Calculus

Problem 24601

Approximate the area under $f(x)=e^{x}+3$ from $x=-2$ to $x=2$ with $n=4$ using left, right, midpoint, and average methods.

See Solution

Problem 24602

Find the limit as $x$ approaches 7 for the expression $\frac{x^{2}}{7} + \frac{7}{x^{2}}$ .

See Solution

Problem 24603

Bestimme die lokale Änderungsrate von $f$ an a mit der h-Methode für: a) $f(x)=x^{2}, a=3$ b) $f(x)=x^{2}+4x, a=2$ c) $f(x)=x^{2}, a=\sqrt{3}$ d) $f(x)=3x^{2}, a=1$ e) $f(x)=x^{2}-2x+1, a=-1$ f) $f(x)=x^{3}, a=-3$ . Vergleiche die Grenzwerte mit dem Differenzenquotienten für $h=0,000001$ .

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Problem 24604

Berechnen Sie die lokale Änderungsrate von $f$ bei a mit der "h-Methode" für die folgenden Funktionen und vergleichen Sie die Grenzwerte mit dem Differenzenquotienten für $h=0,000001$ :
a) $f(x)=x^{2}$ , $a=3$ b) $f(x)=x^{2}+4x$ , $a=2$ c) $f(x)=x^{2}$ , $a=\sqrt{3}$ d) $f(x)=3x^{2}$ , $a=1$ e) $f(x)=x^{2}-2x+1$ , $a=-1$ f) $f(x)=x^{3}$ , $a=-3$

See Solution

Problem 24605

Find the linearization $L(x)$ of $f(x)=x^{4}+4x^{2}$ at $a=-1$ .

See Solution

Problem 24606

Find where the function $f(x)=3 x^{4}-4 x^{3}+3$ is concave up, concave down, and identify inflection points.

See Solution

Problem 24607

Evaluate the integral: $\int_{0}^{\pi}-3 \cos ^{2} x \sin x \, dx = \square(\text{Type an exact answer})$

See Solution

Problem 24608

Find the interval where the function $f$ is increasing given its derivative $f^{\prime}(x)=(x+2)^{2}(x-5)^{5}(x-6)^{4}$ .

See Solution

Problem 24609

Find the interval $x \in \square$ where $\sqrt[4]{1+2x} \approx 1 + \frac{1}{2}x$ .

See Solution

Problem 24610

Find the derivatives of the functions: (a) $y=x^{2} \sin 6x$ , (b) $y=\ln \sqrt{3+t^{2}}$ .

See Solution

Problem 24611

Find where the function $g(t)=\frac{t-5}{t+1}$ is concave up or down and identify inflection points.

See Solution

Problem 24612

Evaluate the integral $\int_{2}^{8} \frac{v^{2}+10}{\sqrt{v^{3}+30 v+3}} d v$ . Type the exact answer with radicals.

See Solution

Problem 24613

Find where the function $f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)$ is concave up or down and identify inflection points on $[-1, \infty)$ .

See Solution

Problem 24614

Find the differential for each function: (a) $y=e^{\tan \pi t}$ , (b) $y=\sqrt{4+\ln z}$ .

See Solution

Problem 24615

Find where the function $f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)$ is concave up/down and identify inflection points on $[-1, \infty)$ .

See Solution

Problem 24616

Estimate $(8.03)^{2 / 3}$ using linear approximation or differentials.

See Solution

Problem 24617

Find the differential $d y$ for $y=\tan x$ , then evaluate it at $x=\pi/4$ and $d x=0.05$ .

See Solution

Problem 24618

Given $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ , find the limits as $r \to 0$ and $r \to \infty$ , and determine the minimizing $r$ .

See Solution

Problem 24619

Evaluate the integral: $\int_{2}^{4} \frac{v^{2}+3}{\sqrt{v^{3}+9 v+6}} d v=\square$ (Provide an exact answer with radicals.)

See Solution

Problem 24620

Find the differential $d y$ for the function $y=\tan x$ .

See Solution

Problem 24621

Evaluate the integral using geometry: $\int_{1}^{6}|2 x-4| d x$ . What is the simplified result?

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Problem 24622

(a) Find dimensions to maximize area of a pen with 900 m of fencing on three sides. (b) Determine dimensions of 4 adjacent pens each with area 25 m² to minimize fencing used.

See Solution

Problem 24623

Untersuche die Funktion $f_{1}(x)=x^{3}-x$ auf lokale Extremstellen.

See Solution

Problem 24624

Evaluate the integral: $\int_{2}^{4}(2 x-3)^{4} d x$ using the substitution $u=2x-3$ . Find $a$ , $b$ , and $f(u)$ .

See Solution

Problem 24625

Find the line equation for the linear approximation of $f(x)=10-3x^2$ at $a=3$ , then estimate $f(2.9)$ and compute percent error.

See Solution

Problem 24626

Find the critical numbers of the function $g(y)=\frac{y-1}{y^{2}-3 y+3}$ . Enter as a comma-separated list or DNE.

See Solution

Problem 24627

Given the function $h(x)=(x+2)^{3}-3x-1$ , find intervals of increase and decrease, local min/max values, inflection point, and concavity.

See Solution

Problem 24628

Find the instantaneous rate of change of $f(x)=-x^{5}-x^{4}+x^{2}+3x$ at $x=-1$ .

See Solution

Problem 24629

Find the derivative $f'(x)$ of the function $f(x) = 5x^{5} + x - 2$ and evaluate it at $x = ?$ .

See Solution

Problem 24630

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (7 x)-7 x}{4 x^{3}}$ . Simplify your answer.

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Problem 24631

Evaluate the derivative $f^{\prime}(-2)$ for the function $f(x)=x^{3}+x$ .

See Solution

Problem 24632

Find the average rate of change of the function $g(x)=-x^{2}+8x+24$ from $x=2$ to $x=10$ .

See Solution

Problem 24633

Evaluate $\frac{d y}{d x}$ for $y=5 x^{2}+x$ at $x=-1$ .

See Solution

Problem 24634

Find the min and max values of $f(t)=t \sqrt{25-t^{2}}$ on the interval $[-1,5]$ .

See Solution

Problem 24635

Evaluate the integral from 1 to 8 of $\sqrt[3]{y}$ dy: $\int_{1}^{8} \sqrt[3]{y} d y=\square($ Type an integer or a simplified fraction.)

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Problem 24636

Verify if $f(x)=2-24x+2x^2$ meets Rolle's Theorem on $[5,7]$ and find $c$ values. List them separated by commas.

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Problem 24637

Find the antiderivative $F$ of $f(x)=x^{6}-2 x^{-3}+1$ such that $F(1)=1$ . What is $F(x)=\square$ ?

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Problem 24638

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{12} x^{-3}-\frac{1}{4} x^{4}-\frac{1}{6} x+2 x^{3}$ .

See Solution

Problem 24639

Find $r$ that minimizes $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ by setting the first derivative to 0.

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Problem 24640

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{15} x^{6}-4 x^{2}-\frac{1}{3} x^{-2}+\frac{3}{8} x$ .

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Problem 24641

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=-\frac{1}{3} x^{3}+\frac{1}{4} x^{2}+\frac{2}{3} x^{-2}-\frac{1}{6} x^{-4}$ .

See Solution

Problem 24642

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ of the function $y=-\frac{2}{3} x^{4}-\frac{1}{6} x^{-2}+\frac{3}{20} x+\frac{1}{5} x^{5}$ .

See Solution

Problem 24643

Check if $f(x)=\sqrt{x}-\frac{1}{9} x$ meets Rolle's Theorem on $[0,81]$ and find all $c$ values. $c=\quad$

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Problem 24644

Evaluate the integral $\int_{4}^{6} \frac{v^{2}+10}{\sqrt{v^{3}+30 v+6}} d v$ using a change of variables.

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Problem 24645

Find the tangent line approximation for $f(x)=-x^{\frac{3}{4}}$ at $x=81$ to estimate $f(81.1)$ .

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Problem 24646

Evaluate the integral $\int_{2}^{3} \frac{v^{2}+16}{\sqrt{v^{3}+48 v+6}} d v=\square($ exact answer, with radicals.)

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Problem 24647

What is the minimum value of $f(6)$ given that $f(4)=7$ and $f^{\prime}(x) \geq 1$ for $4 \leq x \leq 6$ ?

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Problem 24648

Given the function $f(x)=7 \sin (x)+7 \cos (x)$ for $0 \leq x \leq 2\pi$ , find intervals where $f$ is increasing/decreasing, local min/max values, inflection points, and concavity intervals.

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Problem 24649

Check if $f(x)=2 x^{2}-3 x+1$ meets the Mean Value Theorem conditions on the interval $[0,2]$ .

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Problem 24650

A boat is 2 mi from shore, 13 mi from a restaurant.
a. Where should she land to minimize travel time?
b. Find the minimum rowing speed to row directly to the restaurant.

See Solution

Problem 24651

Given the function $f(x)=2 x^{3}+3 x^{2}-180 x$ , find where $f$ is increasing, decreasing, local min/max, inflection point, and concavity.

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Problem 24652

A soda can manufacturer wants to minimize aluminum cost for a can with volume $V$ . Find $S(r)$ for $V=225 \mathrm{~cm}^{3}$ and analyze limits.
(a) Write $S$ as a function of $r$ : $S(r)=\frac{450}{r}+2 \pi r^{2}$
(b) What happens to $S(r)$ as $r \rightarrow \infty$ ?
(c) What happens to $S(r)$ as $r \rightarrow 0$ ?

See Solution

Problem 24653

Find the tangent line at $x=1$ for $f(x)=-5 x^{\frac{2}{5}}$ and approximate $f(0.95)$ .

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Problem 24654

Minimize the aluminum cost $S=2 \pi r h + 2 \pi r^2$ for a can with volume $\pi r^2 h = V$ . Complete parts (a) to (d).

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Problem 24655

A cone has height $10 \mathrm{~cm}$ and diameter $10 \mathrm{~cm}$ . If $\mathrm{h}$ decreases at $-3/10 \mathrm{~cm/hr}$ , find $\frac{dV}{dt}$ when $\mathrm{h}=5$ . Round to three decimal places.

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Problem 24656

Find $f^{\prime}\left(\frac{3}{5}\right)$ if $f(x)=\cos^{-1}(x)$ .

See Solution

Problem 24657

Evaluate the integral $\int_{1}^{4} \frac{v^{2}+3}{\sqrt{v^{3}+9 v+6}} d v$ using a variable change.

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Problem 24658

A soda can manufacturer wants to minimize aluminum cost. Minimize $S=2 \pi r h + 2 \pi r^{2}$ with constraint $\pi r^{2} h=V$ .

See Solution

Problem 24659

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\arccos \left(\frac{\sqrt{5}}{3}+h\right)-\arccos \left(\frac{\sqrt{5}}{3}\right)}{h}$

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Problem 24660

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sin ^{-1}\left(\frac{3}{5}+h\right)-\sin ^{-1}\left(\frac{3}{5}\right)}{h}$ .

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Problem 24661

Find the derivative of $f(x)=\sin^{-1}(3x)$ , denoted as $f^{\prime}(x)$ .

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Problem 24662

A boat is 5 mi from shore, 8 mi from a restaurant. Find the best landing point to minimize travel time and row speed.

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Problem 24663

Find the derivative of the function $f(x)=\tan^{-1}(4x)$ , expressed in simplest form.

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Problem 24664

A soda can manufacturer aims to minimize aluminum cost. Given volume $V=339 \mathrm{~cm}^{3}$ , find $S(r)$ as $S(r)=\frac{678}{r}+2 \pi r^{2}$ . What happens to $S(r)$ as $r \rightarrow \infty$ ?

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Problem 24665

A right triangle has legs 15 in and 20 in. Short leg decreases by $10 \mathrm{in/sec}$ , long leg increases by $4 \mathrm{in/sec}$ . Find area change rate.

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Problem 24666

A boat is $5 \mathrm{mi}$ from shore, $8 \mathrm{mi}$ from a restaurant.
a. Where should she land to minimize travel time if she rows at $2 \mathrm{mi/hr}$ and walks at $3 \mathrm{mi/hr}$ ?
b. What is the minimum rowing speed for direct travel to the restaurant?

See Solution

Problem 24667

A right triangle has legs of 24 in and 32 in. The short leg increases by $4 \mathrm{in/sec}$ , and the long leg by $3 \mathrm{in/sec}$ . Find the rate of change of the hypotenuse.

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Problem 24668

Find the interval where the function $f$ is increasing given that $f^{\prime}(x)=(x+2)^{4}(x-5)^{3}(x-6)^{6}$ .

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Problem 24669

A cylinder's radius decreases at 9 m/min, with a constant volume of 728 m³. Find the height's rate of change when h = 3 m. Use $V=\pi r^{2} h$ and round to three decimal places.

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Problem 24670

A right triangle has legs of 12 in and 16 in, with the short leg increasing by $8 \mathrm{in/sec}$ and the long leg by $6 \mathrm{in/sec}$ . Find the hypotenuse's rate of change.

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Problem 24671

A cylinder's height decreases at 7 in/min with a volume of 501 in³. When the radius is 7 in, find the radius's rate of change. Use $V=\pi r^{2} h$ and round to three decimal places.

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Problem 24672

A cone's height rises at 8 ft/min with a volume of 178 cubic ft. Find the radius change rate when the radius is 4 ft. Use $V=\frac{1}{3} \pi r^{2} h$ and round to three decimal places.

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Problem 24673

What can we conclude about $a$ and $b$ if they are in nested intervals with widths less than $\frac{1}{n}$ ?

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Problem 24674

Find the derivative of $f^{\prime}(x)=\left(3 x^{2}-\frac{3}{2 \sqrt[3]{x}}+3 x^{2} e^{x^{5}}+\frac{1}{\sqrt{3 x}}\right)$ .

See Solution

Problem 24675

Find the tangent line at $x=1$ for $f(x)=3 x^{\frac{2}{3}}$ and use it to approximate $f(0.85)$ .

See Solution

Problem 24676

Find the derivative of the function given by $f^{\prime}(x)=\left(3 x^{3}-\frac{3}{2 \sqrt[3]{x}}+e^{x^{5}}+\frac{1}{\sqrt{3 x}}\right)^{\prime}$ .

See Solution

Problem 24677

Given the function $f(x)=2 x^{3}+3 x^{2}-180 x$ , find where it is increasing, decreasing, local min/max, inflection point, and concavity intervals.

See Solution

Problem 24678

Find the tangent line equation for $y=g(x)$ at $x=4$ where $g(4)=-3$ and $g^{\prime}(4)=6$ .

See Solution

Problem 24679

Solve the differential equation $y'=(\sqrt{x})(y)$ and verify your solution.

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Problem 24680

Find the rate of change of the surface area of a cube with volume 914 cm³ decreasing at 2371 cm³/min. Round to three decimals.

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Problem 24681

Find the absolute extremum of the function $A(r)=\frac{42}{r}+3 \pi r^{2}$ for $r>0$ .

See Solution

Problem 24682

Find the derivative $j'(-1)$ if $j(x) = f(x^3)$ and $f'(-1) = 2$ .

See Solution

Problem 24683

Find the rate of increase of the radius when the diameter is 20 cm, given that the volume increases at $150 \mathrm{~cm}^{3} / \mathrm{s}$ . Use the volume formula $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 24684

Solve the equation $(\sec (x)) \cdot \frac{d y}{d x}=e^{y+\sin (x)}$ .

See Solution

Problem 24685

A 10 ft ladder leans against a wall. If it slides away at 1.1 ft/s, find the angle's rate of change when 8 ft from the wall.

See Solution

Problem 24686

Gravel is dumped at $25 \mathrm{ft}^{3} / \mathrm{min}$ . Find the height increase rate when the cone is $6 \mathrm{ft}$ high. $\mathrm{ft} / \mathrm{min}$

See Solution

Problem 24687

Find the rate of increase of the radius, $\frac{d r}{d t}$ , when the diameter is $20 \mathrm{~cm}$ and $\frac{d V}{d t}=150 \mathrm{~cm}^{3}/\mathrm{s}$ .

See Solution

Problem 24688

Find all antiderivatives of $f(x)=16 x^{15}$ and verify by differentiating. Antiderivatives are $F(x)=\square$

See Solution

Problem 24689

Given $y=\sqrt{2x+1}$ with $x$ and $y$ as functions of $t$ :
(a) If $\frac{dx}{dt}=9$ , find $\frac{dy}{dt}$ when $x=4$ .
(b) If $\frac{dy}{dt}=2$ , find $\frac{dx}{dt}$ when $x=40$ .

See Solution

Problem 24690

Find the rate of increase of the radius when the diameter is $20 \mathrm{~cm}$ , given $\frac{d V}{d t}=150 \mathrm{~cm}^{3}/\mathrm{s}$ .

See Solution

Problem 24691

Find all antiderivatives of $f(x)=9 \csc ^{2} x$ and verify by differentiating. Antiderivatives are $F(x)=\square$ .

See Solution

Problem 24692

A tank with radius $3 \mathrm{~m}$ fills at $2 \mathrm{~m}^{3} / \mathrm{min}$ . Find the height increase rate in $\mathrm{m} / \mathrm{min}$ .

See Solution

Problem 24693

Show how $\frac{d G}{d t}$ relates to $\frac{d M}{d t}$ by differentiating $G=\mathrm{CM}^{0.7}$ with respect to $t$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

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Problem 24694

Find all antiderivatives of $f(x)=6 \cos x-2$ and verify by differentiating. Antiderivatives: $F(x)=\square$

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Problem 24695

Find all antiderivatives of $f(y)=-\frac{30}{y^{31}}$ and verify by differentiating your result.

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Problem 24696

A rectangle's length increases by $5 \mathrm{~cm/s}$ and width by $4 \mathrm{~cm/s}$ . Find the area increase rate when length is $9 \mathrm{~cm}$ and width is $7 \mathrm{~cm}$ .

See Solution

Problem 24697

A rectangle's length increases at $7 \mathrm{~cm/s}$ and width at $9 \mathrm{~cm/s}$ . Find area increase rate when length $15 \mathrm{~cm}$ and width $9 \mathrm{~cm}$ .

See Solution

Problem 24698

Find the derivative of $e^{x^{3}}$ .

See Solution

Problem 24699

Find the average rate of change of $C(x)=2000+8x+0.1x^2$ from $x=100$ to $x=105$ and $x=101$ . Then, find the marginal cost at $x=100$ .

See Solution

Problem 24700

A cylindrical tank with radius $5 \mathrm{~m}$ is filled at $3 \mathrm{~m}^{3}/\mathrm{min}$ . Find the height increase rate in $\mathrm{m}/\mathrm{min}$ .

See Solution

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Calculus

Problem 24601

Approximate the area under f(x)=ex+3f(x)=e^{x}+3f(x)=ex+3 from x=−2x=-2x=−2 to x=2x=2x=2 with n=4n=4n=4 using left, right, midpoint, and average methods.

Problem 24602

Find the limit as xxx approaches 7 for the expression x27+7x2\frac{x^{2}}{7} + \frac{7}{x^{2}}7x2​+x27​.

Problem 24603

Problem 24604

Problem 24605

Find the linearization L(x)L(x)L(x) of f(x)=x4+4x2f(x)=x^{4}+4x^{2}f(x)=x4+4x2 at a=−1a=-1a=−1.

Problem 24606

Find where the function f(x)=3x4−4x3+3f(x)=3 x^{4}-4 x^{3}+3f(x)=3x4−4x3+3 is concave up, concave down, and identify inflection points.

Problem 24607

Evaluate the integral: ∫0π−3cos⁡2xsin⁡x dx=□(Type an exact answer)\int_{0}^{\pi}-3 \cos ^{2} x \sin x \, dx = \square(\text{Type an exact answer})∫0π​−3cos2xsinxdx=□(Type an exact answer)

Problem 24608

Find the interval where the function fff is increasing given its derivative f′(x)=(x+2)2(x−5)5(x−6)4f^{\prime}(x)=(x+2)^{2}(x-5)^{5}(x-6)^{4}f′(x)=(x+2)2(x−5)5(x−6)4.

Problem 24609

Find the interval x∈□x \in \squarex∈□ where 1+2x4≈1+12x\sqrt[4]{1+2x} \approx 1 + \frac{1}{2}x41+2x​≈1+21​x.

Problem 24610

Find the derivatives of the functions: (a) y=x2sin⁡6xy=x^{2} \sin 6xy=x2sin6x, (b) y=ln⁡3+t2y=\ln \sqrt{3+t^{2}}y=ln3+t2​.

Problem 24611

Find where the function g(t)=t−5t+1g(t)=\frac{t-5}{t+1}g(t)=t+1t−5​ is concave up or down and identify inflection points.

Problem 24612

Evaluate the integral ∫28v2+10v3+30v+3dv\int_{2}^{8} \frac{v^{2}+10}{\sqrt{v^{3}+30 v+3}} d v∫28​v3+30v+3​v2+10​dv. Type the exact answer with radicals.

Problem 24613

Find where the function f(x)=4(x+1)52(2x−3)f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)f(x)=4(x+1)25​(2x−3) is concave up or down and identify inflection points on [−1,∞)[-1, \infty)[−1,∞).

Problem 24614

Find the differential for each function: (a) y=etan⁡πty=e^{\tan \pi t}y=etanπt, (b) y=4+ln⁡zy=\sqrt{4+\ln z}y=4+lnz​.

Problem 24615

Find where the function f(x)=4(x+1)52(2x−3)f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)f(x)=4(x+1)25​(2x−3) is concave up/down and identify inflection points on [−1,∞)[-1, \infty)[−1,∞).

Problem 24616

Estimate (8.03)2/3(8.03)^{2 / 3}(8.03)2/3 using linear approximation or differentials.

Problem 24617

Find the differential dyd ydy for y=tan⁡xy=\tan xy=tanx, then evaluate it at x=π/4x=\pi/4x=π/4 and dx=0.05d x=0.05dx=0.05.

Problem 24618

Given V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​, find the limits as r→0r \to 0r→0 and r→∞r \to \inftyr→∞, and determine the minimizing rrr.

Problem 24619

Evaluate the integral: ∫24v2+3v3+9v+6dv=□\int_{2}^{4} \frac{v^{2}+3}{\sqrt{v^{3}+9 v+6}} d v=\square∫24​v3+9v+6​v2+3​dv=□ (Provide an exact answer with radicals.)

Problem 24620

Find the differential dyd ydy for the function y=tan⁡xy=\tan xy=tanx.

Problem 24621

Evaluate the integral using geometry: ∫16∣2x−4∣dx\int_{1}^{6}|2 x-4| d x∫16​∣2x−4∣dx. What is the simplified result?

Problem 24622

(a) Find dimensions to maximize area of a pen with 900 m of fencing on three sides. (b) Determine dimensions of 4 adjacent pens each with area 25 m² to minimize fencing used.

Problem 24623

Untersuche die Funktion f1(x)=x3−xf_{1}(x)=x^{3}-xf1​(x)=x3−x auf lokale Extremstellen.

Problem 24624

Evaluate the integral: ∫24(2x−3)4dx\int_{2}^{4}(2 x-3)^{4} d x∫24​(2x−3)4dx using the substitution u=2x−3u=2x-3u=2x−3. Find aaa, bbb, and f(u)f(u)f(u).

Problem 24625

Find the line equation for the linear approximation of f(x)=10−3x2f(x)=10-3x^2f(x)=10−3x2 at a=3a=3a=3, then estimate f(2.9)f(2.9)f(2.9) and compute percent error.

Problem 24626

Find the critical numbers of the function g(y)=y−1y2−3y+3g(y)=\frac{y-1}{y^{2}-3 y+3}g(y)=y2−3y+3y−1​. Enter as a comma-separated list or DNE.

Problem 24627

Given the function h(x)=(x+2)3−3x−1h(x)=(x+2)^{3}-3x-1h(x)=(x+2)3−3x−1, find intervals of increase and decrease, local min/max values, inflection point, and concavity.

Problem 24628

Find the instantaneous rate of change of f(x)=−x5−x4+x2+3xf(x)=-x^{5}-x^{4}+x^{2}+3xf(x)=−x5−x4+x2+3x at x=−1x=-1x=−1.

Problem 24629

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=5x5+x−2f(x) = 5x^{5} + x - 2f(x)=5x5+x−2 and evaluate it at x=?x = ?x=?.

Problem 24630

Find the limit: lim⁡x→0tan⁡(7x)−7x4x3\lim _{x \rightarrow 0} \frac{\tan (7 x)-7 x}{4 x^{3}}limx→0​4x3tan(7x)−7x​. Simplify your answer.

Problem 24631

Evaluate the derivative f′(−2)f^{\prime}(-2)f′(−2) for the function f(x)=x3+xf(x)=x^{3}+xf(x)=x3+x.

Problem 24632

Find the average rate of change of the function g(x)=−x2+8x+24g(x)=-x^{2}+8x+24g(x)=−x2+8x+24 from x=2x=2x=2 to x=10x=10x=10.

Problem 24633

Evaluate dydx\frac{d y}{d x}dxdy​ for y=5x2+xy=5 x^{2}+xy=5x2+x at x=−1x=-1x=−1.

Problem 24634

Find the min and max values of f(t)=t25−t2f(t)=t \sqrt{25-t^{2}}f(t)=t25−t2​ on the interval [−1,5][-1,5][−1,5].

Problem 24635

Evaluate the integral from 1 to 8 of y3\sqrt[3]{y}3y​ dy: ∫18y3dy=□(\int_{1}^{8} \sqrt[3]{y} d y=\square(∫18​3y​dy=□( Type an integer or a simplified fraction.)

Problem 24636

Verify if f(x)=2−24x+2x2f(x)=2-24x+2x^2f(x)=2−24x+2x2 meets Rolle's Theorem on [5,7][5,7][5,7] and find ccc values. List them separated by commas.

Problem 24637

Find the antiderivative FFF of f(x)=x6−2x−3+1f(x)=x^{6}-2 x^{-3}+1f(x)=x6−2x−3+1 such that F(1)=1F(1)=1F(1)=1. What is F(x)=□F(x)=\squareF(x)=□?

Problem 24638

Find the third derivative f′′′(x)f^{\prime \prime \prime}(x)f′′′(x) of the function f(x)=112x−3−14x4−16x+2x3f(x)=\frac{1}{12} x^{-3}-\frac{1}{4} x^{4}-\frac{1}{6} x+2 x^{3}f(x)=121​x−3−41​x4−61​x+2x3.

Problem 24639

Find rrr that minimizes V(r)=1r6−Ar3V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}V(r)=r61​−r3A​ by setting the first derivative to 0.

Problem 24640

Find the third derivative f′′′(x)f^{\prime \prime \prime}(x)f′′′(x) of the function f(x)=115x6−4x2−13x−2+38xf(x)=\frac{1}{15} x^{6}-4 x^{2}-\frac{1}{3} x^{-2}+\frac{3}{8} xf(x)=151​x6−4x2−31​x−2+83​x.

Problem 24641

Approximate the area under $f(x)=e^{x}+3$ from $x=-2$ to $x=2$ with $n=4$ using left, right, midpoint, and average methods.

Find the limit as $x$ approaches 7 for the expression $\frac{x^{2}}{7} + \frac{7}{x^{2}}$ .

Find the linearization $L(x)$ of $f(x)=x^{4}+4x^{2}$ at $a=-1$ .

Find where the function $f(x)=3 x^{4}-4 x^{3}+3$ is concave up, concave down, and identify inflection points.

Evaluate the integral: $\int_{0}^{\pi}-3 \cos ^{2} x \sin x \, dx = \square(\text{Type an exact answer})$

Find the interval where the function $f$ is increasing given its derivative $f^{\prime}(x)=(x+2)^{2}(x-5)^{5}(x-6)^{4}$ .

Find the interval $x \in \square$ where $\sqrt[4]{1+2x} \approx 1 + \frac{1}{2}x$ .

Find the derivatives of the functions: (a) $y=x^{2} \sin 6x$ , (b) $y=\ln \sqrt{3+t^{2}}$ .

Find where the function $g(t)=\frac{t-5}{t+1}$ is concave up or down and identify inflection points.

Evaluate the integral $\int_{2}^{8} \frac{v^{2}+10}{\sqrt{v^{3}+30 v+3}} d v$ . Type the exact answer with radicals.

Find where the function $f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)$ is concave up or down and identify inflection points on $[-1, \infty)$ .

Find the differential for each function: (a) $y=e^{\tan \pi t}$ , (b) $y=\sqrt{4+\ln z}$ .

Find where the function $f(x)=4(x+1)^{\frac{5}{2}}(2 x-3)$ is concave up/down and identify inflection points on $[-1, \infty)$ .

Estimate $(8.03)^{2 / 3}$ using linear approximation or differentials.

Find the differential $d y$ for $y=\tan x$ , then evaluate it at $x=\pi/4$ and $d x=0.05$ .

Given $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ , find the limits as $r \to 0$ and $r \to \infty$ , and determine the minimizing $r$ .

Evaluate the integral: $\int_{2}^{4} \frac{v^{2}+3}{\sqrt{v^{3}+9 v+6}} d v=\square$ (Provide an exact answer with radicals.)

Find the differential $d y$ for the function $y=\tan x$ .

Evaluate the integral using geometry: $\int_{1}^{6}|2 x-4| d x$ . What is the simplified result?

Untersuche die Funktion $f_{1}(x)=x^{3}-x$ auf lokale Extremstellen.

Evaluate the integral: $\int_{2}^{4}(2 x-3)^{4} d x$ using the substitution $u=2x-3$ . Find $a$ , $b$ , and $f(u)$ .

Find the line equation for the linear approximation of $f(x)=10-3x^2$ at $a=3$ , then estimate $f(2.9)$ and compute percent error.

Find the critical numbers of the function $g(y)=\frac{y-1}{y^{2}-3 y+3}$ . Enter as a comma-separated list or DNE.

Given the function $h(x)=(x+2)^{3}-3x-1$ , find intervals of increase and decrease, local min/max values, inflection point, and concavity.

Find the instantaneous rate of change of $f(x)=-x^{5}-x^{4}+x^{2}+3x$ at $x=-1$ .

Find the derivative $f'(x)$ of the function $f(x) = 5x^{5} + x - 2$ and evaluate it at $x = ?$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (7 x)-7 x}{4 x^{3}}$ . Simplify your answer.

Evaluate the derivative $f^{\prime}(-2)$ for the function $f(x)=x^{3}+x$ .

Find the average rate of change of the function $g(x)=-x^{2}+8x+24$ from $x=2$ to $x=10$ .

Evaluate $\frac{d y}{d x}$ for $y=5 x^{2}+x$ at $x=-1$ .

Find the min and max values of $f(t)=t \sqrt{25-t^{2}}$ on the interval $[-1,5]$ .

Evaluate the integral from 1 to 8 of $\sqrt[3]{y}$ dy: $\int_{1}^{8} \sqrt[3]{y} d y=\square($ Type an integer or a simplified fraction.)

Verify if $f(x)=2-24x+2x^2$ meets Rolle's Theorem on $[5,7]$ and find $c$ values. List them separated by commas.

Find the antiderivative $F$ of $f(x)=x^{6}-2 x^{-3}+1$ such that $F(1)=1$ . What is $F(x)=\square$ ?

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{12} x^{-3}-\frac{1}{4} x^{4}-\frac{1}{6} x+2 x^{3}$ .

Find $r$ that minimizes $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ by setting the first derivative to 0.

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=\frac{1}{15} x^{6}-4 x^{2}-\frac{1}{3} x^{-2}+\frac{3}{8} x$ .

Find the third derivative $f^{\prime \prime \prime}(x)$ of the function $f(x)=-\frac{1}{3} x^{3}+\frac{1}{4} x^{2}+\frac{2}{3} x^{-2}-\frac{1}{6} x^{-4}$ .

Find the third derivative $\frac{d^{3} y}{d x^{3}}$ of the function $y=-\frac{2}{3} x^{4}-\frac{1}{6} x^{-2}+\frac{3}{20} x+\frac{1}{5} x^{5}$ .

Check if $f(x)=\sqrt{x}-\frac{1}{9} x$ meets Rolle's Theorem on $[0,81]$ and find all $c$ values. $c=\quad$

Evaluate the integral $\int_{4}^{6} \frac{v^{2}+10}{\sqrt{v^{3}+30 v+6}} d v$ using a change of variables.

Find the tangent line approximation for $f(x)=-x^{\frac{3}{4}}$ at $x=81$ to estimate $f(81.1)$ .

Evaluate the integral $\int_{2}^{3} \frac{v^{2}+16}{\sqrt{v^{3}+48 v+6}} d v=\square($ exact answer, with radicals.)

What is the minimum value of $f(6)$ given that $f(4)=7$ and $f^{\prime}(x) \geq 1$ for $4 \leq x \leq 6$ ?

Given the function $f(x)=7 \sin (x)+7 \cos (x)$ for $0 \leq x \leq 2\pi$ , find intervals where $f$ is increasing/decreasing, local min/max values, inflection points, and concavity intervals.

Check if $f(x)=2 x^{2}-3 x+1$ meets the Mean Value Theorem conditions on the interval $[0,2]$ .

A boat is 2 mi from shore, 13 mi from a restaurant.
a. Where should she land to minimize travel time?
b. Find the minimum rowing speed to row directly to the restaurant.

Given the function $f(x)=2 x^{3}+3 x^{2}-180 x$ , find where $f$ is increasing, decreasing, local min/max, inflection point, and concavity.

Find the tangent line at $x=1$ for $f(x)=-5 x^{\frac{2}{5}}$ and approximate $f(0.95)$ .

Minimize the aluminum cost $S=2 \pi r h + 2 \pi r^2$ for a can with volume $\pi r^2 h = V$ . Complete parts (a) to (d).

A cone has height $10 \mathrm{~cm}$ and diameter $10 \mathrm{~cm}$ . If $\mathrm{h}$ decreases at $-3/10 \mathrm{~cm/hr}$ , find $\frac{dV}{dt}$ when $\mathrm{h}=5$ . Round to three decimal places.

Find $f^{\prime}\left(\frac{3}{5}\right)$ if $f(x)=\cos^{-1}(x)$ .

Evaluate the integral $\int_{1}^{4} \frac{v^{2}+3}{\sqrt{v^{3}+9 v+6}} d v$ using a variable change.

A soda can manufacturer wants to minimize aluminum cost. Minimize $S=2 \pi r h + 2 \pi r^{2}$ with constraint $\pi r^{2} h=V$ .

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\arccos \left(\frac{\sqrt{5}}{3}+h\right)-\arccos \left(\frac{\sqrt{5}}{3}\right)}{h}$

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\sin ^{-1}\left(\frac{3}{5}+h\right)-\sin ^{-1}\left(\frac{3}{5}\right)}{h}$ .

Find the derivative of $f(x)=\sin^{-1}(3x)$ , denoted as $f^{\prime}(x)$ .

Find the derivative of the function $f(x)=\tan^{-1}(4x)$ , expressed in simplest form.

A soda can manufacturer aims to minimize aluminum cost. Given volume $V=339 \mathrm{~cm}^{3}$ , find $S(r)$ as $S(r)=\frac{678}{r}+2 \pi r^{2}$ . What happens to $S(r)$ as $r \rightarrow \infty$ ?

A right triangle has legs 15 in and 20 in. Short leg decreases by $10 \mathrm{in/sec}$ , long leg increases by $4 \mathrm{in/sec}$ . Find area change rate.

A boat is $5 \mathrm{mi}$ from shore, $8 \mathrm{mi}$ from a restaurant.
a. Where should she land to minimize travel time if she rows at $2 \mathrm{mi/hr}$ and walks at $3 \mathrm{mi/hr}$ ?
b. What is the minimum rowing speed for direct travel to the restaurant?

A right triangle has legs of 24 in and 32 in. The short leg increases by $4 \mathrm{in/sec}$ , and the long leg by $3 \mathrm{in/sec}$ . Find the rate of change of the hypotenuse.

Find the interval where the function $f$ is increasing given that $f^{\prime}(x)=(x+2)^{4}(x-5)^{3}(x-6)^{6}$ .

A cylinder's radius decreases at 9 m/min, with a constant volume of 728 m³. Find the height's rate of change when h = 3 m. Use $V=\pi r^{2} h$ and round to three decimal places.

A right triangle has legs of 12 in and 16 in, with the short leg increasing by $8 \mathrm{in/sec}$ and the long leg by $6 \mathrm{in/sec}$ . Find the hypotenuse's rate of change.

A cylinder's height decreases at 7 in/min with a volume of 501 in³. When the radius is 7 in, find the radius's rate of change. Use $V=\pi r^{2} h$ and round to three decimal places.

A cone's height rises at 8 ft/min with a volume of 178 cubic ft. Find the radius change rate when the radius is 4 ft. Use $V=\frac{1}{3} \pi r^{2} h$ and round to three decimal places.

What can we conclude about $a$ and $b$ if they are in nested intervals with widths less than $\frac{1}{n}$ ?

Find the derivative of $f^{\prime}(x)=\left(3 x^{2}-\frac{3}{2 \sqrt[3]{x}}+3 x^{2} e^{x^{5}}+\frac{1}{\sqrt{3 x}}\right)$ .