Calculus

Problem 19001

Find the indefinite integral of $g(x) = \sqrt[3]{x}$ .

See Solution

Problem 19002

Evaluate the integral: $\int_{0}^{1} x(\sqrt[3]{x}+\sqrt[5]{x}) d x=\square-\square=\square$ . Enter simplified fractions only.

See Solution

Problem 19003

Estimate $f(5.2)$ using the linear approximation given $f(5)=5.1$ and $f'(5)=3.2$ . Provide three decimal places.

See Solution

Problem 19004

A mosquito colony grows uninhibitedly. If $\mathrm{N}(t)=N_{0} \cdot e^{kt}$ , find $\mathrm{N}(2)$ given $\mathrm{N}_{0}=1000$ , $\mathrm{N}(1)=1500$ .

See Solution

Problem 19005

True or false: The linear approximation of $f(x)=\sin (x)$ at $x=\frac{\pi}{2}$ is $L(x)=1+\cos (x)(x-\frac{\pi}{2})$ .

See Solution

Problem 19006

Melanie invests \$7600 at 4.0\% annual interest, compounded continuously. What is the investment value after 2 years?

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Problem 19007

Calculate the integral $\int_{0}^{\pi / 3} \frac{\tan (x)}{\cos (x)} d x=\square-\square=\square x$ .

See Solution

Problem 19008

Find the absolute minimum and maximum of the function $f(x)=\frac{x^{2}-64}{x^{2}+64}$ on the interval $[-64,64]$ .

See Solution

Problem 19009

True or false: If $f$ is differentiable and $f'(c)=0$ , does $f$ have a local max or min at $x=c$ ? True or False?

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Problem 19010

True or false: The linear approximation of $f$ at $x=a$ equals the tangent line to $y=f(x)$ at $(a, f(a))$ . Explain.

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Problem 19011

True or false: If $f$ is continuous on $[a, b]$ , does it attain an absolute maximum and minimum on $[a, b]$ ?

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Problem 19012

Does Fermat's Theorem guarantee that if $f(x)$ has a max at $f(a)$ , then $f'(a)=0$ ?

See Solution

Problem 19013

Berechne die Stammfunktionen: $\int \frac{1}{x^{3}} dx$ , $\int \sqrt[3]{x} dx$ , $\int \frac{1}{x} dx$ .

See Solution

Problem 19014

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

See Solution

Problem 19015

121 rabbits grow to 1694 in 1 year. Estimate the population after another 3 months using exponential growth. Round to nearest rabbit.

See Solution

Problem 19016

Find the area between $f(x)=x^{2}$ and $g(x)=x^{3}$ from $x=0$ to $x=2$ .

See Solution

Problem 19017

Determine the units of $\int_{a}^{b} f(t) dt$ if $f(t)$ is in m/s² and $t$ is in seconds. Options: m²/s, s, m, m/s.

See Solution

Problem 19018

Determine where the function $f(x)=-12 x^{5}+105 x^{4}-240 x^{3}$ is increasing and decreasing.

See Solution

Problem 19019

Evaluate the integral $\int x^{3} \sqrt{x^{2}+23} \, dx$ .

See Solution

Problem 19020

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=\cot 7 x$ using $d y=f(x) dx$ .

See Solution

Problem 19021

A person travels $D$ miles at speed $(10+x) \mathrm{mi/hr}$ . Time $T(x)=60D(10+x)^{-1}$ . Show $T(x) \approx L(x)=3D\left(2-\frac{x}{5}\right)$ near $x=0$ .

See Solution

Problem 19022

Express the change in $y$ for the function $f(x)=3 x^{3}-2 x$ as $d y=f^{\prime}(x) d x$ .

See Solution

Problem 19023

Express the change in $y$ for $f(x)=e^{3x}$ as dy = $f^{\prime}(x) dx$ .

See Solution

Problem 19024

Find the limit:
$\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)$
as $x \rightarrow 0^{+}$ , $0^{-}$ , $\sqrt[3]{12}$ , and $-1$ .

See Solution

Problem 19025

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=3x+1$ using $d y=f^{\prime}(x) d x$ .

See Solution

Problem 19026

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and determine the asymptotes using limits. Domain: $\square$ .

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Problem 19027

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{4} x^{2} d x$ , b) $\int_{2}^{4} x^{2} d x$ , c) $\int_{-1}^{5} 2 x d x$ , d) $\int_{10}^{11} 0,5 x d x$ .

See Solution

Problem 19028

Find the limit: $\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)$ as $x \rightarrow 0^{+}, 0^{-}, \sqrt[3]{12}, -1$ .

See Solution

Problem 19029

Find where the function $y=x \ln \left(\frac{x}{8}\right)-x$ is increasing, decreasing, and locate relative extrema.

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Problem 19030

Find points where the tangent slope is -1 or 0 for these functions:
a) $f(x)=\frac{x^{2}+1}{x-1}$
b) $f(x)=\frac{x^{2}+2}{x}$
c) $f(x)=\frac{x+3}{1+x}$
d) $f(x)=\frac{x^{2}}{x+1}$ .
Calculate the derivative and set it to -1 or 0 to find x-values.

See Solution

Problem 19031

Find the limit as $\theta$ approaches 0 from the left: $\lim _{\theta \rightarrow 0^{-}}(4+\csc \theta)$ .

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Problem 19032

Find the derivative of $f(y)=\tan^{-1}(7y^5 + 3)$ .

See Solution

Problem 19033

Find the derivative of $y=(3t-1)(6t-6)^{-1}$ .

See Solution

Problem 19034

Estimate $\sqrt{86}$ using linear approximation with a chosen value of $a$ for minimal error.

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Problem 19035

Estimate $e^{0.04}$ using linear approximation with a suitable value of $a$ for minimal error.

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Problem 19036

Estimate $\ln(1.04)$ using linear approximation with a value of $a$ for minimal error.

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Problem 19037

Find critical points and absolute extremes of $f(x)=(x+1)^{\frac{2}{3}}$ on $[-2,0]$ . Confirm with a graph.

See Solution

Problem 19038

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{e^{x}}{8 x^{2}+7}$ .

See Solution

Problem 19039

Evaluate the integral using symmetry: $\int_{-2}^{2}\left(8 x^{4}-9\right) d x$ .

See Solution

Problem 19040

Find intervals where the function $f(x)=\frac{x^{2}+1}{x^{2}}$ is concave upward.

See Solution

Problem 19041

Evaluate the integral using symmetry: $\int_{-2 \pi}^{2 \pi} 3 \sin x \, dx$ .

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Problem 19042

A tank has 180 L of fluid with 40 g of salt. Brine with 1 g/L is added at 3 L/min. Find $A(t)$ , grams of salt at time $t$ .

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Problem 19043

Find the limit as $x$ approaches 0 for the expression $x^{2}+2$ .

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Problem 19044

A tank has 180 L of fluid with 40 g of salt. Brine (1 g/L) is added at 3 L/min. Find $A(t)$ , the salt in grams at time $t$ . $A(t)=180-140 e^{\left(-\frac{t}{30}\right)} \quad g$

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Problem 19045

A tank has 150L of fluid with 50g of salt. Brine with 1g/L is added at 5L/min. Find $A(t)$ , grams of salt at time $t$ .

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Problem 19046

Find the speed of a ball at $\mathrm{t}=5$ seconds, given $\mathrm{x}(\mathrm{t})=\mathrm{t}^{2}$ . Calculate $\frac{d x}{d t}$ .

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Problem 19047

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and use limits to determine the asymptotes. Domain: $(, \quad)$ .

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Problem 19048

Find the volume of the solid with a base in the second quadrant, bounded by $y=24-6x^{2}$ and the $x$ -axis, using square cross-sections.

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Problem 19049

Find the derivative $f^{\prime}(t)$ of the function $f(t)=\frac{5^{t}}{\pi}+\ln t$ .

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Problem 19050

Find the derivatives: a. $f(t)=\frac{5^{t}}{\pi}+\ln t$ , $f^{\prime}(t)=?$ b. $g(x)=x \ln \left(x^{2} \sqrt{x}\right)$ , $g^{\prime}(x)=?$ c. $h(x)=e^{5 \ln x+2}$ , $h^{\prime}(x)=?$

See Solution

Problem 19051

Find the rate of change in kg/hr from 50 kg at 30 hr to 20 kg at 10 hr using $\frac{{50 - 20}}{{30 - 10}}$ .

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Problem 19052

Evaluate $f(x)=(1+x)^{1/x}$ for $x$ values near 0 and find $\lim_{x \rightarrow 0} f(x) \approx \square$ .

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Problem 19053

Find the volume of the solid with a base in the first quadrant, bounded by $y=12-3 x^{2}$ and the $x$ -axis, using square cross-sections.

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Problem 19054

Calculate the integral $\int_{10}^{20} 5 \, dx$ .

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Problem 19055

What can you conclude about the integral $\int_{-9}^{1} f(x) d x$ if $3 \leq f(x) \leq 6$ for $-9 \leq x \leq 1$ ?

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Problem 19056

If $\int_{3}^{7}(4 f(x)+8) d x=17$ , find $\int_{3}^{7} f(x) d x$ .

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Problem 19057

A wheel with a constant angular acceleration of $0.9 \mathrm{rad} / \mathrm{s}^{2}$ turns 63 rad in 2.2 s. Find the time it was in motion before this interval.

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Problem 19058

Find the differential of $y=\sqrt{9+t^{2}}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} t$ ?

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Problem 19059

Find $d y$ for $y=2 x^{2}+5 x+2$ at $x=5$ , for $d x=0.2$ and $d x=0.4$ .

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Problem 19060

Find the differential of $y=(9+2r)^{-4}$ . What is $\mathrm{d}y$ in terms of $\mathrm{d}r$ ?

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Problem 19061

A forest fire starts at 1200 acres and grows at $8 \sqrt{t}$ acres/hour. How many acres are covered after 11 hours?

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Problem 19062

Check if Rolle's Theorem applies to $f(x)=x^{2}-x-2$ on $[a,b]$ . If yes, find $c$ where $f'(c)=0$ . If no, explain.

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Problem 19063

Check if Rolle's Theorem applies to $f$ on $[a, b]$ . If yes, find $c$ where $f'(c)=0$ . If no, explain why.
2. $f(x)=-x^{2}+3x,[0,3]$
3. $f(x)=x^{2/3}-1,[-8,8]$

See Solution

Problem 19064

Find $f'(x)$ if $f(x)=\int_{2}^{x^{2}} t^{3} dt$ .

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Problem 19065

Find the derivative of $g(x)=\int_{4x}^{6x} \frac{u+2}{u-8} du$ . What is $g'(x)$ ?

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Problem 19066

Find the derivative of $h(x)=\int_{-2}^{\sin (x)}(\cos (t^{2})+t) dt$ . What is $h'(x)$ ?

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Problem 19067

Graph $y=x-\tan(\pi x)$ on $[-\frac{1}{4}, \frac{1}{4}]$ . Check if Rolle's Theorem applies and find $c$ where $f'(c)=0$ .

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Problem 19068

Calculate the integral from $b$ to $2b$ of $x^4$ with respect to $x$ : $\int_{b}^{2b} x^{4} dx$ .

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Problem 19069

Estimate the area under $f(x)=x^{2}+4x$ from $x=4$ to $x=7$ using 3 left endpoint rectangles.

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Problem 19070

Find the critical numbers of the function $f(x)=2 x^{3}-33 x^{2}+144 x-10$ . What are the smaller and larger values?

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Problem 19071

Find the critical number of the function $f(x)=(7 x-4) e^{2 x}$ .

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Problem 19072

Find the production level $x$ that minimizes the average cost given $c(x)=2x^{3}-20x^{2}+19,000x$ .

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Problem 19073

Find the critical number of the function $g(x)=x^{1/3}-x^{-2/3}$ .

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Problem 19074

Find where the function $y=-\frac{x^{3}}{3}-4 x^{2}+9 x+4$ is increasing, decreasing, and where relative extrema occur.

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Problem 19075

Evaluate the integral $\int_{-3}^{0}(7x-4x^{2})dx$ using the limit definition of the integral. Result: $\square($ Type an integer or a simplified fraction. $)$

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Problem 19076

Find the critical number of the function $f(x)=4 x \ln (x)$ .

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Problem 19077

Find the three critical numbers of the function $F(x)=x^{4/5}(4x-16)^{2}$ in increasing order.

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Problem 19078

A flywheel starts at 40 rpm and accelerates at 0.9 rad/s² until it reaches 75 rpm. How many revolutions in 25 s?

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Problem 19079

For $y=48 x-3 x^{5}$ , find where the function increases/decreases, concavity, maxima/minima, inflection points, symmetry, and intercepts. Then sketch the graph. Where is the function decreasing? A. The function is decreasing on $\square$ .

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Problem 19080

Given $f(x)=x^{2}-12 x+15$ , find the critical point $c$ , compute $f(c)$ , and determine min/max on $[0,12]$ and $[0,1]$ .

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Problem 19081

Calculate the integrals: $\int_{0}^{2 \pi} \sin x \, dx$ and $\int_{0}^{2 \pi} \cos x \, dx$ .

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Problem 19082

Sketch the function $f(x)=6-6 x^{2}$ from $0$ to $4$ and find the net area using left, right, and midpoint Riemann sums with $n=4$ .

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Problem 19083

Sketch the function $f(x)=6-6x^{2}$ on $[0,4]$ . Then, find the net area using left, right, and midpoint Riemann sums with $n=4$ . Identify positive and negative contributions to the area from the sketch.

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Problem 19084

Find the max and min of $f(x)=\frac{\ln (x)}{x}$ on $[1,3]$ . Min value =, Max value =.

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Problem 19085

Find the average rate of change of $h(x)=-x^{2}+5x+8$ from $x=0$ to $x=7$ .

See Solution

Problem 19086

Find the general antiderivative of $f(x)=17 x+17 x^{-2}$ and verify by differentiation. $\int f(x) dx =$

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Problem 19087

Find the account balance after 8 years for a \$40,000 deposit at 2.5\% continuous interest. Options: \$48,736.12, \$48,486.34, \$48,856.11, \$49,123.52.

See Solution

Problem 19088

Evaluate the integral: $\int x(48 x^{2}-48) \, dx = C$

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Problem 19089

Evaluate the integral: $\int e^{9t-5} dt = C$

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Problem 19090

Graph the integrand for $\int_{1}^{9}\left(\frac{2}{x}+1\right) d x$ ; find Riemann sums for $n=4$ and identify under/overestimates.

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Problem 19091

Find the end behavior of the function $f(x)=2x^{4}-x^{2}+3x-3$ as $x \rightarrow -\infty$ and $x \rightarrow \infty$ .

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Problem 19092

The function $r$ has a hole at $(1, \frac{1}{2})$ . Which statement about the limits as $x$ approaches 1 is true?

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Problem 19093

Solve the equation $\frac{d y}{d x}=\sqrt{x+49}$ with the initial condition $y(0)=0$ . Find $y$ .

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Problem 19094

Given $g(x) = f(x^2 - x)$ , find $g'(3)$ using $f$ and $f'$ values from the table.

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Problem 19095

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{3}{\sqrt[3]{7 k+2}}$ converges. Which conditions apply? A. Negative B. Positive C. Continuous D. $a_{k}=f(k)$ E. Decreasing F. Increasing.

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Problem 19096

Given $g(x) = f(x^2 - x)$ and $g'(0) = -1$ , find $f''(0)$ using $f(x)$ and $f'(x)$ values.

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Problem 19097

Determine if the series converges or diverges and find the integral $\int_{1}^{\infty} \frac{3}{\sqrt[3]{7 x+2}} d x$ .

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Problem 19098

Find upper and lower bounds for $\int_{0}^{1} \frac{1}{1+x^{2}} d x$ using the Max-Min Inequality.

See Solution

Problem 19099

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{3}{\sqrt[3]{7 k+2}}$ converges.

See Solution

Problem 19100

Find the average rate of change of $h(x)=x^{2}+5x-1$ from $x=-6$ to $x=0$ .

See Solution

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Calculus

Problem 19001

Find the indefinite integral of g(x)=x3g(x) = \sqrt[3]{x}g(x)=3x​.

Problem 19002

Evaluate the integral: ∫01x(x3+x5)dx=□−□=□\int_{0}^{1} x(\sqrt[3]{x}+\sqrt[5]{x}) d x=\square-\square=\square∫01​x(3x​+5x​)dx=□−□=□. Enter simplified fractions only.

Problem 19003

Estimate f(5.2)f(5.2)f(5.2) using the linear approximation given f(5)=5.1f(5)=5.1f(5)=5.1 and f′(5)=3.2f'(5)=3.2f′(5)=3.2. Provide three decimal places.

Problem 19004

A mosquito colony grows uninhibitedly. If N(t)=N0⋅ekt\mathrm{N}(t)=N_{0} \cdot e^{kt}N(t)=N0​⋅ekt, find N(2)\mathrm{N}(2)N(2) given N0=1000\mathrm{N}_{0}=1000N0​=1000, N(1)=1500\mathrm{N}(1)=1500N(1)=1500.

Problem 19005

True or false: The linear approximation of f(x)=sin⁡(x)f(x)=\sin (x)f(x)=sin(x) at x=π2x=\frac{\pi}{2}x=2π​ is L(x)=1+cos⁡(x)(x−π2)L(x)=1+\cos (x)(x-\frac{\pi}{2})L(x)=1+cos(x)(x−2π​).

Problem 19006

Melanie invests \$7600 at 4.0\% annual interest, compounded continuously. What is the investment value after 2 years?

Problem 19007

Calculate the integral ∫0π/3tan⁡(x)cos⁡(x)dx=□−□=□x\int_{0}^{\pi / 3} \frac{\tan (x)}{\cos (x)} d x=\square-\square=\square x∫0π/3​cos(x)tan(x)​dx=□−□=□x.

Problem 19008

Find the absolute minimum and maximum of the function f(x)=x2−64x2+64f(x)=\frac{x^{2}-64}{x^{2}+64}f(x)=x2+64x2−64​ on the interval [−64,64][-64,64][−64,64].

Problem 19009

True or false: If fff is differentiable and f′(c)=0f'(c)=0f′(c)=0, does fff have a local max or min at x=cx=cx=c? True or False?

Problem 19010

True or false: The linear approximation of fff at x=ax=ax=a equals the tangent line to y=f(x)y=f(x)y=f(x) at (a,f(a))(a, f(a))(a,f(a)). Explain.

Problem 19011

True or false: If fff is continuous on [a,b][a, b][a,b], does it attain an absolute maximum and minimum on [a,b][a, b][a,b]?

Problem 19012

Does Fermat's Theorem guarantee that if f(x)f(x)f(x) has a max at f(a)f(a)f(a), then f′(a)=0f'(a)=0f′(a)=0?

Problem 19013

Berechne die Stammfunktionen: ∫1x3dx\int \frac{1}{x^{3}} dx∫x31​dx, ∫x3dx\int \sqrt[3]{x} dx∫3x​dx, ∫1xdx\int \frac{1}{x} dx∫x1​dx.

Problem 19014

Find the limit: =lim⁡x→1+x2+3−2xx−1=\lim _{x \rightarrow 1^{+}} \frac{\sqrt{x^{2}+3}-2 x}{x-1}=limx→1+​x−1x2+3​−2x​.

Problem 19015

121 rabbits grow to 1694 in 1 year. Estimate the population after another 3 months using exponential growth. Round to nearest rabbit.

Problem 19016

Find the area between f(x)=x2f(x)=x^{2}f(x)=x2 and g(x)=x3g(x)=x^{3}g(x)=x3 from x=0x=0x=0 to x=2x=2x=2.

Problem 19017

Determine the units of ∫abf(t)dt\int_{a}^{b} f(t) dt∫ab​f(t)dt if f(t)f(t)f(t) is in m/s² and ttt is in seconds. Options: m²/s, s, m, m/s.

Problem 19018

Determine where the function f(x)=−12x5+105x4−240x3f(x)=-12 x^{5}+105 x^{4}-240 x^{3}f(x)=−12x5+105x4−240x3 is increasing and decreasing.

Problem 19019

Evaluate the integral ∫x3x2+23 dx\int x^{3} \sqrt{x^{2}+23} \, dx∫x3x2+23​dx.

Problem 19020

Find the relationship between a small change in xxx and the change in yyy for f(x)=cot⁡7xf(x)=\cot 7 xf(x)=cot7x using dy=f(x)dxd y=f(x) dxdy=f(x)dx.

Problem 19021

A person travels DDD miles at speed (10+x)mi/hr(10+x) \mathrm{mi/hr}(10+x)mi/hr. Time T(x)=60D(10+x)−1T(x)=60D(10+x)^{-1}T(x)=60D(10+x)−1. Show T(x)≈L(x)=3D(2−x5)T(x) \approx L(x)=3D\left(2-\frac{x}{5}\right)T(x)≈L(x)=3D(2−5x​) near x=0x=0x=0.

Problem 19022

Express the change in yyy for the function f(x)=3x3−2xf(x)=3 x^{3}-2 xf(x)=3x3−2x as dy=f′(x)dxd y=f^{\prime}(x) d xdy=f′(x)dx.

Problem 19023

Express the change in yyy for f(x)=e3xf(x)=e^{3x}f(x)=e3x as dy = f′(x)dxf^{\prime}(x) dxf′(x)dx.

Problem 19024

Find the limit: lim⁡(x24−3x)\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)lim(4x2​−x3​) as x→0+x \rightarrow 0^{+}x→0+, 0−0^{-}0−, 123\sqrt[3]{12}312​, and −1-1−1.

Problem 19025

Find the relationship between a small change in xxx and the change in yyy for f(x)=3x+1f(x)=3x+1f(x)=3x+1 using dy=f′(x)dxd y=f^{\prime}(x) d xdy=f′(x)dx.

Problem 19026

Find the domain of y=9−ex3+exy=\frac{9-e^{x}}{3+e^{x}}y=3+ex9−ex​ and determine the asymptotes using limits. Domain: □\square□.

Problem 19027

Berechnen Sie die folgenden Integrale: a) ∫04x2dx\int_{0}^{4} x^{2} d x∫04​x2dx, b) ∫24x2dx\int_{2}^{4} x^{2} d x∫24​x2dx, c) ∫−152xdx\int_{-1}^{5} 2 x d x∫−15​2xdx, d) ∫10110,5xdx\int_{10}^{11} 0,5 x d x∫1011​0,5xdx.

Problem 19028

Find the limit: lim⁡(x24−3x)\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)lim(4x2​−x3​) as x→0+,0−,123,−1x \rightarrow 0^{+}, 0^{-}, \sqrt[3]{12}, -1x→0+,0−,312​,−1.

Problem 19029

Find where the function y=xln⁡(x8)−xy=x \ln \left(\frac{x}{8}\right)-xy=xln(8x​)−x is increasing, decreasing, and locate relative extrema.

Problem 19030

Problem 19031

Find the limit as θ\thetaθ approaches 0 from the left: lim⁡θ→0−(4+csc⁡θ)\lim _{\theta \rightarrow 0^{-}}(4+\csc \theta)limθ→0−​(4+cscθ).

Problem 19032

Find the derivative of f(y)=tan⁡−1(7y5+3)f(y)=\tan^{-1}(7y^5 + 3)f(y)=tan−1(7y5+3).

Problem 19033

Find the derivative of y=(3t−1)(6t−6)−1y=(3t-1)(6t-6)^{-1}y=(3t−1)(6t−6)−1.

Problem 19034

Estimate 86\sqrt{86}86​ using linear approximation with a chosen value of aaa for minimal error.

Problem 19035

Estimate e0.04e^{0.04}e0.04 using linear approximation with a suitable value of aaa for minimal error.

Problem 19036

Estimate ln⁡(1.04)\ln(1.04)ln(1.04) using linear approximation with a value of aaa for minimal error.

Problem 19037

Find critical points and absolute extremes of f(x)=(x+1)23f(x)=(x+1)^{\frac{2}{3}}f(x)=(x+1)32​ on [−2,0][-2,0][−2,0]. Confirm with a graph.

Problem 19038

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=ex8x2+7f(x)=\frac{e^{x}}{8 x^{2}+7}f(x)=8x2+7ex​.

Problem 19039

Evaluate the integral using symmetry: ∫−22(8x4−9)dx\int_{-2}^{2}\left(8 x^{4}-9\right) d x∫−22​(8x4−9)dx.

Problem 19040

Find intervals where the function f(x)=x2+1x2f(x)=\frac{x^{2}+1}{x^{2}}f(x)=x2x2+1​ is concave upward.

Find the indefinite integral of $g(x) = \sqrt[3]{x}$ .

Evaluate the integral: $\int_{0}^{1} x(\sqrt[3]{x}+\sqrt[5]{x}) d x=\square-\square=\square$ . Enter simplified fractions only.

Estimate $f(5.2)$ using the linear approximation given $f(5)=5.1$ and $f'(5)=3.2$ . Provide three decimal places.

A mosquito colony grows uninhibitedly. If $\mathrm{N}(t)=N_{0} \cdot e^{kt}$ , find $\mathrm{N}(2)$ given $\mathrm{N}_{0}=1000$ , $\mathrm{N}(1)=1500$ .

True or false: The linear approximation of $f(x)=\sin (x)$ at $x=\frac{\pi}{2}$ is $L(x)=1+\cos (x)(x-\frac{\pi}{2})$ .

Calculate the integral $\int_{0}^{\pi / 3} \frac{\tan (x)}{\cos (x)} d x=\square-\square=\square x$ .

Find the absolute minimum and maximum of the function $f(x)=\frac{x^{2}-64}{x^{2}+64}$ on the interval $[-64,64]$ .

True or false: If $f$ is differentiable and $f'(c)=0$ , does $f$ have a local max or min at $x=c$ ? True or False?

True or false: The linear approximation of $f$ at $x=a$ equals the tangent line to $y=f(x)$ at $(a, f(a))$ . Explain.

True or false: If $f$ is continuous on $[a, b]$ , does it attain an absolute maximum and minimum on $[a, b]$ ?

Does Fermat's Theorem guarantee that if $f(x)$ has a max at $f(a)$ , then $f'(a)=0$ ?

Berechne die Stammfunktionen: $\int \frac{1}{x^{3}} dx$ , $\int \sqrt[3]{x} dx$ , $\int \frac{1}{x} dx$ .

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

Find the area between $f(x)=x^{2}$ and $g(x)=x^{3}$ from $x=0$ to $x=2$ .

Determine the units of $\int_{a}^{b} f(t) dt$ if $f(t)$ is in m/s² and $t$ is in seconds. Options: m²/s, s, m, m/s.

Determine where the function $f(x)=-12 x^{5}+105 x^{4}-240 x^{3}$ is increasing and decreasing.

Evaluate the integral $\int x^{3} \sqrt{x^{2}+23} \, dx$ .

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=\cot 7 x$ using $d y=f(x) dx$ .

A person travels $D$ miles at speed $(10+x) \mathrm{mi/hr}$ . Time $T(x)=60D(10+x)^{-1}$ . Show $T(x) \approx L(x)=3D\left(2-\frac{x}{5}\right)$ near $x=0$ .

Express the change in $y$ for the function $f(x)=3 x^{3}-2 x$ as $d y=f^{\prime}(x) d x$ .

Express the change in $y$ for $f(x)=e^{3x}$ as dy = $f^{\prime}(x) dx$ .

Find the limit:
$\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)$
as $x \rightarrow 0^{+}$ , $0^{-}$ , $\sqrt[3]{12}$ , and $-1$ .

Find the relationship between a small change in $x$ and the change in $y$ for $f(x)=3x+1$ using $d y=f^{\prime}(x) d x$ .

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and determine the asymptotes using limits. Domain: $\square$ .

Berechnen Sie die folgenden Integrale: a) $\int_{0}^{4} x^{2} d x$ , b) $\int_{2}^{4} x^{2} d x$ , c) $\int_{-1}^{5} 2 x d x$ , d) $\int_{10}^{11} 0,5 x d x$ .

Find the limit: $\lim \left(\frac{x^{2}}{4}-\frac{3}{x}\right)$ as $x \rightarrow 0^{+}, 0^{-}, \sqrt[3]{12}, -1$ .

Find where the function $y=x \ln \left(\frac{x}{8}\right)-x$ is increasing, decreasing, and locate relative extrema.

Find the limit as $\theta$ approaches 0 from the left: $\lim _{\theta \rightarrow 0^{-}}(4+\csc \theta)$ .

Find the derivative of $f(y)=\tan^{-1}(7y^5 + 3)$ .

Find the derivative of $y=(3t-1)(6t-6)^{-1}$ .

Estimate $\sqrt{86}$ using linear approximation with a chosen value of $a$ for minimal error.

Estimate $e^{0.04}$ using linear approximation with a suitable value of $a$ for minimal error.

Estimate $\ln(1.04)$ using linear approximation with a value of $a$ for minimal error.

Find critical points and absolute extremes of $f(x)=(x+1)^{\frac{2}{3}}$ on $[-2,0]$ . Confirm with a graph.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{e^{x}}{8 x^{2}+7}$ .

Evaluate the integral using symmetry: $\int_{-2}^{2}\left(8 x^{4}-9\right) d x$ .

Find intervals where the function $f(x)=\frac{x^{2}+1}{x^{2}}$ is concave upward.

Evaluate the integral using symmetry: $\int_{-2 \pi}^{2 \pi} 3 \sin x \, dx$ .

A tank has 180 L of fluid with 40 g of salt. Brine with 1 g/L is added at 3 L/min. Find $A(t)$ , grams of salt at time $t$ .

Find the limit as $x$ approaches 0 for the expression $x^{2}+2$ .

A tank has 180 L of fluid with 40 g of salt. Brine (1 g/L) is added at 3 L/min. Find $A(t)$ , the salt in grams at time $t$ . $A(t)=180-140 e^{\left(-\frac{t}{30}\right)} \quad g$

A tank has 150L of fluid with 50g of salt. Brine with 1g/L is added at 5L/min. Find $A(t)$ , grams of salt at time $t$ .

Find the speed of a ball at $\mathrm{t}=5$ seconds, given $\mathrm{x}(\mathrm{t})=\mathrm{t}^{2}$ . Calculate $\frac{d x}{d t}$ .

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and use limits to determine the asymptotes. Domain: $(, \quad)$ .

Find the volume of the solid with a base in the second quadrant, bounded by $y=24-6x^{2}$ and the $x$ -axis, using square cross-sections.

Find the derivative $f^{\prime}(t)$ of the function $f(t)=\frac{5^{t}}{\pi}+\ln t$ .

Find the rate of change in kg/hr from 50 kg at 30 hr to 20 kg at 10 hr using $\frac{{50 - 20}}{{30 - 10}}$ .

Evaluate $f(x)=(1+x)^{1/x}$ for $x$ values near 0 and find $\lim_{x \rightarrow 0} f(x) \approx \square$ .

Find the volume of the solid with a base in the first quadrant, bounded by $y=12-3 x^{2}$ and the $x$ -axis, using square cross-sections.

Calculate the integral $\int_{10}^{20} 5 \, dx$ .

What can you conclude about the integral $\int_{-9}^{1} f(x) d x$ if $3 \leq f(x) \leq 6$ for $-9 \leq x \leq 1$ ?

If $\int_{3}^{7}(4 f(x)+8) d x=17$ , find $\int_{3}^{7} f(x) d x$ .

A wheel with a constant angular acceleration of $0.9 \mathrm{rad} / \mathrm{s}^{2}$ turns 63 rad in 2.2 s. Find the time it was in motion before this interval.

Find the differential of $y=\sqrt{9+t^{2}}$ . What is $\mathrm{d} y$ in terms of $\mathrm{d} t$ ?

Find $d y$ for $y=2 x^{2}+5 x+2$ at $x=5$ , for $d x=0.2$ and $d x=0.4$ .

Find the differential of $y=(9+2r)^{-4}$ . What is $\mathrm{d}y$ in terms of $\mathrm{d}r$ ?

A forest fire starts at 1200 acres and grows at $8 \sqrt{t}$ acres/hour. How many acres are covered after 11 hours?

Check if Rolle's Theorem applies to $f(x)=x^{2}-x-2$ on $[a,b]$ . If yes, find $c$ where $f'(c)=0$ . If no, explain.

Check if Rolle's Theorem applies to $f$ on $[a, b]$ . If yes, find $c$ where $f'(c)=0$ . If no, explain why.
2. $f(x)=-x^{2}+3x,[0,3]$
3. $f(x)=x^{2/3}-1,[-8,8]$

Find $f'(x)$ if $f(x)=\int_{2}^{x^{2}} t^{3} dt$ .

Find the derivative of $g(x)=\int_{4x}^{6x} \frac{u+2}{u-8} du$ . What is $g'(x)$ ?

Find the derivative of $h(x)=\int_{-2}^{\sin (x)}(\cos (t^{2})+t) dt$ . What is $h'(x)$ ?

Graph $y=x-\tan(\pi x)$ on $[-\frac{1}{4}, \frac{1}{4}]$ . Check if Rolle's Theorem applies and find $c$ where $f'(c)=0$ .

Calculate the integral from $b$ to $2b$ of $x^4$ with respect to $x$ : $\int_{b}^{2b} x^{4} dx$ .

Estimate the area under $f(x)=x^{2}+4x$ from $x=4$ to $x=7$ using 3 left endpoint rectangles.

Find the critical numbers of the function $f(x)=2 x^{3}-33 x^{2}+144 x-10$ . What are the smaller and larger values?

Find the critical number of the function $f(x)=(7 x-4) e^{2 x}$ .

Find the production level $x$ that minimizes the average cost given $c(x)=2x^{3}-20x^{2}+19,000x$ .

Find the critical number of the function $g(x)=x^{1/3}-x^{-2/3}$ .

Find where the function $y=-\frac{x^{3}}{3}-4 x^{2}+9 x+4$ is increasing, decreasing, and where relative extrema occur.