Calculus

Problem 24901

Differentiate $y=(2 x)^{\ln (6 x)}$ using logarithmic differentiation to find $y^{\prime}$ .

See Solution

Problem 24902

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

See Solution

Problem 24903

Find all values of $c$ satisfying the Mean Value Theorem for $f(x) = x^3 + 13x^2 + 47x + 30$ on $[-5, 0]$ .

See Solution

Problem 24904

A $0.65 \mathrm{~kg}$ balloon is launched at $60^{\circ}$ from a $52 \mathrm{~m}$ building with $12 \mathrm{~m/s}$ . Find its speed on impact using energy methods.

See Solution

Problem 24905

A roller coaster starts from rest at 35 m. Find speeds at points 2 (0 m), 3 (28 m), and 4 (15 m) using energy equations.

See Solution

Problem 24906

Evaluate $\int_{6}^{7} f(x) \mathrm{d} x$ given $\int_{3}^{7} f(x)=9$ and $\int_{3}^{6} 2 f(x) \mathrm{d} x=4$ .

See Solution

Problem 24907

Find the area $A$ of the shaded region under the standard normal distribution with mean 0 and std dev 1. Round to four decimal places.

See Solution

Problem 24908

Soit $f(x)=\left\{\begin{array}{lll}x^{3}+1 & \text { si } & x<1 \\ \sqrt{x-1} & \text { si } & x \geq 1\end{array}\right.$ . Vérifiez la continuité en $x=1$ , les nombres de transition et les points d'inflexion.

See Solution

Problem 24909

Evaluate $\int_{0}^{2} x f\left(x^{2}\right) - \sqrt{4 - x^{2}} \, dx$ given $\int_{0}^{4} f(x) \, dx = 6$ .

See Solution

Problem 24910

Find the 3rd-degree Maclaurin polynomial for $f(x)=\frac{1}{7x+3}$ .
Your answer is: $p_{3}(x)=$

See Solution

Problem 24911

Find the 3rd-degree Maclaurin polynomial for $f(x)=\cos(4x)$ .
Provide your answer below: $p_{3}(x)=$

See Solution

Problem 24912

Find the Maclaurin polynomial $p_{2}(x)$ for $f(x)=\cos \left(\frac{x}{3}\right)$ .
Answer: $\dot{p}_{2}(x)=\square$

See Solution

Problem 24913

Find the first and second degree Maclaurin polynomials for $f(x)=5 e^{x}+3 e^{-x}$ at $a=0$ .

See Solution

Problem 24914

Find the Maclaurin polynomial $p_{1}(x)$ for the function $f(x)=e^{2x}$ .
Provide your answer below: $p_{1}(x)=\square$

See Solution

Problem 24915

Find the third-degree Maclaurin polynomial $p_{3}(x)$ for the function $f(x)=5^{x}$ .
Provide your answer below: $p_{3}(x)=\square$

See Solution

Problem 24916

Find the rate of change of the function $f(x)=x^{3}-x^{2}+x-5$ at $x=2$ . Options: 6, 9, -12, -10.

See Solution

Problem 24917

Find the second Maclaurin polynomial $p_{2}(x)$ for $f(x)=\frac{1}{6x+8}$ .
Provide your answer below: $p_{2}(x)=\square$

See Solution

Problem 24918

Find the limit: $\lim _{x \rightarrow 5} \frac{\sqrt{x}-\sqrt{5}}{x-5}=$

See Solution

Problem 24919

Calculate the length of the curve defined by $x=1+3 t^{2}$ and $y=4+2 t^{3}$ for $0 \leq t \leq 1$ .

See Solution

Problem 24920

Find the gradient of the curve $3 x^{2} - y^{2} = 11$ at the point $(2, 3)$ .

See Solution

Problem 24921

Find the value of $a$ if $\lim _{x \rightarrow 1} \frac{a\left(x^{2}-1\right)}{x^{4}-1}=2$ .

See Solution

Problem 24922

Estimate $\sin \left(\frac{\pi}{25}\right)$ using the second Maclaurin polynomial for $f(x)=\sin (x)$ . Round to four decimal places.
$\sin \left(\frac{\pi}{25}\right) \approx \square$

See Solution

Problem 24923

If $x^{2}+y^{2}=9$ , find $\frac{dy}{dx}$ in terms of $x$ and $y$ . Choices include $2x+2y$ , $9-x^{2}$ , $(3-x)/y$ , and $-x/y$ .

See Solution

Problem 24924

Estimate $e^{0.5}$ using the fourth Maclaurin polynomial for $f(x)=e^{x}$ . Round to four decimal places.

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Problem 24925

Evaluate the integral $\int 2 x\left(x^{2}+1\right)^{8} d x$ using the substitution $u=x^{2}+1$ .

See Solution

Problem 24926

Find the area of the surface formed by rotating $x=\frac{1}{3}(y^{2}+2)^{3/2}$ , for $1 \leq y \leq 3$ , around the x-axis.

See Solution

Problem 24927

A cylinder's radius grows at 3 m/s, volume at 108 m³/s. With height 6 m and volume 33 m³, find height's rate of change. Use $V=\pi r^{2} h$ . Round to three decimal places.

See Solution

Problem 24928

Estimate $e^{7}$ using the second Taylor polynomial of $f(x)=e^{x}$ at $x=6$ . Round to four decimal places.
$e^{7} \approx \square$

See Solution

Problem 24929

Find the integral $\int(4 x^{3}+9) \sqrt{x^{4}+9 x} \, dx$ using the substitution $u=x^{4}+9 x$ and verify by differentiation.

See Solution

Problem 24930

Evaluate the integral $\int \sqrt{9 x+2} \, dx$ using the substitution $u = a x + b$ .

See Solution

Problem 24931

Find the one-sided limit: $\lim _{x \rightarrow 1^{-}} \frac{4}{x-1}$ . Options: -2, -4, $\infty$ , $-\infty$ .

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Problem 24932

Estimate $\sin(-1 + 4\pi)$ using the fourth Taylor polynomial for $f(x) = \sin(x)$ at $x = 4\pi$ . Round to four decimal places.
Your answer: $\sin(-1 + 4\pi) \approx$

See Solution

Problem 24933

Estimate $\cos(-1 + 2\pi)$ using the second Taylor polynomial of $f(x) = \cos(x)$ at $x = 2\pi$ . Round to four decimal places.
$\cos(-1 + 2\pi) \approx \square$

See Solution

Problem 24934

Estimate $\cos(1+\pi)$ using the fourth Taylor polynomial of $f(x)=\cos(x)$ at $x=\pi$ . Round to four decimal places.
Your answer: $\cos(1+\pi) \approx \square$

See Solution

Problem 24935

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{x^{2}+5 x}{|x|}$ . What is the answer?

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Problem 24936

Find the integral $\int -16 x \sin(8 x^{2} - 1) \, dx$ using $u = 8 x^{2} - 1$ and check by differentiating.

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Problem 24937

Estimate $\sin \left(\frac{\pi}{27}\right)$ using the second Maclaurin polynomial for $f(x)=\sin (x)$ . Round to four decimal places.
Your answer: $\sin \left(\frac{\pi}{27}\right) \approx \square$

See Solution

Problem 24938

A conical cup is 10 cm tall and 10 cm radius. Water level drops at 3 cm/sec. Find volume change rate when water is 3 cm deep. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to nearest thousandth or give exact in terms of $\pi$ .

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Problem 24939

Evaluate the integral $\int x^{12} e^{x^{13}} d x$ using a change of variables or a table.

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Problem 24940

Find the surface area of the curve $x=5 \cos^3 \theta$ , $y=5 \sin^3 \theta$ for $0 \leq \theta \leq \frac{\pi}{2}$ , rotated about the x-axis.

See Solution

Problem 24941

Estimate $e^{-1}$ using the fourth Maclaurin polynomial for $f(x)=e^{x}$ . Round to four decimal places.
$e^{-1} \approx$

See Solution

Problem 24942

A balloon deflates at $\frac{32 \pi}{3} \mathrm{~cm}^{3} / \mathrm{sec}$ . Find the radius change rate when $r=3 \mathrm{~cm}$ . Use $V=\frac{4}{3} \pi r^{3}$ .

See Solution

Problem 24943

Find the tangent line equation to the curve $y=4 \sqrt{x}$ at $x=4$ . Options: $y=x-4$ , $y=x+4$ , $y=-x+6$ , $y=x-6$ .

See Solution

Problem 24944

Evaluate the integral: $\int \frac{4 x^{3}}{\sqrt{3-16 x^{4}}} d x$ using a change of variables or a table.

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Problem 24945

Find $f^{\prime \prime}(2)$ for the function $f(x)=1-\frac{1}{x}$ .

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Problem 24946

Find the third derivative of the function $y=5 x^{3}+7 x-16$ . What is $\frac{d^{3} y}{d x^{3}}$ ?

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Problem 24947

Evaluate the integral: $\int\left(x^{3}+x^{2}\right)^{10}\left(3 x^{2}+2 x\right) dx$ using a change of variables.

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Problem 24948

Find $g^{\prime}(3)$ for the function $g(x)=(4x-11)^{10}$ .

See Solution

Problem 24949

A 5-m ladder leans against a wall, top slides down at $2 \mathrm{~m/s}$ . Find base's speed from wall when it's $4 \mathrm{~m}$ away.

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Problem 24950

Estimate $\sqrt[3]{5}$ with the fourth Taylor polynomial at $x=4$ and bound the error using Taylor's Theorem. Round to six decimal places. Provide your answer: $\left|R_{4}(x)\right| \leq$

See Solution

Problem 24951

Find the second derivative of the function $f(x)=(x-2)(x+8)$ . What is it?

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Problem 24952

Evaluate the integral $\int_{0}^{2} \frac{2 x}{(x^{2}+3)^{3}} dx$ using a change of variables or a table.

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Problem 24953

Find the Maclaurin polynomial degree for $10 \cos (x)$ so that the error in estimating $10 \cos (0.15)$ is < 0.001.
Answer: $n=$

See Solution

Problem 24954

Evaluate the integral $\int_{0}^{8} \frac{p}{\sqrt{225+p^{2}}} d p$ using a change of variables or a table.

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Problem 24955

Find the indefinite integral $\int(\sin 8 y+\cos 3 y) dy$ and verify by differentiating your result.

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Problem 24956

Evaluate the integral: $\int \frac{1}{2 x+1} d x$ , assuming $u>0$ when $\ln u$ appears.

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Problem 24957

Evaluate the integral: $\int \frac{x^{2}}{\left(2-x^{3}\right)^{4}} d x$ , assuming $u>0$ for $\ln u$ .

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Problem 24958

Bestimme das Integral $\int_{1}^{8} \left( x^{3}+\frac{1}{x} \right) dx$ .

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Problem 24959

Calculate the integral from 1 to 2 of the function $x^{3}+\frac{1}{x}$ .

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Problem 24960

Find the Maclaurin polynomial degree for $10 \cos (x)$ so that the error in $10 \cos (0.15)$ is < 0.001.

See Solution

Problem 24961

Bestimme die Ableitung von $f(x) = \frac{1}{\left(\frac{1}{2} x - 3\right)^{2}}$ .

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Problem 24962

1. Löse die Gleichung $\sqrt{x-5,5}-\sqrt{x+3,25}=0$ .
2. Berechne das Integral $\int_{1}^{8} x^{3}+\frac{1}{x} d x$ .
3. Erkläre in Stichpunkten: a) Fläche zwischen zwei Funktionen, b) "orientierte Fläche" und Integralrechnung.
4. Finde die Stammfunktionen von $\int 3 x^{5}+\sqrt{x^{5}}+\frac{1}{x^{5}} d x$ .

See Solution

Problem 24963

Evaluate the integral: $\int \frac{(\sqrt{x}-2)^{5}}{2 \sqrt{x}} d x$ using a change of variables.

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Problem 24964

Evaluate the integral: $\int 3 \sin x(6+3 \cos x)^{3} d x$

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Problem 24965

Find the Maclaurin polynomial degree needed for the error in estimating $f(0.11)$ (where $f(x)=6 \ln (x+1)$ ) to be < 0.001.
Enter your answer as a numerical value.
Provide your answer below: $n=$

See Solution

Problem 24966

Find the Maclaurin polynomial degree needed for the error in estimating $f(0.31)$ with $f(x)=2 \ln (x+1)$ to be < 0.001.
Enter your answer as a numerical value.
Provide your answer below: $n=$

See Solution

Problem 24967

Finde alle Stammfunktionen für die Funktion $3 x^{5}+\sqrt{x^{5}}+\frac{1}{x^{5}}$ .

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Problem 24968

Find the integral using substitution: $\int \tan 10 x \sec ^{2} 10 x \, dx$ .

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Problem 24969

Bestimme die Ableitung von $f(x) = \sin \left( \sqrt[3]{\frac{4}{3x - 2}} \right)$ .

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Problem 24970

Find the distance a particle travels from hour 2 to hour 5 given $v(t)=9 t^{2}+10 t$ .

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Problem 24971

Use the divergence test on the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{2}+3}{\sqrt[3]{8 n^{3}+3 n^{2}-7}}\right)$ and choose the best result.

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Problem 24972

A town's population starts at 400 and grows at $P^{\prime}(t)=30(1+\sqrt{t})$ . Find population after 30 years and $P(t)$ .

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Problem 24973

A eucalyptus tree grows at $0.5 + \frac{6}{(t+3)^{3}}$ feet/year. Find growth in years 2 and 3.

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Problem 24974

Berechnen Sie die Fläche A zwischen den Graphen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ im Intervall $[-1, 1,5]$ .

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Problem 24975

Estimate $\sin(0.3)$ with the third Maclaurin polynomial and bound the error using Taylor's Theorem. Round to six decimals.
Provide your answer below: $\left|R_{3}(x)\right| \leq \square$

See Solution

Problem 24976

Given the function $v(t)=t^{3}-9 t^{2}+14 t$ on $[0,9]$ , find when the motion is positive, displacement, and distance traveled.

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Problem 24977

Estimate the distance traveled in cm using left-endpoint and right-endpoint values from the given velocity data over 10 sec.

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Problem 24978

Estimate $\sqrt{8}$ using the fourth Taylor polynomial at $x=7$ and bound the error with Taylor's Theorem. Round to six decimals.
Provide your answer below: $\left|R_{4}(x)\right| \leq \square$

See Solution

Problem 24979

Find the sum of the series $\sum_{n=1}^{\infty} e^{2 n} 11^{1-n}$ as a fraction in terms of $e$ .

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Problem 24980

Check if the series $\sum_{n=1}^{\infty} \frac{3(-1)^{n+1} n^{3}}{7 n^{3}+6 n^{2}+7 n+5}$ converges or diverges using tests.

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Problem 24981

Find the ball's displacement from $t=0$ to $t=4$ given $v(t)=-32t+142$ ft/s. Displacement is $\square$ feet.

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Problem 24982

Use the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{n}$ to check if $\sum_{n=1}^{\infty} \frac{4 e^{1/n}}{n}$ converges or diverges.

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Problem 24983

Estimate the distance traveled in cm using left and right endpoint values with ten 1 sec intervals from the given velocity data.

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Problem 24984

Find the Maclaurin polynomial degree needed for the error in $f(0.21)$ with $f(x)=13 \ln (x+1)$ to be < 0.001.
Enter your answer as a numerical value.
Provide your answer below: $n=$

See Solution

Problem 24985

Find the sum of the series: $\sum_{n=1}^{\infty}\left(\frac{19}{n^{8}}-\frac{19}{(n+1)^{8}}\right)$ . If it diverges, enter $\varnothing$ .

See Solution

Problem 24986

Find the displacement of a ball with velocity $v(t)=-32 t+142$ from $t=0$ to $t=4$ , then its position at $t=4$ given $s(0)=7$ .

See Solution

Problem 24987

Evaluate the integral: $\int \frac{5}{(5 x+1) \ln (5 x+1)} d x$

See Solution

Problem 24988

Bestimme die Fläche zwischen den Graphen der Funktionen $f(x)=x^{3}-3 x^{2}+1$ und $g(x)=x-2$ .

See Solution

Problem 24989

Find the critical numbers of the function $g(y)=\frac{y-5}{y^{2}-3y+15}$ . Enter answers as a comma-separated list or DNE.

See Solution

Problem 24990

Find the critical numbers of the function $f(\theta)=14 \cos \theta+7 \sin^{2} \theta$ . Enter as a comma-separated list.

See Solution

Problem 24991

Bestimme die Fläche zwischen den Graphen der Funktionen $f(x)=x^{3}-3 x^{2}+1$ und $g(x)=x-2$ .

See Solution

Problem 24992

Find the max and min of $f(x)=x-\ln (x)$ on the interval $[1/6, 6]$ .

See Solution

Problem 24993

Find the max and min of $f(x)=x-\ln(x)$ on the interval $[1/6, 6]$ .

See Solution

Problem 24994

Untersuchen Sie die Konvergenz oder Divergenz der Funktionen für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ :
a) $f: x \mapsto \frac{0,5 x}{2 x-2}$ b) $f: x \mapsto \frac{3 x-1}{1-7 x}$ c) $f: x \mapsto \frac{4 x-6}{0,2 x^{2}}$ d) $f: x \mapsto \frac{2 x^{2}-6 x+1}{112 x}$ e) $f: x \mapsto \frac{-8}{2 x+3 x^{2}}$ f) $f: x \mapsto \frac{-3 x^{2}-5 x+12}{-0,25 x^{2}+6 x}$ g) $f: x \mapsto \frac{x^{2}+2 x}{-9 x+3}$ h) $f: x \mapsto \frac{x(3-x)}{-8 x-7}$

See Solution

Problem 24995

Find the critical numbers of the function $h(p)=\frac{p-4}{p^{2}+6}$ . Enter answers as a comma-separated list or DNE.

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Problem 24996

Find the critical numbers of the function $f(x)=x^{3}+12 x^{2}-27 x$ . Enter answers as a comma-separated list or DNE.

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Problem 24997

Find the tangent line equation using implicit differentiation for $x^{2}+2xy-y^{2}+x=74$ at the point (6,8). $y=$

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Problem 24998

Given $f(x)+x^{2}[f(x)]^{4}=18$ and $f(1)=2$ , find $f^{\prime}(1)$ .

See Solution

Problem 24999

Find $dy/dx$ using implicit differentiation for $5 \cos x \sin y = 4$ . What is $y'$ ?

See Solution

Problem 25000

Find $dy/dx$ using implicit differentiation for the equation: $6x^3 + x^2y - xy^3 = 5$ .

See Solution

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Calculus

Problem 24901

Differentiate y=(2x)ln⁡(6x)y=(2 x)^{\ln (6 x)}y=(2x)ln(6x) using logarithmic differentiation to find y′y^{\prime}y′.

Problem 24902

Find the derivative of y=xx2+1(x+1)23y=\frac{x \sqrt{x^{2}+1}}{(x+1)^{\frac{2}{3}}}y=(x+1)32​xx2+1​​ using logarithmic differentiation.

Problem 24903

Find all values of ccc satisfying the Mean Value Theorem for f(x)=x3+13x2+47x+30f(x) = x^3 + 13x^2 + 47x + 30f(x)=x3+13x2+47x+30 on [−5,0][-5, 0][−5,0].

Problem 24904

A 0.65 kg0.65 \mathrm{~kg}0.65 kg balloon is launched at 60∘60^{\circ}60∘ from a 52 m52 \mathrm{~m}52 m building with 12 m/s12 \mathrm{~m/s}12 m/s. Find its speed on impact using energy methods.

Problem 24905

A roller coaster starts from rest at 35 m. Find speeds at points 2 (0 m), 3 (28 m), and 4 (15 m) using energy equations.

Problem 24906

Evaluate ∫67f(x)dx\int_{6}^{7} f(x) \mathrm{d} x∫67​f(x)dx given ∫37f(x)=9\int_{3}^{7} f(x)=9∫37​f(x)=9 and ∫362f(x)dx=4\int_{3}^{6} 2 f(x) \mathrm{d} x=4∫36​2f(x)dx=4.

Problem 24907

Find the area AAA of the shaded region under the standard normal distribution with mean 0 and std dev 1. Round to four decimal places.

Problem 24908

Soit f(x)={x3+1 si x<1x−1 si x≥1f(x)=\left\{\begin{array}{lll}x^{3}+1 & \text { si } & x<1 \\ \sqrt{x-1} & \text { si } & x \geq 1\end{array}\right.f(x)={x3+1x−1​​ si si ​x<1x≥1​. Vérifiez la continuité en x=1x=1x=1, les nombres de transition et les points d'inflexion.

Problem 24909

Evaluate ∫02xf(x2)−4−x2 dx\int_{0}^{2} x f\left(x^{2}\right) - \sqrt{4 - x^{2}} \, dx∫02​xf(x2)−4−x2​dx given ∫04f(x) dx=6\int_{0}^{4} f(x) \, dx = 6∫04​f(x)dx=6.

Problem 24910

Find the 3rd-degree Maclaurin polynomial for f(x)=17x+3f(x)=\frac{1}{7x+3}f(x)=7x+31​. Your answer is: p3(x)= p_{3}(x)= p3​(x)=

Problem 24911

Find the 3rd-degree Maclaurin polynomial for f(x)=cos⁡(4x)f(x)=\cos(4x)f(x)=cos(4x). Provide your answer below: p3(x)= p_{3}(x)= p3​(x)=

Problem 24912

Find the Maclaurin polynomial p2(x)p_{2}(x)p2​(x) for f(x)=cos⁡(x3)f(x)=\cos \left(\frac{x}{3}\right)f(x)=cos(3x​). Answer: p˙2(x)=□ \dot{p}_{2}(x)=\square p˙​2​(x)=□

Problem 24913

Find the first and second degree Maclaurin polynomials for f(x)=5ex+3e−xf(x)=5 e^{x}+3 e^{-x}f(x)=5ex+3e−x at a=0a=0a=0.

Problem 24914

Find the Maclaurin polynomial p1(x)p_{1}(x)p1​(x) for the function f(x)=e2xf(x)=e^{2x}f(x)=e2x. Provide your answer below: p1(x)=□ p_{1}(x)=\square p1​(x)=□

Problem 24915

Find the third-degree Maclaurin polynomial p3(x)p_{3}(x)p3​(x) for the function f(x)=5xf(x)=5^{x}f(x)=5x. Provide your answer below: p3(x)=□ p_{3}(x)=\square p3​(x)=□

Problem 24916

Find the rate of change of the function f(x)=x3−x2+x−5f(x)=x^{3}-x^{2}+x-5f(x)=x3−x2+x−5 at x=2x=2x=2. Options: 6, 9, -12, -10.

Problem 24917

Find the second Maclaurin polynomial p2(x)p_{2}(x)p2​(x) for f(x)=16x+8f(x)=\frac{1}{6x+8}f(x)=6x+81​. Provide your answer below: p2(x)=□ p_{2}(x)=\square p2​(x)=□

Problem 24918

Find the limit: lim⁡x→5x−5x−5=\lim _{x \rightarrow 5} \frac{\sqrt{x}-\sqrt{5}}{x-5}=limx→5​x−5x​−5​​=

Problem 24919

Calculate the length of the curve defined by x=1+3t2x=1+3 t^{2}x=1+3t2 and y=4+2t3y=4+2 t^{3}y=4+2t3 for 0≤t≤10 \leq t \leq 10≤t≤1.

Problem 24920

Find the gradient of the curve 3x2−y2=113 x^{2} - y^{2} = 113x2−y2=11 at the point (2,3)(2, 3)(2,3).

Problem 24921

Find the value of aaa if lim⁡x→1a(x2−1)x4−1=2\lim _{x \rightarrow 1} \frac{a\left(x^{2}-1\right)}{x^{4}-1}=2limx→1​x4−1a(x2−1)​=2.

Problem 24922

Estimate sin⁡(π25)\sin \left(\frac{\pi}{25}\right)sin(25π​) using the second Maclaurin polynomial for f(x)=sin⁡(x)f(x)=\sin (x)f(x)=sin(x). Round to four decimal places.sin⁡(π25)≈□ \sin \left(\frac{\pi}{25}\right) \approx \square sin(25π​)≈□

Problem 24923

If x2+y2=9x^{2}+y^{2}=9x2+y2=9, find dydx\frac{dy}{dx}dxdy​ in terms of xxx and yyy. Choices include 2x+2y2x+2y2x+2y, 9−x29-x^{2}9−x2, (3−x)/y(3-x)/y(3−x)/y, and −x/y-x/y−x/y.

Problem 24924

Estimate e0.5e^{0.5}e0.5 using the fourth Maclaurin polynomial for f(x)=exf(x)=e^{x}f(x)=ex. Round to four decimal places.

Problem 24925

Evaluate the integral ∫2x(x2+1)8dx\int 2 x\left(x^{2}+1\right)^{8} d x∫2x(x2+1)8dx using the substitution u=x2+1u=x^{2}+1u=x2+1.

Problem 24926

Find the area of the surface formed by rotating x=13(y2+2)3/2x=\frac{1}{3}(y^{2}+2)^{3/2}x=31​(y2+2)3/2, for 1≤y≤31 \leq y \leq 31≤y≤3, around the x-axis.

Problem 24927

A cylinder's radius grows at 3 m/s, volume at 108 m³/s. With height 6 m and volume 33 m³, find height's rate of change. Use V=πr2hV=\pi r^{2} hV=πr2h. Round to three decimal places.

Problem 24928

Estimate e7e^{7}e7 using the second Taylor polynomial of f(x)=exf(x)=e^{x}f(x)=ex at x=6x=6x=6. Round to four decimal places. e7≈□ e^{7} \approx \square e7≈□

Problem 24929

Find the integral ∫(4x3+9)x4+9x dx\int(4 x^{3}+9) \sqrt{x^{4}+9 x} \, dx∫(4x3+9)x4+9x​dx using the substitution u=x4+9xu=x^{4}+9 xu=x4+9x and verify by differentiation.

Problem 24930

Evaluate the integral ∫9x+2 dx\int \sqrt{9 x+2} \, dx∫9x+2​dx using the substitution u=ax+bu = a x + bu=ax+b.

Problem 24931

Find the one-sided limit: lim⁡x→1−4x−1\lim _{x \rightarrow 1^{-}} \frac{4}{x-1}limx→1−​x−14​. Options: -2, -4, ∞\infty∞, −∞-\infty−∞.

Problem 24932

Estimate sin⁡(−1+4π)\sin(-1 + 4\pi)sin(−1+4π) using the fourth Taylor polynomial for f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) at x=4πx = 4\pix=4π. Round to four decimal places. Your answer: sin⁡(−1+4π)≈ \sin(-1 + 4\pi) \approx sin(−1+4π)≈

Problem 24933

Estimate cos⁡(−1+2π)\cos(-1 + 2\pi)cos(−1+2π) using the second Taylor polynomial of f(x)=cos⁡(x)f(x) = \cos(x)f(x)=cos(x) at x=2πx = 2\pix=2π. Round to four decimal places. cos⁡(−1+2π)≈□ \cos(-1 + 2\pi) \approx \square cos(−1+2π)≈□

Problem 24934

Estimate cos⁡(1+π)\cos(1+\pi)cos(1+π) using the fourth Taylor polynomial of f(x)=cos⁡(x)f(x)=\cos(x)f(x)=cos(x) at x=πx=\pix=π. Round to four decimal places. Your answer: cos⁡(1+π)≈□ \cos(1+\pi) \approx \square cos(1+π)≈□

Problem 24935

Find the one-sided limit: lim⁡x→0−x2+5x∣x∣\lim _{x \rightarrow 0^{-}} \frac{x^{2}+5 x}{|x|}limx→0−​∣x∣x2+5x​. What is the answer?

Problem 24936

Find the integral ∫−16xsin⁡(8x2−1) dx\int -16 x \sin(8 x^{2} - 1) \, dx∫−16xsin(8x2−1)dx using u=8x2−1u = 8 x^{2} - 1u=8x2−1 and check by differentiating.

Problem 24937

Estimate sin⁡(π27)\sin \left(\frac{\pi}{27}\right)sin(27π​) using the second Maclaurin polynomial for f(x)=sin⁡(x)f(x)=\sin (x)f(x)=sin(x). Round to four decimal places.Your answer: sin⁡(π27)≈□ \sin \left(\frac{\pi}{27}\right) \approx \square sin(27π​)≈□

Problem 24938

A conical cup is 10 cm tall and 10 cm radius. Water level drops at 3 cm/sec. Find volume change rate when water is 3 cm deep. Use V=13πr2hV=\frac{1}{3} \pi r^{2} hV=31​πr2h. Round to nearest thousandth or give exact in terms of π\piπ.

Problem 24939

Evaluate the integral ∫x12ex13dx\int x^{12} e^{x^{13}} d x∫x12ex13dx using a change of variables or a table.

Problem 24940

Differentiate $y=(2 x)^{\ln (6 x)}$ using logarithmic differentiation to find $y^{\prime}$ .

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

Find all values of $c$ satisfying the Mean Value Theorem for $f(x) = x^3 + 13x^2 + 47x + 30$ on $[-5, 0]$ .

A $0.65 \mathrm{~kg}$ balloon is launched at $60^{\circ}$ from a $52 \mathrm{~m}$ building with $12 \mathrm{~m/s}$ . Find its speed on impact using energy methods.

Evaluate $\int_{6}^{7} f(x) \mathrm{d} x$ given $\int_{3}^{7} f(x)=9$ and $\int_{3}^{6} 2 f(x) \mathrm{d} x=4$ .

Find the area $A$ of the shaded region under the standard normal distribution with mean 0 and std dev 1. Round to four decimal places.

Soit $f(x)=\left\{\begin{array}{lll}x^{3}+1 & \text { si } & x<1 \\ \sqrt{x-1} & \text { si } & x \geq 1\end{array}\right.$ . Vérifiez la continuité en $x=1$ , les nombres de transition et les points d'inflexion.

Evaluate $\int_{0}^{2} x f\left(x^{2}\right) - \sqrt{4 - x^{2}} \, dx$ given $\int_{0}^{4} f(x) \, dx = 6$ .

Find the 3rd-degree Maclaurin polynomial for $f(x)=\frac{1}{7x+3}$ .
Your answer is: $p_{3}(x)=$

Find the 3rd-degree Maclaurin polynomial for $f(x)=\cos(4x)$ .
Provide your answer below: $p_{3}(x)=$

Find the Maclaurin polynomial $p_{2}(x)$ for $f(x)=\cos \left(\frac{x}{3}\right)$ .
Answer: $\dot{p}_{2}(x)=\square$

Find the first and second degree Maclaurin polynomials for $f(x)=5 e^{x}+3 e^{-x}$ at $a=0$ .

Find the Maclaurin polynomial $p_{1}(x)$ for the function $f(x)=e^{2x}$ .
Provide your answer below: $p_{1}(x)=\square$

Find the third-degree Maclaurin polynomial $p_{3}(x)$ for the function $f(x)=5^{x}$ .
Provide your answer below: $p_{3}(x)=\square$

Find the rate of change of the function $f(x)=x^{3}-x^{2}+x-5$ at $x=2$ . Options: 6, 9, -12, -10.

Find the second Maclaurin polynomial $p_{2}(x)$ for $f(x)=\frac{1}{6x+8}$ .
Provide your answer below: $p_{2}(x)=\square$

Find the limit: $\lim _{x \rightarrow 5} \frac{\sqrt{x}-\sqrt{5}}{x-5}=$

Calculate the length of the curve defined by $x=1+3 t^{2}$ and $y=4+2 t^{3}$ for $0 \leq t \leq 1$ .

Find the gradient of the curve $3 x^{2} - y^{2} = 11$ at the point $(2, 3)$ .

Find the value of $a$ if $\lim _{x \rightarrow 1} \frac{a\left(x^{2}-1\right)}{x^{4}-1}=2$ .

Estimate $\sin \left(\frac{\pi}{25}\right)$ using the second Maclaurin polynomial for $f(x)=\sin (x)$ . Round to four decimal places.
$\sin \left(\frac{\pi}{25}\right) \approx \square$

If $x^{2}+y^{2}=9$ , find $\frac{dy}{dx}$ in terms of $x$ and $y$ . Choices include $2x+2y$ , $9-x^{2}$ , $(3-x)/y$ , and $-x/y$ .

Estimate $e^{0.5}$ using the fourth Maclaurin polynomial for $f(x)=e^{x}$ . Round to four decimal places.

Evaluate the integral $\int 2 x\left(x^{2}+1\right)^{8} d x$ using the substitution $u=x^{2}+1$ .

Find the area of the surface formed by rotating $x=\frac{1}{3}(y^{2}+2)^{3/2}$ , for $1 \leq y \leq 3$ , around the x-axis.

A cylinder's radius grows at 3 m/s, volume at 108 m³/s. With height 6 m and volume 33 m³, find height's rate of change. Use $V=\pi r^{2} h$ . Round to three decimal places.

Estimate $e^{7}$ using the second Taylor polynomial of $f(x)=e^{x}$ at $x=6$ . Round to four decimal places.
$e^{7} \approx \square$

Find the integral $\int(4 x^{3}+9) \sqrt{x^{4}+9 x} \, dx$ using the substitution $u=x^{4}+9 x$ and verify by differentiation.

Evaluate the integral $\int \sqrt{9 x+2} \, dx$ using the substitution $u = a x + b$ .

Find the one-sided limit: $\lim _{x \rightarrow 1^{-}} \frac{4}{x-1}$ . Options: -2, -4, $\infty$ , $-\infty$ .

Estimate $\sin(-1 + 4\pi)$ using the fourth Taylor polynomial for $f(x) = \sin(x)$ at $x = 4\pi$ . Round to four decimal places.
Your answer: $\sin(-1 + 4\pi) \approx$

Estimate $\cos(-1 + 2\pi)$ using the second Taylor polynomial of $f(x) = \cos(x)$ at $x = 2\pi$ . Round to four decimal places.
$\cos(-1 + 2\pi) \approx \square$

Estimate $\cos(1+\pi)$ using the fourth Taylor polynomial of $f(x)=\cos(x)$ at $x=\pi$ . Round to four decimal places.
Your answer: $\cos(1+\pi) \approx \square$

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{x^{2}+5 x}{|x|}$ . What is the answer?

Find the integral $\int -16 x \sin(8 x^{2} - 1) \, dx$ using $u = 8 x^{2} - 1$ and check by differentiating.

Estimate $\sin \left(\frac{\pi}{27}\right)$ using the second Maclaurin polynomial for $f(x)=\sin (x)$ . Round to four decimal places.
Your answer: $\sin \left(\frac{\pi}{27}\right) \approx \square$

A conical cup is 10 cm tall and 10 cm radius. Water level drops at 3 cm/sec. Find volume change rate when water is 3 cm deep. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to nearest thousandth or give exact in terms of $\pi$ .

Evaluate the integral $\int x^{12} e^{x^{13}} d x$ using a change of variables or a table.

Find the surface area of the curve $x=5 \cos^3 \theta$ , $y=5 \sin^3 \theta$ for $0 \leq \theta \leq \frac{\pi}{2}$ , rotated about the x-axis.

Estimate $e^{-1}$ using the fourth Maclaurin polynomial for $f(x)=e^{x}$ . Round to four decimal places.
$e^{-1} \approx$

A balloon deflates at $\frac{32 \pi}{3} \mathrm{~cm}^{3} / \mathrm{sec}$ . Find the radius change rate when $r=3 \mathrm{~cm}$ . Use $V=\frac{4}{3} \pi r^{3}$ .

Find the tangent line equation to the curve $y=4 \sqrt{x}$ at $x=4$ . Options: $y=x-4$ , $y=x+4$ , $y=-x+6$ , $y=x-6$ .

Evaluate the integral: $\int \frac{4 x^{3}}{\sqrt{3-16 x^{4}}} d x$ using a change of variables or a table.

Find $f^{\prime \prime}(2)$ for the function $f(x)=1-\frac{1}{x}$ .

Find the third derivative of the function $y=5 x^{3}+7 x-16$ . What is $\frac{d^{3} y}{d x^{3}}$ ?

Evaluate the integral: $\int\left(x^{3}+x^{2}\right)^{10}\left(3 x^{2}+2 x\right) dx$ using a change of variables.

Find $g^{\prime}(3)$ for the function $g(x)=(4x-11)^{10}$ .

A 5-m ladder leans against a wall, top slides down at $2 \mathrm{~m/s}$ . Find base's speed from wall when it's $4 \mathrm{~m}$ away.

Estimate $\sqrt[3]{5}$ with the fourth Taylor polynomial at $x=4$ and bound the error using Taylor's Theorem. Round to six decimal places. Provide your answer: $\left|R_{4}(x)\right| \leq$

Find the second derivative of the function $f(x)=(x-2)(x+8)$ . What is it?

Evaluate the integral $\int_{0}^{2} \frac{2 x}{(x^{2}+3)^{3}} dx$ using a change of variables or a table.

Find the Maclaurin polynomial degree for $10 \cos (x)$ so that the error in estimating $10 \cos (0.15)$ is < 0.001.
Answer: $n=$

Evaluate the integral $\int_{0}^{8} \frac{p}{\sqrt{225+p^{2}}} d p$ using a change of variables or a table.

Find the indefinite integral $\int(\sin 8 y+\cos 3 y) dy$ and verify by differentiating your result.

Evaluate the integral: $\int \frac{1}{2 x+1} d x$ , assuming $u>0$ when $\ln u$ appears.

Evaluate the integral: $\int \frac{x^{2}}{\left(2-x^{3}\right)^{4}} d x$ , assuming $u>0$ for $\ln u$ .

Bestimme das Integral $\int_{1}^{8} \left( x^{3}+\frac{1}{x} \right) dx$ .

Calculate the integral from 1 to 2 of the function $x^{3}+\frac{1}{x}$ .

Find the Maclaurin polynomial degree for $10 \cos (x)$ so that the error in $10 \cos (0.15)$ is < 0.001.

Bestimme die Ableitung von $f(x) = \frac{1}{\left(\frac{1}{2} x - 3\right)^{2}}$ .

Evaluate the integral: $\int \frac{(\sqrt{x}-2)^{5}}{2 \sqrt{x}} d x$ using a change of variables.

Evaluate the integral: $\int 3 \sin x(6+3 \cos x)^{3} d x$

Find the Maclaurin polynomial degree needed for the error in estimating $f(0.11)$ (where $f(x)=6 \ln (x+1)$ ) to be < 0.001.
Enter your answer as a numerical value.
Provide your answer below: $n=$

Find the Maclaurin polynomial degree needed for the error in estimating $f(0.31)$ with $f(x)=2 \ln (x+1)$ to be < 0.001.
Enter your answer as a numerical value.
Provide your answer below: $n=$

Finde alle Stammfunktionen für die Funktion $3 x^{5}+\sqrt{x^{5}}+\frac{1}{x^{5}}$ .

Find the integral using substitution: $\int \tan 10 x \sec ^{2} 10 x \, dx$ .

Bestimme die Ableitung von $f(x) = \sin \left( \sqrt[3]{\frac{4}{3x - 2}} \right)$ .

Find the distance a particle travels from hour 2 to hour 5 given $v(t)=9 t^{2}+10 t$ .

Use the divergence test on the series $\sum_{n=1}^{\infty}\left(\frac{3 n^{2}+3}{\sqrt[3]{8 n^{3}+3 n^{2}-7}}\right)$ and choose the best result.

A town's population starts at 400 and grows at $P^{\prime}(t)=30(1+\sqrt{t})$ . Find population after 30 years and $P(t)$ .

A eucalyptus tree grows at $0.5 + \frac{6}{(t+3)^{3}}$ feet/year. Find growth in years 2 and 3.

Berechnen Sie die Fläche A zwischen den Graphen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ im Intervall $[-1, 1,5]$ .