Calculus

Problem 6801

Find the derivative of $y=\frac{5}{4} x^{\frac{2}{3}}$ .

See Solution

Problem 6802

Maximize profit by finding $q$ from $C(q)=100+20 q e^{-0.02 q}$ and $p=80 e^{-0.02 q}$ .

See Solution

Problem 6803

Find the Taylor expansion of $f(x, y)=e^{x^{2} y} \sin (3 y)$ and determine the coefficient of the $x^{4} y^{5}$ term.

See Solution

Problem 6804

Find the points where these functions have horizontal tangents: a) $f(x)=x^{3} e^{2 x}$ , b) $f(x)=2 \sin x+\sin ^{2} x$ , c) $f(x)=\frac{x^{2}}{\sqrt{x^{2}-1}}$ .

See Solution

Problem 6805

Maximize profit by finding price $P$ per unit from cost $C(q)=100+20 q e^{-0.02 q}$ and price $p=80 e^{-0.02 q}$ .

See Solution

Problem 6806

Find the second derivative of $f(x)=\frac{x^{2}}{6+x}$ and calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(5)$ .

See Solution

Problem 6807

Find the second derivative $f^{\prime \prime}(x)$ of $f(x)=\sqrt{x^{2}+100}$ , then compute $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

See Solution

Problem 6808

Find the derivative of the function $y=x^{2/3}$ .

See Solution

Problem 6809

Find the tangent line to $y=(x+1)^{2}(x-3)^{3}$ at $x=1$ using point-slope form.

See Solution

Problem 6810

Differentiate $(x^{3}+2) \sqrt{x^{3}+3 x}$ .

See Solution

Problem 6811

Find the critical points of the function $f(x)=(x-3)^{2}(x+2)$ , given one is at $x=-\frac{1}{3}$ .

See Solution

Problem 6812

Find the tangent line to $y=(x-2) \sqrt{x-5}$ at $x=6$ in slope-intercept form.

See Solution

Problem 6813

Find the derivative $\frac{d y}{d x}$ of the function $y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)$ and simplify.

See Solution

Problem 6814

Find the derivative $\frac{d y}{d x}$ and simplify for $y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)$ .

See Solution

Problem 6815

Find the derivative of $y = \sin(4x^3) + \sin^5(3x^2)$ and simplify it.

See Solution

Problem 6816

Find the derivative of $y=\left(\frac{5 x+7}{3-2 x}\right)^{4}$ and simplify it.

See Solution

Problem 6817

Find the derivative of $y=\frac{x^{2}+3}{\sqrt{2 x-5}}$ and simplify your answer.

See Solution

Problem 6818

Find the derivative of $y=\sqrt{\frac{x^{2}+5}{x^{2}-5}}$ and simplify it.

See Solution

Problem 6819

Find the derivative of $y=\sqrt{\sec ^{3} 5 x}+\sec \sqrt{5 x}$ and simplify it.

See Solution

Problem 6820

Find $a$ and $b$ for the function $f(x)=\begin{cases} 2x^3-2x^2+6, & x<-2 \\ ax+b, & x \geq-2 \end{cases}$ to be continuous and differentiable.

See Solution

Problem 6821

Find the points where the functions have horizontal tangents: a) $f(x)=x^{3}-6 x^{2}+9 x+4$ b) $f(x)=(2 x+1)^{3}(x-3)^{4}$ .

See Solution

Problem 6822

Find the second derivative of $f(x)=\frac{\ln x}{11 x}$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(9)$ .

See Solution

Problem 6823

Find the points where the functions have horizontal tangents: a) $f(x)=x^{3}-6 x^{2}+9 x+4$ b) $f(x)=(2 x+1)^{3}(x-3)^{4}$ .

See Solution

Problem 6824

Find the limit: $\lim _{x \rightarrow-12} \frac{x^{2}-144}{x+12}$ . State if it does not exist.

See Solution

Problem 6825

Find the limit: $\lim _{x \rightarrow-5} \frac{6 x}{(x+5)^{3}}$ . State if it does not exist.

See Solution

Problem 6826

Ein Stau entsteht um 06:00 Uhr und löst sich bis 10:00 Uhr auf. Analysiere die Funktion $f(x) = -\frac{5}{16} x^{4}+3 x^{3}-9 x^{2}+8 x$ . Bestimme Nullstellen, Bedeutung von $f(2)<0$ , Zeitpunkt maximaler Staulänge und wann der Stau am längsten ist.

See Solution

Problem 6827

Calculate the limit: $\lim _{x \rightarrow 3^{+}} \sqrt{x^{2}-9}$ .

See Solution

Problem 6828

Find the value of $a$ for the substitution $x=a \sec \theta$ to simplify the integral $\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}$ . What new integral do you get?

See Solution

Problem 6829

Find the limits of the function $f(x)$ given that it levels out at $y=-1$ as $x \to \infty$ and decreases as $x \to -\infty$ . Calculate:
$\lim _{x \rightarrow \infty} f(x) \quad \text{and} \quad \lim _{x \rightarrow-\infty} f(x)$

See Solution

Problem 6830

Find the limits as $x \rightarrow \infty$ and $x \rightarrow -\infty$ for $\frac{1}{2 x^{2}}+3$ .

See Solution

Problem 6831

Find the derivative and roots of the function $f(x) = a_{k} x^{k} - \frac{1}{3125} x^{6} + \frac{54}{3125} x^{4} - \frac{1296}{3125} x^{2} + \frac{11664}{3125}$ .

See Solution

Problem 6832

Calculate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}=$ .

See Solution

Problem 6833

Find the instantaneous velocity of the object at $t=3$ for the position function $s(t)=4 t^{2}+3 t+5$ .

See Solution

Problem 6834

Evaluate the integral: $\int \frac{x}{\sqrt{1+x^{2}}} d x=$ . What is the result?

See Solution

Problem 6835

Find the average rate of change for $f(x)=x^{2}+10x$ from $x=0$ to $x=4$ . Which formula is correct? A, B, C, or D?

See Solution

Problem 6836

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for $\sqrt{x y}=9+x^{2} y^{6}$ .

See Solution

Problem 6837

Find the limits of $\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}$ as $x \to \infty$ and $x \to -\infty$ .

See Solution

Problem 6838

Solve the integral $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ and choose the correct option from the list.

See Solution

Problem 6839

Find the rate of change of $f(x)=x^{3}-2 e^{x}$ at $x=1$ and estimate $g^{\prime}(2)$ for $g(x)=\ln(3x)$ over $[1.9,2.1]$ .

See Solution

Problem 6840

Find the second derivative of $h(x)=(6-x^{2})(5x+18)$ .

See Solution

Problem 6841

Find terminal velocity $v_{\text{term}}$ in terms of $g$ and $D$ from $\frac{d^{2}s}{dt^{2}}=g-\frac{0.008}{D}\left(\frac{ds}{dt}\right)^{2}$ .
Then calculate $v_{\text{term}}(10^{-3})$ and $v_{\text{term}}(10^{-4})$ to three decimal places.

See Solution

Problem 6842

Is $f^{\prime}(100)=2$ means each extra \$1 spent on ads increases revenue by \$2? True or False?

See Solution

Problem 6843

Evaluate the limit: $f^{\prime}(3)=\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$ . What is $f^{\prime}(3)$ ?

See Solution

Problem 6844

If $\lim _{x \rightarrow 1^{-}} f(x)=3$ and $\lim _{x \rightarrow 1^{+}} f(x)=7$ , does $\lim _{x \rightarrow 1} f(x)$ exist? True or False?

See Solution

Problem 6845

Help with the function $h(x)=-\log (x)$ : graph it, find its derivative, or calculate its value for a specific $x$ .

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Problem 6846

Find the limit as $x$ approaches 1 for the expression $\frac{x-1}{x^{2}-1}$ . Choices: 0.5, 0, 2, 1.

See Solution

Problem 6847

A feverish person's temperature had a positive first derivative and negative second derivative for three minutes. What happened?

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Problem 6848

Find the limit as $x$ approaches 0 from the left for the expression $\frac{7 x}{x}$ .

See Solution

Problem 6849

Select the correct interpretation of $f^{\prime}(2012)<0$ and $f^{\prime \prime}(2012)<0$ for population $P=f(t)$ .

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Problem 6850

True or False: $f^{\prime}(a)$ represents the average rate of change of $f$ on $[a, a+h]$ for small $h$ .

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Problem 6851

Find the limit as $x$ approaches 0 for the expression $\frac{7 x}{x}$ .

See Solution

Problem 6852

Find the integral: $\int \frac{x}{\sqrt{1+x^{2}}} d x$

See Solution

Problem 6853

Calculate the integral $\int \tan ^{5}\left(\frac{x}{6}\right) d x$ .

See Solution

Problem 6854

Find the value of $a$ for the substitution $x=a \sec \theta$ to simplify the integral $\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}$ . What is the resulting integral?

See Solution

Problem 6855

Calculate the volume from rotating the area between $y=4x-x^{2}$ and $y=x$ around the y-axis using cylindrical shells.

See Solution

Problem 6856

A grindstone with radius $6.5 \mathrm{~cm}$ rotates at $6500 \mathrm{rpm}$ . Find centripetal acceleration in multiples of $g$ and linear speed in $\mathrm{m/s}$ .

See Solution

Problem 6857

Find the volume using cylindrical shells for these curves rotated around the $y$ -axis:
1. $y=\sqrt[3]{x}$ , $y=0$ , $x=1$
2. $y=x^{3}$ , $y=0$ , $x=1$ , $x=2$
3. $y=e^{-x^{2}}$ , $y=0$ , $x=0$ , $x=1$
4. $y=4x-x^{2}$ , $y=x$
5. $y=x^{2}$ , $y=6x-2x^{2}$ .

See Solution

Problem 6858

A car moves at $v=14.3 \mathrm{~m/s}$ with tire diameter $d=0.71 \mathrm{~m}$ . Find centripetal acceleration $a_{c}$ and the ratio $a_{n}/a_{c}$ if speed doubles.

See Solution

Problem 6859

Calculate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ .

See Solution

Problem 6860

Determine if the series converges or diverges and find the sum if it converges: $\sum_{k=1}^{\infty} 4\left(\frac{3}{2}\right)^{k-1}$

See Solution

Problem 6861

Find $f^{\prime}(2)$ for $f(x)=g(x)^{x}$ given $g(2)=3$ and $g^{\prime}(2)=2$ .

See Solution

Problem 6862

Calculate the volume of revolution for the area between $y=x^{3}$ , $y=8$ , $x=0$ , $x=3$ around $x=3$ using cylindrical shells.

See Solution

Problem 6863

Sketch the graph of the function $f(x)$ given: $f(0) = 3$ , $\lim_{x \to \pm\infty} f(x) = 0$ , $\lim_{x \to 0^{+}} f(x) = 2$ , and $\lim_{x \to 0^{-}} f(x) = -2$ .

See Solution

Problem 6864

Find the rate of change of power $P(R)=\frac{2.5 R}{(R+0.5)^{2}}$ W at $R=10 \Omega$ . Round to four decimal places.

See Solution

Problem 6865

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=6x^{2}$ , simplify your answer.

See Solution

Problem 6866

Find the centripetal acceleration of a satellite at a distance of $42,250.0 \mathrm{~km}$ from Earth's center. $a_{\mathrm{c}}=$

See Solution

Problem 6867

Find $f^{\prime}(2)$ for $f(x)=g(x)^{x}$ , given $g(2)=3$ and $g^{\prime}(2)=2$ .

See Solution

Problem 6868

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-x+2$ , with $h \neq 0$ . Simplify your answer.

See Solution

Problem 6869

Find the rate of change of power $P(R)=\frac{2.5 R}{(R+0.5)^{2}}$ at $R=10 \Omega$ . Round to four decimal places.

See Solution

Problem 6870

Find the 1st, 2nd, and 3rd derivatives of $y(x)=\sqrt[9]{x}$ . $y'(x)=$ $y''(x)=$ $y'''(x)=$

See Solution

Problem 6871

Calculate the volume of the solid formed by rotating the area between $y=4-2x$ , $y=0$ , and $x=0$ around $x=-1$ using cylindrical shells.

See Solution

Problem 6872

Differentiate $\sin (2 x)+\cos (8 y)=\sin (2 x) \cos (8 y)$ implicitly to find $\frac{d y}{d x}$ .

See Solution

Problem 6873

Find the ball's velocity, acceleration, jerk, and snap at $t=1.0$ s from the motion equation $s(t)=-2.0 t^{4}+1.0 t^{3}+3.5 t^{2}+2.6 t+0.2$ . Round to one decimal place.

See Solution

Problem 6874

Differentiate implicitly: $\sin(2x) + \cos(8y) = \sin(2x) \cos(8y)$ to find $\frac{dy}{dx}$ .

See Solution

Problem 6875

Find $y^{\prime \prime}$ and $y^{\prime \prime \prime}$ for $y(t)=2 t^{-2}+4 t^{-5/3}$ .

See Solution

Problem 6876

Sketch a graph of $F(x)$ where $F'(x) < 0$ on $(-\infty,-5)$ and $(-2,7)$ , and $F'(x) > 0$ on $(-5,-2)$ and $(7, \infty)$ .

See Solution

Problem 6877

Find the fourth derivative $\frac{d^{4} y}{d x^{4}}$ of the function $\frac{y^{3} x^{2}+\tan (y x)}{3 x y^{3}+2 \tan \left(x y^{6}\right)}$ .

See Solution

Problem 6878

Find the second and third derivatives of $y(x)=\frac{5 e^{x}}{x}$ . Calculate $y^{\prime \prime}(x)$ and $y^{\prime \prime \prime}(x)$ .

See Solution

Problem 6879

Evaluate the integral $\int_{-\pi / 4}^{\pi / 4}\left(x^{2}+\ln |\cos x|\right) \sin \frac{x}{2} \, dx$ .

See Solution

Problem 6880

Find the third derivative of $f(t)=5 t^{6}-t$ at $t=4$ : $f^{\prime \prime \prime}(4)=$

See Solution

Problem 6881

Find a formula for $f^{(n)}$ where $f(x)=(x+8)^{-1}$ . What is $f^{(n)}(x)=?$

See Solution

Problem 6882

Determine a formula for the $n$ -th derivative $f^{(n)}$ of $f(x)=5 x^{2} e^{x}$ .

See Solution

Problem 6883

Differentiate $y$ implicitly from $\ln(7y) = 4xy$ . Find $\frac{d^2y}{dx^2}$ , set it to 0, and solve for $(x, y)$ .

See Solution

Problem 6884

Calculate the integral: $I = \int 6 x^{2} e^{x^{3}} \, dx$ .

See Solution

Problem 6885

Find the integral $f(x)=\int_{0}^{x} \ln (\cosh (t)) d t$ .

See Solution

Problem 6886

Find the tangent line equation to $g(x)=f^{\prime}(x)$ at $x=-1$ for $f(x)=5 x^{3}+3 x^{2}+2 x$ .

See Solution

Problem 6887

Evaluate the integral: $\int \frac{-2 x^{5}+12 x^{4}-3 x^{3}+15 x^{2}-93 x+47}{6 x^{6}-2 x^{5}+6 x^{4}-50 x^{3}+16 x^{2}-48 x+16} d x$

See Solution

Problem 6888

Find the second derivative of $h(x) = (7 - x^2)(5x + 13)$ . What is $h''(x)$ ?

See Solution

Problem 6889

Find the tangent line equation for $g(x)=f^{\prime}(x)$ at $x=-1$ , where $f(x)=5 x^{3}+3 x^{2}+2 x$ .

See Solution

Problem 6890

Evaluate the Riemann sum for $I=\int_{0}^{2} x \, dx$ and determine which error statement is optimal as $n \rightarrow+\infty$ .

See Solution

Problem 6891

Bestimme die Stammfunktion von $\frac{4}{e} x$ .

See Solution

Problem 6892

Find terminal velocity $v_{\text{term}}$ in terms of $g$ and $D$ , then calculate $v_{\text{term}}$ for $D=10^{-3}$ m and $D=10^{-4}$ m.

See Solution

Problem 6893

Soit $f(x)=\frac{x}{e^{x}}$ . Quelle est la dérivée $f^{\prime}(x)$ parmi les options suivantes ? a. $f^{\prime}(x)=\mathrm{e}^{-x}$ b. $f^{\prime}(x)=x \mathrm{e}^{-x}$ c. $f^{\prime}(x)=(1-x) \mathrm{e}^{-x}$ d. $f^{\prime}(x)=(1+x) \mathrm{e}^{-x}$

See Solution

Problem 6894

Find the derivative of $\ln(x e^{x^{3}})$ .

See Solution

Problem 6895

Find $F^{\prime}(0)$ for $F(x)=f(g(x))$ given $f(1)=7$ , $f^{\prime}(1)=2$ , $f^{\prime}(0)=3$ , $g(0)=1$ , $g^{\prime}(0)=5$ .

See Solution

Problem 6896

Gegeben ist die Funktion $f(x)=x^{3}-6 x^{2}+8 x$ . Warum ist $A=A_{0}(4)-A_{0}(0)=0$ nicht korrekt? Bestimmen Sie den echten Flächeninhalt.

See Solution

Problem 6897

Find $k$ such that $\int_{1}^{k}(4 x^{3}-18 x) dx=-12$ .

See Solution

Problem 6898

Bestimmen Sie, wo die lokale Änderungsrate von $f(x)$ negativ oder positiv ist, und skizzieren Sie $f^{\prime}$ . Funktion: $f(x)=\frac{1}{3} x^{3}-3,5 x^{2}+10 x$ .

See Solution

Problem 6899

Bestimme die Integrale: a) $\int_{2}^{5} x \, dx$ , b) $\int_{-1}^{1}(2x+1) \, dx$ , c) $\int_{-1}^{2}-2t \, dt$ , d) $\int_{0}^{4}-2 \, dx$ , e) $\int_{-5}^{0}(-t-5) \, dt$ .

See Solution

Problem 6900

Berechne die Oberstufe von $f(x)=x^{2}$ im Intervall $[0, 1]$ mit 16 Rechtecken.

See Solution

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Calculus

Problem 6801

Find the derivative of y=54x23y=\frac{5}{4} x^{\frac{2}{3}}y=45​x32​.

Problem 6802

Maximize profit by finding qqq from C(q)=100+20qe−0.02qC(q)=100+20 q e^{-0.02 q}C(q)=100+20qe−0.02q and p=80e−0.02qp=80 e^{-0.02 q}p=80e−0.02q.

Problem 6803

Find the Taylor expansion of f(x,y)=ex2ysin⁡(3y)f(x, y)=e^{x^{2} y} \sin (3 y)f(x,y)=ex2ysin(3y) and determine the coefficient of the x4y5x^{4} y^{5}x4y5 term.

Problem 6804

Find the points where these functions have horizontal tangents: a) f(x)=x3e2xf(x)=x^{3} e^{2 x}f(x)=x3e2x, b) f(x)=2sin⁡x+sin⁡2xf(x)=2 \sin x+\sin ^{2} xf(x)=2sinx+sin2x, c) f(x)=x2x2−1f(x)=\frac{x^{2}}{\sqrt{x^{2}-1}}f(x)=x2−1​x2​.

Problem 6805

Maximize profit by finding price PPP per unit from cost C(q)=100+20qe−0.02qC(q)=100+20 q e^{-0.02 q}C(q)=100+20qe−0.02q and price p=80e−0.02qp=80 e^{-0.02 q}p=80e−0.02q.

Problem 6806

Find the second derivative of f(x)=x26+xf(x)=\frac{x^{2}}{6+x}f(x)=6+xx2​ and calculate f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(5)f^{\prime \prime}(5)f′′(5).

Problem 6807

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) of f(x)=x2+100f(x)=\sqrt{x^{2}+100}f(x)=x2+100​, then compute f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(7)f^{\prime \prime}(7)f′′(7).

Problem 6808

Find the derivative of the function y=x2/3y=x^{2/3}y=x2/3.

Problem 6809

Find the tangent line to y=(x+1)2(x−3)3y=(x+1)^{2}(x-3)^{3}y=(x+1)2(x−3)3 at x=1x=1x=1 using point-slope form.

Problem 6810

Differentiate (x3+2)x3+3x(x^{3}+2) \sqrt{x^{3}+3 x}(x3+2)x3+3x​.

Problem 6811

Find the critical points of the function f(x)=(x−3)2(x+2)f(x)=(x-3)^{2}(x+2)f(x)=(x−3)2(x+2), given one is at x=−13x=-\frac{1}{3}x=−31​.

Problem 6812

Find the tangent line to y=(x−2)x−5y=(x-2) \sqrt{x-5}y=(x−2)x−5​ at x=6x=6x=6 in slope-intercept form.

Problem 6813

Find the derivative dydx\frac{d y}{d x}dxdy​ of the function y=3x4+(3x)4−4x3+cot⁡(3x)−sec⁡(π)y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)y=3x4+(3x)4−3x​4​+cot(3x)−sec(π) and simplify.

Problem 6814

Find the derivative dydx\frac{d y}{d x}dxdy​ and simplify for y=3x4+(3x)4−4x3+cot⁡(3x)−sec⁡(π)y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)y=3x4+(3x)4−3x​4​+cot(3x)−sec(π).

Problem 6815

Find the derivative of y=sin⁡(4x3)+sin⁡5(3x2)y = \sin(4x^3) + \sin^5(3x^2)y=sin(4x3)+sin5(3x2) and simplify it.

Problem 6816

Find the derivative of y=(5x+73−2x)4y=\left(\frac{5 x+7}{3-2 x}\right)^{4}y=(3−2x5x+7​)4 and simplify it.

Problem 6817

Find the derivative of y=x2+32x−5y=\frac{x^{2}+3}{\sqrt{2 x-5}}y=2x−5​x2+3​ and simplify your answer.

Problem 6818

Find the derivative of y=x2+5x2−5y=\sqrt{\frac{x^{2}+5}{x^{2}-5}}y=x2−5x2+5​​ and simplify it.

Problem 6819

Find the derivative of y=sec⁡35x+sec⁡5xy=\sqrt{\sec ^{3} 5 x}+\sec \sqrt{5 x}y=sec35x​+sec5x​ and simplify it.

Problem 6820

Find aaa and bbb for the function f(x)={2x3−2x2+6,x<−2ax+b,x≥−2f(x)=\begin{cases} 2x^3-2x^2+6, & x<-2 \\ ax+b, & x \geq-2 \end{cases}f(x)={2x3−2x2+6,ax+b,​x<−2x≥−2​ to be continuous and differentiable.

Problem 6821

Find the points where the functions have horizontal tangents: a) f(x)=x3−6x2+9x+4f(x)=x^{3}-6 x^{2}+9 x+4f(x)=x3−6x2+9x+4 b) f(x)=(2x+1)3(x−3)4f(x)=(2 x+1)^{3}(x-3)^{4}f(x)=(2x+1)3(x−3)4.

Problem 6822

Find the second derivative of f(x)=ln⁡x11xf(x)=\frac{\ln x}{11 x}f(x)=11xlnx​, then calculate f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(9)f^{\prime \prime}(9)f′′(9).

Problem 6823

Find the points where the functions have horizontal tangents: a) f(x)=x3−6x2+9x+4f(x)=x^{3}-6 x^{2}+9 x+4f(x)=x3−6x2+9x+4 b) f(x)=(2x+1)3(x−3)4f(x)=(2 x+1)^{3}(x-3)^{4}f(x)=(2x+1)3(x−3)4.

Problem 6824

Find the limit: lim⁡x→−12x2−144x+12\lim _{x \rightarrow-12} \frac{x^{2}-144}{x+12}x→−12lim​x+12x2−144​. State if it does not exist.

Problem 6825

Find the limit: lim⁡x→−56x(x+5)3\lim _{x \rightarrow-5} \frac{6 x}{(x+5)^{3}}x→−5lim​(x+5)36x​. State if it does not exist.

Problem 6826

Problem 6827

Calculate the limit: lim⁡x→3+x2−9\lim _{x \rightarrow 3^{+}} \sqrt{x^{2}-9}limx→3+​x2−9​.

Problem 6828

Find the value of aaa for the substitution x=asec⁡θx=a \sec \thetax=asecθ to simplify the integral ∫x4dxx2−4\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}∫x2−4​x4dx​. What new integral do you get?

Problem 6829

Problem 6830

Find the limits as x→∞x \rightarrow \inftyx→∞ and x→−∞x \rightarrow -\inftyx→−∞ for 12x2+3\frac{1}{2 x^{2}}+32x21​+3.

Problem 6831

Find the derivative and roots of the function f(x)=akxk−13125x6+543125x4−12963125x2+116643125f(x) = a_{k} x^{k} - \frac{1}{3125} x^{6} + \frac{54}{3125} x^{4} - \frac{1296}{3125} x^{2} + \frac{11664}{3125}f(x)=ak​xk−31251​x6+312554​x4−31251296​x2+312511664​.

Problem 6832

Calculate the integral: ∫x3dx9−x2=\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}=∫9−x2​x3dx​=.

Problem 6833

Find the instantaneous velocity of the object at t=3t=3t=3 for the position function s(t)=4t2+3t+5s(t)=4 t^{2}+3 t+5s(t)=4t2+3t+5.

Problem 6834

Evaluate the integral: ∫x1+x2dx=\int \frac{x}{\sqrt{1+x^{2}}} d x=∫1+x2​x​dx=. What is the result?

Problem 6835

Find the average rate of change for f(x)=x2+10xf(x)=x^{2}+10xf(x)=x2+10x from x=0x=0x=0 to x=4x=4x=4. Which formula is correct? A, B, C, or D?

Problem 6836

Find the derivative dydx\frac{d y}{d x}dxdy​ using implicit differentiation for xy=9+x2y6\sqrt{x y}=9+x^{2} y^{6}xy​=9+x2y6.

Problem 6837

Find the limits of (x−1)(x2+x)(x+2)2(x−3)\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}(x+2)2(x−3)(x−1)(x2+x)​ as x→∞x \to \inftyx→∞ and x→−∞x \to -\inftyx→−∞.

Problem 6838

Solve the integral ∫x3dx9−x2\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}∫9−x2​x3dx​ and choose the correct option from the list.

Problem 6839

Find the rate of change of f(x)=x3−2exf(x)=x^{3}-2 e^{x}f(x)=x3−2ex at x=1x=1x=1 and estimate g′(2)g^{\prime}(2)g′(2) for g(x)=ln⁡(3x)g(x)=\ln(3x)g(x)=ln(3x) over [1.9,2.1][1.9,2.1][1.9,2.1].

Problem 6840

Find the second derivative of h(x)=(6−x2)(5x+18)h(x)=(6-x^{2})(5x+18)h(x)=(6−x2)(5x+18).

Problem 6841

Find the derivative of $y=\frac{5}{4} x^{\frac{2}{3}}$ .

Maximize profit by finding $q$ from $C(q)=100+20 q e^{-0.02 q}$ and $p=80 e^{-0.02 q}$ .

Find the Taylor expansion of $f(x, y)=e^{x^{2} y} \sin (3 y)$ and determine the coefficient of the $x^{4} y^{5}$ term.

Find the points where these functions have horizontal tangents: a) $f(x)=x^{3} e^{2 x}$ , b) $f(x)=2 \sin x+\sin ^{2} x$ , c) $f(x)=\frac{x^{2}}{\sqrt{x^{2}-1}}$ .

Maximize profit by finding price $P$ per unit from cost $C(q)=100+20 q e^{-0.02 q}$ and price $p=80 e^{-0.02 q}$ .

Find the second derivative of $f(x)=\frac{x^{2}}{6+x}$ and calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(5)$ .

Find the second derivative $f^{\prime \prime}(x)$ of $f(x)=\sqrt{x^{2}+100}$ , then compute $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

Find the derivative of the function $y=x^{2/3}$ .

Find the tangent line to $y=(x+1)^{2}(x-3)^{3}$ at $x=1$ using point-slope form.

Differentiate $(x^{3}+2) \sqrt{x^{3}+3 x}$ .

Find the critical points of the function $f(x)=(x-3)^{2}(x+2)$ , given one is at $x=-\frac{1}{3}$ .

Find the tangent line to $y=(x-2) \sqrt{x-5}$ at $x=6$ in slope-intercept form.

Find the derivative $\frac{d y}{d x}$ of the function $y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)$ and simplify.

Find the derivative $\frac{d y}{d x}$ and simplify for $y=3 x^{4}+(3 x)^{4}-\frac{4}{\sqrt[3]{x}}+\cot (3 x)-\sec (\pi)$ .

Find the derivative of $y = \sin(4x^3) + \sin^5(3x^2)$ and simplify it.

Find the derivative of $y=\left(\frac{5 x+7}{3-2 x}\right)^{4}$ and simplify it.

Find the derivative of $y=\frac{x^{2}+3}{\sqrt{2 x-5}}$ and simplify your answer.

Find the derivative of $y=\sqrt{\frac{x^{2}+5}{x^{2}-5}}$ and simplify it.

Find the derivative of $y=\sqrt{\sec ^{3} 5 x}+\sec \sqrt{5 x}$ and simplify it.

Find $a$ and $b$ for the function $f(x)=\begin{cases} 2x^3-2x^2+6, & x<-2 \\ ax+b, & x \geq-2 \end{cases}$ to be continuous and differentiable.

Find the points where the functions have horizontal tangents: a) $f(x)=x^{3}-6 x^{2}+9 x+4$ b) $f(x)=(2 x+1)^{3}(x-3)^{4}$ .

Find the second derivative of $f(x)=\frac{\ln x}{11 x}$ , then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(9)$ .

Find the points where the functions have horizontal tangents: a) $f(x)=x^{3}-6 x^{2}+9 x+4$ b) $f(x)=(2 x+1)^{3}(x-3)^{4}$ .

Find the limit: $\lim _{x \rightarrow-12} \frac{x^{2}-144}{x+12}$ . State if it does not exist.

Find the limit: $\lim _{x \rightarrow-5} \frac{6 x}{(x+5)^{3}}$ . State if it does not exist.

Calculate the limit: $\lim _{x \rightarrow 3^{+}} \sqrt{x^{2}-9}$ .

Find the value of $a$ for the substitution $x=a \sec \theta$ to simplify the integral $\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}$ . What new integral do you get?

Find the limits as $x \rightarrow \infty$ and $x \rightarrow -\infty$ for $\frac{1}{2 x^{2}}+3$ .

Find the derivative and roots of the function $f(x) = a_{k} x^{k} - \frac{1}{3125} x^{6} + \frac{54}{3125} x^{4} - \frac{1296}{3125} x^{2} + \frac{11664}{3125}$ .

Calculate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}=$ .

Find the instantaneous velocity of the object at $t=3$ for the position function $s(t)=4 t^{2}+3 t+5$ .

Evaluate the integral: $\int \frac{x}{\sqrt{1+x^{2}}} d x=$ . What is the result?

Find the average rate of change for $f(x)=x^{2}+10x$ from $x=0$ to $x=4$ . Which formula is correct? A, B, C, or D?

Find the derivative $\frac{d y}{d x}$ using implicit differentiation for $\sqrt{x y}=9+x^{2} y^{6}$ .

Find the limits of $\frac{(x-1)(x^{2}+x)}{(x+2)^{2}(x-3)}$ as $x \to \infty$ and $x \to -\infty$ .

Solve the integral $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ and choose the correct option from the list.

Find the rate of change of $f(x)=x^{3}-2 e^{x}$ at $x=1$ and estimate $g^{\prime}(2)$ for $g(x)=\ln(3x)$ over $[1.9,2.1]$ .

Find the second derivative of $h(x)=(6-x^{2})(5x+18)$ .

Is $f^{\prime}(100)=2$ means each extra \$1 spent on ads increases revenue by \$2? True or False?

Evaluate the limit: $f^{\prime}(3)=\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}$ . What is $f^{\prime}(3)$ ?

If $\lim _{x \rightarrow 1^{-}} f(x)=3$ and $\lim _{x \rightarrow 1^{+}} f(x)=7$ , does $\lim _{x \rightarrow 1} f(x)$ exist? True or False?

Help with the function $h(x)=-\log (x)$ : graph it, find its derivative, or calculate its value for a specific $x$ .

Find the limit as $x$ approaches 1 for the expression $\frac{x-1}{x^{2}-1}$ . Choices: 0.5, 0, 2, 1.

Find the limit as $x$ approaches 0 from the left for the expression $\frac{7 x}{x}$ .

Select the correct interpretation of $f^{\prime}(2012)<0$ and $f^{\prime \prime}(2012)<0$ for population $P=f(t)$ .

True or False: $f^{\prime}(a)$ represents the average rate of change of $f$ on $[a, a+h]$ for small $h$ .

Find the limit as $x$ approaches 0 for the expression $\frac{7 x}{x}$ .

Find the integral: $\int \frac{x}{\sqrt{1+x^{2}}} d x$

Calculate the integral $\int \tan ^{5}\left(\frac{x}{6}\right) d x$ .

Find the value of $a$ for the substitution $x=a \sec \theta$ to simplify the integral $\int \frac{x^{4} d x}{\sqrt{x^{2}-4}}$ . What is the resulting integral?

Calculate the volume from rotating the area between $y=4x-x^{2}$ and $y=x$ around the y-axis using cylindrical shells.

A grindstone with radius $6.5 \mathrm{~cm}$ rotates at $6500 \mathrm{rpm}$ . Find centripetal acceleration in multiples of $g$ and linear speed in $\mathrm{m/s}$ .

A car moves at $v=14.3 \mathrm{~m/s}$ with tire diameter $d=0.71 \mathrm{~m}$ . Find centripetal acceleration $a_{c}$ and the ratio $a_{n}/a_{c}$ if speed doubles.

Calculate the integral: $\int \frac{x^{3} d x}{\sqrt{9-x^{2}}}$ .

Determine if the series converges or diverges and find the sum if it converges: $\sum_{k=1}^{\infty} 4\left(\frac{3}{2}\right)^{k-1}$

Find $f^{\prime}(2)$ for $f(x)=g(x)^{x}$ given $g(2)=3$ and $g^{\prime}(2)=2$ .

Calculate the volume of revolution for the area between $y=x^{3}$ , $y=8$ , $x=0$ , $x=3$ around $x=3$ using cylindrical shells.

Sketch the graph of the function $f(x)$ given: $f(0) = 3$ , $\lim_{x \to \pm\infty} f(x) = 0$ , $\lim_{x \to 0^{+}} f(x) = 2$ , and $\lim_{x \to 0^{-}} f(x) = -2$ .

Find the rate of change of power $P(R)=\frac{2.5 R}{(R+0.5)^{2}}$ W at $R=10 \Omega$ . Round to four decimal places.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=6x^{2}$ , simplify your answer.

Find the centripetal acceleration of a satellite at a distance of $42,250.0 \mathrm{~km}$ from Earth's center. $a_{\mathrm{c}}=$

Find $f^{\prime}(2)$ for $f(x)=g(x)^{x}$ , given $g(2)=3$ and $g^{\prime}(2)=2$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-x+2$ , with $h \neq 0$ . Simplify your answer.

Find the rate of change of power $P(R)=\frac{2.5 R}{(R+0.5)^{2}}$ at $R=10 \Omega$ . Round to four decimal places.

Find the 1st, 2nd, and 3rd derivatives of $y(x)=\sqrt[9]{x}$ . $y'(x)=$ $y''(x)=$ $y'''(x)=$

Calculate the volume of the solid formed by rotating the area between $y=4-2x$ , $y=0$ , and $x=0$ around $x=-1$ using cylindrical shells.

Differentiate $\sin (2 x)+\cos (8 y)=\sin (2 x) \cos (8 y)$ implicitly to find $\frac{d y}{d x}$ .

Find the ball's velocity, acceleration, jerk, and snap at $t=1.0$ s from the motion equation $s(t)=-2.0 t^{4}+1.0 t^{3}+3.5 t^{2}+2.6 t+0.2$ . Round to one decimal place.

Differentiate implicitly: $\sin(2x) + \cos(8y) = \sin(2x) \cos(8y)$ to find $\frac{dy}{dx}$ .