{6+e^{x}}$ , find $f^{\prime}(x)$ , intervals of increase/decrease, local min/max, $f^{\prime \prime}(x)$ , and concavity.

Calculate the curve length for $x=\frac{1}{3} \sqrt{y}(y-3)$ from $y=1$ to $y=9$ .

Given the function $f(x)=e^{-3.5 x^{2}}$ , find critical values, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

Find the surface area when the curve $y=\frac{1}{3} x^{3/2}$ , for $0 \leq x \leq 12$ , is rotated about the $y$ -axis.

Find the area of the surface formed by rotating the curve $y=\frac{x^{3}}{6}+\frac{1}{2 x}$ from $x=\frac{1}{2}$ to $x=1$ around the $x$ -axis.

A turkey cools from $185^{\circ}$ F to $140^{\circ}$ F in 30 min.
(a) Find its temp after 45 min.
(b) When will it reach $100^{\circ}$ F?

Calculate the limit: $\lim _{x \rightarrow 0} \frac{2 x^{3}+x \cos x-\sin x}{x^{2}}$ .

Evaluate the limit $\lim _{t \rightarrow 0}(11 \sin (12 t) \ln (12 t))=\square$ using L'Hôpital's Rule.

Find the limit: $\lim _{x \rightarrow-\infty}\left(\frac{3 x-4}{\sqrt{x^{2}+2}}\right)$

Show algebraically that as $x$ approaches 0, the slope of the tangent line $h^{\prime}(x)=\frac{2 x^{3}+x \cdot \cos x-\sin x}{x^{2}}$ approaches 0.

Find the limit as $x$ approaches 0 from the right of $(\sqrt[4]{x}-1)$ . What is the value?

Find the limit: $\lim _{x \rightarrow 49} \frac{x-49}{\sqrt{x}-7}$ using algebraic methods.

Find the limit as $x$ approaches infinity for $\frac{5+2x}{3-x}$ . Options: a) 0, b) $0-2$ , c) $\frac{5}{3}$ , d) 0, Does not exist.

Find $\lim _{x \rightarrow 0^{-}}(\sqrt[4]{x}-3)$ . What is the limit? a) 00 b) 01 c) -1 d) Does not exist

Calculate limits for the speed $v$ of a dropped object: (a) $\lim_{t \to \infty} v$ , (b) $\lim_{m \to \infty} v$ using L'Hospital's rule.

Find the limit as $x$ approaches -1.5 from the right of $\sqrt{3+2x} + x$ . Options: a) 1.5 b) $0-1.5$ c) 00 d) 1

Find the bacteria population after 3 hours if it grows at $16\%$ per hour from an initial count of 129. Round to the nearest tenth.

As $x$ approaches 0.5, find the limit of $f(x)=\frac{1}{x^{2}}$ . Choices: a) 16 b) 04 c) 0.025 d) 0.25

Find the limit as $x$ approaches $-\infty$ for $\mathbf{f}(x) = \frac{x+4}{x^2+16}$ . Options: a) $\frac{1}{4}$ b) $0-\frac{1}{4}$ c) 0 d) Does not exist.