Calculus

Problem 18701

Find the limit: $\lim _{x \rightarrow \infty} \frac{5 x-x^{2}}{3 x^{2}+2 x-4}$ .

See Solution

Problem 18702

Find the general antiderivative of $f(x)=3 x^{\frac{1}{3}}+5 x^{-\frac{1}{3}}+7$ . Let $F(x)=\square$ (use C for constant).

See Solution

Problem 18703

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x}{e^{x}}$ .

See Solution

Problem 18704

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x}$ .

See Solution

Problem 18705

Find the general antiderivative of $f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}$ .

See Solution

Problem 18706

Find the general antiderivative of $f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}$ , denoted as $F(x)=\square$ (use C for constant).

See Solution

Problem 18707

Find the general antiderivative of $f(x)=\frac{3 x^{5}+10 x^{4}-x^{2}}{x^{3}}$ . Let $F(x)=\square$ (use C for the constant).

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Problem 18708

Bestimme die ersten beiden Ableitungen von $f$ für: a) $f(x)=3+e^{x}$ , b) $f(x)=3 e^{x}$ , c) $f(x)=e^{3 x}$ , d) $f(x)=e^{-x}$ .

See Solution

Problem 18709

Which statement about the function $f(x)=(x+5)(x+1)^{3}(x-2)^{2}$ is true?

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Problem 18710

Insect population size $P(t)=400 e^{0.04 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P(t)=560$ , (e) when $P(t)$ doubles.

See Solution

Problem 18711

Bestimme PR oder KR für die Ableitung der Funktionen und berechne die erste Ableitung:
a) $f(x)=2 \cdot e^{x+1}$
b) $f(x)=2 x \cdot e^{x}+1$
c) $f(x)=2+e^{3 x}$
d) $f(x)=\frac{x}{e^{x}}$

See Solution

Problem 18712

Problema: Dibuja el gráfico de $T=25+\mathrm{e}^{0.4 t}$ para $0 \leq t \leq 12$ . Encuentra $T$ a la mitad y el tiempo para $T=100^{\circ}C$ .

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Problem 18713

Insect population $P(t)=500 e^{0.02 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P(t)=700$ , (e) when $P(t)$ doubles.

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Problem 18714

Evaluate the integral $\int \frac{3 x^{4}-x^{3}+6 x^{2}}{x^{4}} d x$ and find functions $f(x)$ and $g(x)$ such that $\left(\frac{f(x)}{g(x)}\right)^{\prime}=\frac{3 x^{4}-x^{3}+6 x^{2}}{x^{4}}$ .

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Problem 18715

A flu epidemic starts with 5 cases on day $t=0$ . The growth rate is $r(t)=22 e^{0.04 t}$ .
(a) Find the total cases formula $F(t)$ .
(b) Calculate $F(25)$ (round to the nearest whole number).

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Problem 18716

How long does it take for a bacteria population to double with a growth rate of $8.9\%$ per hour? Round to the nearest hundredth.

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Problem 18717

Una moneda de un euro cae desde reposo. ¿Cuál es su posición y velocidad tras $1 \mathrm{~s}$ ? Opciones: a) b) c) d) e)

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Problem 18718

Find the relative rate of change of the Van der Waals equation $P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}$ with respect to $V$ . What is $\frac{\partial P}{\partial V}$ ?

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Problem 18719

Find the relative rate of change of pressure $P$ in the Van der Waals equation: $P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}$ . What is $\frac{\partial P}{\partial T}$ ? (A) (B) (C) (D) (E) (F) (G)

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Problem 18720

Find the derivative of $f(x)=\int_{-4}^{x} \sqrt{t^{3}+64} \, dt$ . What is $f'(x)$ ?

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Problem 18721

Evaluate the integral from 1 to 5 of $6 x^{-2} - 5$ . What is $\int_{1}^{5}(6 x^{-2}-5) d x$ ?

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Problem 18722

Evaluate the integral $\int_{0}^{2} 20 x^{4} \, dx$ .

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Problem 18723

Find the derivative of $g(y)=\int_{1}^{y} t^{15} \sin t \, dt$ . What is $g'(y)$ ?

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Problem 18724

Evaluate the integral: $\int_{2}^{12} 4 \sqrt{x} \, dx$ using the Fundamental Theorem of Calculus.

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Problem 18725

Find the partial derivative $\frac{\partial f}{\partial x}$ for $f(x, y, z)=x y^{2}-2 z y+x$ .

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Problem 18726

Find the partial derivative $\frac{\partial f}{\partial y}$ for $f(x, y, z)=x y^{2}-2 z y+x$ . Options: (A) $y+1$ , (B) $y^{2}$ , (C) $y^{2}-2 z y+1$ , (D) $2 x y-2 z$ , (E) $y^{2}+1$ , (F) $2 x y-2 y$ , (G) Not listed.

See Solution

Problem 18727

Find the partial derivatives $\frac{\delta f}{\delta x}$ and $\frac{\delta f}{\delta y}$ for $f(x, y)=x e^{x y}$ .

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Problem 18728

Find the partial derivative $\frac{\partial f}{\partial x}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ .

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Problem 18729

Evaluate the Laplacian of $\psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}}$ in Cartesian and spherical coordinates and prove they are equal.

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Problem 18730

Find $\frac{\partial f}{\partial t}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ . Choose the correct option.

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Problem 18731

Find $\frac{\partial f}{\partial y}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ . Choose from options (A) to (G).

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Problem 18732

Find the second partial derivative $\frac{\partial^{2} z}{\partial x^{2}}$ for $z(x, y)=\sin (x y)$ . Options: (A) $x \sin (x y)$ (B) $-x^{2} \sin (x y)$ (C) $x^{2} \sin (x y)$ (D) $-y^{2} \sin (x y)$ (E) $y^{2} \sin (x y)$ (F) $y \sin (x y)$ (G) Not listed.

See Solution

Problem 18733

Find the second partial derivative $\frac{\partial^{2} z}{\partial y^{2}}$ for $z(x, y)=\sin (x y)$ .

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Problem 18734

Find the intervals where the function $f(x)=x^{3}-9 x^{2}+5$ is increasing, decreasing, or constant. Use interval notation.

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Problem 18735

Déterminez les dimensions du rectangle d'aire maximale inscrit sous la courbe $f(x)=\sqrt{4-x^{2}}$ et au-dessus de l'axe.

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Problem 18736

Find the second mixed partial derivative $\frac{\partial^{2} f}{\partial x \partial y}$ for $f(x, y)=\cos (x y)$ .

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Problem 18737

Find the time for \$ 4125 to double at a continuous compounding rate of 9.5\%.

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Problem 18738

Find the second partial derivative $\frac{\partial^{2} f}{\partial x^{2}}$ for $f(x, y)=\cos (x y)$ . Choices: (A) (B) (C) (D) (E) (F) (G)

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Problem 18739

Find local maxima, minima, and saddle points of $f(x, y)=4+x^{3}+y^{3}-3 x y$ . Options: (A) (-1,-1) min, (B) (0,0) saddle, (1,1) min, (C) (0,0) max, (D) (0,0) min, (1,1) max, (E) (1,1) saddle, (0,0) min, (F) (0,0) saddle, (1,1) max, (G) Not listed.

See Solution

Problem 18740

Find the absolute max and min of $f(x, y)=x^{2}+4 y^{2}-2 x^{2} y+4$ on $-1 \leq x, y \leq 1$ .

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Problem 18741

Find the intervals where the function $f(x)=\frac{x^{2}+4 x+4}{x+1}$ is increasing, decreasing, or constant. Use interval notation.

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Problem 18742

Identify the true statement about the function $f(x, y)=4y-1+x^{2}-y^{2}$ from the options provided.

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Problem 18743

Match the differential equations with their solutions. Get all answers correct for credit.
1. $\frac{d^{2} y}{d x^{2}}+16 y=0$
2. $\frac{d y}{d x}=\frac{-2 x y}{x^{2}-4 y^{2}}$
3. $\frac{d^{2} y}{d x^{2}}+8 \frac{d y}{d x}+16 y=0$
4. $\frac{d y}{d x}=8 x y$
5. $\frac{d y}{d x}+12 x^{2} y=12 x^{2}$
A. $y=C e^{-4 x^{3}}+1$ B. $y=A e^{-4 x}+B x e^{-4 x}$ C. $y=A e^{4 x^{2}}$ D. $3 y x^{2}-4 y^{3}=C$ E. $y_{-}=A \cos (4 x)+B \sin (4 x)$

See Solution

Problem 18744

Find the volume of the solid formed by rotating the area between $xy=1$ , $y=0$ , $x=1$ , $x=5$ around $x=-1$ using the shell method.

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Problem 18745

Evaluate the integrals using given values: (A) $\int_{0}^{2}(f(x)+g(x)) d x$ , (B) $\int_{0}^{3}(f(x)-g(x)) d x$ , (C) $\int_{2}^{3}(3 f(x)+2 g(x)) d x$ , (D) Find $a$ for $\int_{0}^{3}(a f(x)+g(x)) d x=0$ . $a=-\frac{5}{9}$

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Problem 18746

Find $Q$ at $t=0$ in $Q=55 e^{1.03 t}$ , determine if $Q$ is increasing or decreasing, and find the growth rate.

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Problem 18747

Calculate the volume $V$ of the solid formed by rotating the area between $x=y^{2}$ and $x=1-y^{2}$ around $x=4$ .

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Problem 18748

Determine which two of the following series converge:
1. $\sum_{n=1}^{\infty}\left(\frac{n-1}{n+1}\right)^{n^{2}+n}$
2. $\sum_{n=1}^{\infty} \frac{n^{3}+1}{\sqrt[3]{n^{10}+n}}$
3. $\sum_{n=1}^{\infty}\left(\frac{n^{2}}{2 n+1}\right)^{n}$
4. $\sum_{n=1}^{\infty}\left(1+\frac{1}{n^{2}}\right)^{n^{3}}$
5. $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{n !}$
6. $\sum_{n=1}^{\infty} \frac{3^{n}}{n^{3}}$
7. $\sum_{n=1}^{\infty} \frac{n !}{(2 n) !}$

See Solution

Problem 18749

Find the asymptotic behavior of $a=\left(\frac{n(n+1)}{n^{2}+4}\right)$ as $n \to \infty$ and check series convergence.

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Problem 18750

Which differential equations are separable? Select all that apply: A. $\frac{d y}{d t}=-3 y$ B. $\frac{d y}{d t}=t+1$ C. $\frac{d y}{d t}=t y-y$ D. $\frac{d y}{d t}=t^{2}-y^{2}$ E. None of the above
Which functions $f(x)$ satisfy $\int f^{\prime}(x) d x=\int x d x$ ? Select all that apply: A. $f(x)=7 x^{2}$ B. $f(x)=\frac{1}{2} x^{2}$ C. $f(x)=x$ D. $f(x)=x-7$ E. $f(x)=7 x$ F. $f(x)=\frac{1}{2} x^{2}+7$ G. $f(x)=\frac{1}{2} x^{2}-7$ H. $f(x)=x+7$

See Solution

Problem 18751

Sea $f(x)=3 \ln (x-2)+1$ a) Encuentra el dominio de $f$ . b) ¿Cuál es la asíntota vertical de $f$ ? c) Halla la intersección con el eje $x$ . d) Encuentra la tangente a $f$ paralela a $y=x$ .

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Problem 18752

Identify the two true statements about the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n}$ .

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Problem 18753

Calculate the volume of the region bounded by $y=3x$ , $y=3$ , $y=3\sqrt{x}$ , $x=0$ , $x=1$ rotated about $y=3$ .

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Problem 18754

Find the poster dimensions that minimize area, given 150 sq in of printed matter and margins: 3 in top/bottom, 2 in sides.

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Problem 18755

Which series from the list converges by the root test? Choose 2:
1. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln ^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
2. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
3. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n / 2}$
4. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n^{2}}$
5. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$
6. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$

See Solution

Problem 18756

How long to double \ $3975 at a continuous$ 6\%$ interest rate?

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Problem 18757

Solve the equation $(4+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}$ using separation of variables. Find $y^{2}=\mathbf{C}$ .

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Problem 18758

Calculate the volume of the rectangle with length 1 and width $e$ rotated around the axis $O A$ .

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Problem 18759

Calculate the total in a savings account after investing \$9675 at 7.0\% for 7 years with continuous compounding. Round to two decimals.

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Problem 18760

Identify two series from the list where the ratio test indicates divergence:
1. $\sum_{n=1}^{\infty} \frac{(n+1)^{2 n}}{(n+1) !}$
2. $\sum_{n=1}^{\infty} \frac{\left(n^{2}\right) !}{(2 n) !}$
3. $\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 5}+\cdots$
4. $\sum_{n=1}^{\infty} \frac{n !}{2^{n !}}$
5. $\frac{3}{1}+\frac{3 \cdot 5}{1 \cdot 3}+\cdots$
6. $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$

See Solution

Problem 18761

Set up the integral for the volume of the solid formed by rotating $y = 2\cos(x^2)$ around the $y$ -axis using cylindrical shells.

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Problem 18762

Find the volume using cylindrical shells for $y=x^{3}$ , $y=0$ , $x=1$ , and $x=4$ .

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Problem 18763

Find the integral of the expression: $\int(1-\cot^{2} x) \, dx$ .

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Problem 18764

Find the positive critical number of the function $f(x)=(x^{2}-2)^{3}$ . Round your answer to three decimal places.

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Problem 18765

Find the volume using cylindrical shells for the region bounded by $y=\frac{1}{x}$ , $y=0$ , $x=1$ , and $x=3$ .

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Problem 18766

Evaluate the integral from 4 to 9: $\int_{4}^{9}(6 \sqrt{t}+4 t) d t$ .

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Problem 18767

Estimate $f(9.2)$ using linear approximation given $f(9)=6.4$ and $f^{\prime}(9)=7.4$ . Round to three decimal places.

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Problem 18768

Set up an integral for the volume of the region bounded by $y=\ln(x)$ , $y=0$ , and $x=5$ about the $y$ -axis.

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Problem 18769

Find the volume $V$ of the solid formed by rotating the area between $x=y^{2}$ and $x=1-y^{2}$ around $x=4$ .

See Solution

Problem 18770

Find the function $g(x)$ where $g^{\prime}(x)=6 x^{5}-3 x^{2}+\frac{2}{x}-6$ and $g(1)=14. g(x)=\square$

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Problem 18771

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis from $x=2$ to $x=9$ using right-endpoint Riemann sums.

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Problem 18772

Find the derivative of $f(x)=9+3 \tan^{-1}(4.3 \ln(x))$ and evaluate $f'(8)$ . Round to three decimal places.

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Problem 18773

Solve the initial value problem $(1+x^{20}) y' + 20 x^{19} y = 18 x^{20}$ with $y(0)=2$ .

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Problem 18774

Find the rate of change of area for a circle with radius 8 ft, where the radius increases at 5 ft/s. Round to the nearest thousandth.

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Problem 18775

Find $\frac{d y}{d x}$ from the equation $V=-4 x-2 \ln (x)+1.4 y-2.2 \ln (y)$ , then evaluate at $x=6$ , $y=2$ .

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Problem 18776

Find the asymptotic behavior of $a=\left(\frac{n(n+1)}{n^{2}+4}\right)$ as $n \to \infty$ and check series convergence.

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Problem 18777

Find the function $f(x)$ where $f^{\prime}(x)=x^{3}-3 x^{-4}+2$ and $f(1)=2$ . What is $f(x)=\square$ ?

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Problem 18778

Find the positive $x$ -coordinate of the inflection point for $f(x)=x \exp \left(-x^{2} / 4\right)$ . Round to three decimal places.

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Problem 18779

Justifiez vos réponses. Calculez les dérivées des fonctions suivantes : a) $f(x)=\sin \left(e^{x}\right)$ b) $g(x)=e^{\left(x^{2}\right)}+3^{\left(x^{2}\right)}$ c) $h(x)=\pi^{\sin (x)}+\log (\cos x)$ d) $s(x)=\tan ^{2}\left(x^{2}\right)$ e) $k(x)=\arcsin \left(\frac{1}{x}\right)+\arccos \left(\frac{1}{2 x}\right)+\cos (-x)$

See Solution

Problem 18780

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis on $[2,9]$ using right-endpoint Riemann sums.
a. Find $\Delta x$ in terms of $n$ : $\Delta x=\frac{7}{n}$
b. Determine right endpoints $x_{1}, x_{2}, x_{3}$ in terms of $n$ : $x_{1}, x_{2}, x_{3}=\square$
c. General expression for right endpoint $x_{k}$ : $x_{k}=\square$

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Problem 18781

Prove that the $(n+1)^{th}$ derivative of $f(x) g(x)$ follows Pascal's triangle pattern, then show it for the $100^{th}$ derivative.

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Problem 18782

A conical cup is 20 cm tall and 10 cm radius. Water fills at $\frac{16 \pi}{3 V} \mathrm{~cm}^{3} / \mathrm{sec}$ . Find the rise rate when the water is 3 cm deep.

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Problem 18783

Which two series ensure convergence using the root test?
1. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$
2. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n^{2}}$
3. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
4. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln ^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
5. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n / 2}$
6. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$

See Solution

Problem 18784

Determine which two of the following sequences converge:
1. $\sum_{n=1}^{\infty} \frac{n !}{(2 n) !}$
2. $\sum_{n=1}^{\infty}\left(\frac{n-1}{n+1}\right)^{n^{2}+n}$
3. $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{n !}$
4. $\sum_{n=1}^{\infty}\left(\frac{n^{2}}{2 n+1}\right)^{n}$
5. $\sum_{n=1}^{\infty} \frac{3^{n}}{n^{3}}$
6. $\sum_{n=1}^{\infty}\left(1+\frac{1}{n^{2}}\right)^{n^{3}}$
7. $\sum_{n=1}^{\infty} \frac{n^{3}+1}{\sqrt[3]{n^{10}+n}}$

See Solution

Problem 18785

Determine which two series converge using the root test:
1. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$
2. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n^{2}}$
3. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
4. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln ^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
5. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n / 2}$
6. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$

See Solution

Problem 18786

Find the function $f(x)$ where $f^{\prime}(x)=x^{4}-4 x^{-2}+3$ and $f(1)=0$ . What is $f(x)$ ?

See Solution

Problem 18787

Which two series below diverge according to the ratio test?
1. $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$
2. $\sum_{n=1}^{\infty} \frac{\left(n^{2}\right) !}{(2 n) !}$
3. $\sum_{n=1}^{\infty} \frac{(n+1)^{2 n}}{(n+1) !}$
4. $\sum_{n=1}^{\infty} \frac{n !}{2^{n !}}$
5. Series with odd products and denominators.

See Solution

Problem 18788

Find the function $f$ such that $f^{\prime}(t)=\sin t+8t$ and $f(0)=-3$ . What is $f(t)$ ?

See Solution

Problem 18789

Find the volume of the solid formed by revolving the area between $x=(y-4)^{2}$ and $x=1$ around $y=3$ .

See Solution

Problem 18790

Find the function $y$ such that $\frac{d y}{d x}=8 x\left(x^{7}-\frac{1}{8}\right)$ with $y(1)=2$ .

See Solution

Problem 18791

Find the rate of change of a rider's position when 30 m high on a 53 m Ferris wheel completing 3 revolutions in 12 min.

See Solution

Problem 18792

Find the right-endpoint Riemann sum for $f\left(x_{k}\right)=\left(2+\frac{7 k}{n}\right)^{2}$ with $\Delta x = \frac{7}{n}$ , then its limit as $n \to \infty$ .

See Solution

Problem 18793

Find $y$ where $\frac{d y}{d x}=5 x^{2}+3 x-8+2 e^{x}$ and $y(0)=6$ . What is $y$ ?

See Solution

Problem 18794

Calculate the average value $f_{\text {ave }}$ of the function $f(z)=\frac{e^{1 / z}}{z^{2}}$ over the interval [3,8].

See Solution

Problem 18795

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis from $x=2$ to $x=9$ using Riemann sums.

See Solution

Problem 18796

Find the average value $f_{\text{ave}}$ of $f(x)=\sqrt{x}$ over the interval $[0,16]$ .

See Solution

Problem 18797

Evaluate the integral: $\int (12 x^{3/7} - 10 x^{6/7}) \, dx = \square($ Type an exact answer. $)$

See Solution

Problem 18798

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis from $x=2$ to $x=9$ using right-endpoint Riemann sums.

See Solution

Problem 18799

Subdivide $[2,9]$ into $n$ intervals. Find $\Delta x$ , right endpoints $x_1, x_2, x_3$ , and Riemann sum limits.
1. $\Delta x=\frac{7}{n}$
2. $x_1=2+\frac{7}{n}, x_2=2+\frac{14}{n}, x_3=2+\frac{21}{n}$
3. $x_k=2+\frac{7k}{n}$
4. $f(x_k)=(2+\frac{7k}{n})^2$
5. $f(x_k) \Delta x=(2+\frac{7k}{n})^2 \cdot \frac{7}{n}$
6. Riemann sum: $\sum_{k=1}^{n} f(x_k) \Delta x$
7. Limit: $\lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta x=0$

See Solution

Problem 18800

Evaluate the integral: $\int(5 x^{3}+7 x^{2}-3 x+2) dx = \square($ Type an exact answer. $)$

See Solution

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Calculus

Problem 18701

Find the limit: lim⁡x→∞5x−x23x2+2x−4\lim _{x \rightarrow \infty} \frac{5 x-x^{2}}{3 x^{2}+2 x-4}limx→∞​3x2+2x−45x−x2​.

Problem 18702

Find the general antiderivative of f(x)=3x13+5x−13+7f(x)=3 x^{\frac{1}{3}}+5 x^{-\frac{1}{3}}+7f(x)=3x31​+5x−31​+7. Let F(x)=□F(x)=\squareF(x)=□ (use C for constant).

Problem 18703

Find the limit: lim⁡x→−∞xex\lim _{x \rightarrow-\infty} \frac{x}{e^{x}}limx→−∞​exx​.

Problem 18704

Find the limit: lim⁡x→0sin⁡3xsin⁡5x\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x}limx→0​sin5xsin3x​.

Problem 18705

Find the general antiderivative of f(x)=3x6+10x5−x3x4f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}f(x)=x43x6+10x5−x3​.

Problem 18706

Find the general antiderivative of f(x)=3x6+10x5−x3x4f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}f(x)=x43x6+10x5−x3​, denoted as F(x)=□F(x)=\squareF(x)=□ (use C for constant).

Problem 18707

Find the general antiderivative of f(x)=3x5+10x4−x2x3f(x)=\frac{3 x^{5}+10 x^{4}-x^{2}}{x^{3}}f(x)=x33x5+10x4−x2​. Let F(x)=□F(x)=\squareF(x)=□ (use C for the constant).

Problem 18708

Bestimme die ersten beiden Ableitungen von fff für: a) f(x)=3+exf(x)=3+e^{x}f(x)=3+ex, b) f(x)=3exf(x)=3 e^{x}f(x)=3ex, c) f(x)=e3xf(x)=e^{3 x}f(x)=e3x, d) f(x)=e−xf(x)=e^{-x}f(x)=e−x.

Problem 18709

Which statement about the function f(x)=(x+5)(x+1)3(x−2)2f(x)=(x+5)(x+1)^{3}(x-2)^{2}f(x)=(x+5)(x+1)3(x−2)2 is true?

Problem 18710

Insect population size P(t)=400e0.04tP(t)=400 e^{0.04 t}P(t)=400e0.04t: (a) Find P(0)P(0)P(0), (b) growth rate, (c) P(10)P(10)P(10), (d) when P(t)=560P(t)=560P(t)=560, (e) when P(t)P(t)P(t) doubles.

Problem 18711

Bestimme PR oder KR für die Ableitung der Funktionen und berechne die erste Ableitung:a) f(x)=2⋅ex+1f(x)=2 \cdot e^{x+1}f(x)=2⋅ex+1b) f(x)=2x⋅ex+1f(x)=2 x \cdot e^{x}+1f(x)=2x⋅ex+1c) f(x)=2+e3xf(x)=2+e^{3 x}f(x)=2+e3xd) f(x)=xexf(x)=\frac{x}{e^{x}}f(x)=exx​

Problem 18712

Problema: Dibuja el gráfico de T=25+e0.4tT=25+\mathrm{e}^{0.4 t}T=25+e0.4t para 0≤t≤120 \leq t \leq 120≤t≤12. Encuentra TTT a la mitad y el tiempo para T=100∘CT=100^{\circ}CT=100∘C.

Problem 18713

Insect population P(t)=500e0.02tP(t)=500 e^{0.02 t}P(t)=500e0.02t: (a) Find P(0)P(0)P(0), (b) growth rate, (c) P(10)P(10)P(10), (d) when P(t)=700P(t)=700P(t)=700, (e) when P(t)P(t)P(t) doubles.

Problem 18714

Problem 18715

A flu epidemic starts with 5 cases on day t=0t=0t=0. The growth rate is r(t)=22e0.04tr(t)=22 e^{0.04 t}r(t)=22e0.04t. (a) Find the total cases formula F(t)F(t)F(t). (b) Calculate F(25)F(25)F(25) (round to the nearest whole number).

Problem 18716

How long does it take for a bacteria population to double with a growth rate of 8.9%8.9\%8.9% per hour? Round to the nearest hundredth.

Problem 18717

Una moneda de un euro cae desde reposo. ¿Cuál es su posición y velocidad tras 1 s1 \mathrm{~s}1 s? Opciones: a) b) c) d) e)

Problem 18718

Find the relative rate of change of the Van der Waals equation P(V,T)=RTV−b−aV2P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}P(V,T)=V−bRT​−V2a​ with respect to VVV. What is ∂P∂V\frac{\partial P}{\partial V}∂V∂P​?

Problem 18719

Find the relative rate of change of pressure PPP in the Van der Waals equation: P(V,T)=RTV−b−aV2P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}P(V,T)=V−bRT​−V2a​. What is ∂P∂T\frac{\partial P}{\partial T}∂T∂P​? (A) (B) (C) (D) (E) (F) (G)

Problem 18720

Find the derivative of f(x)=∫−4xt3+64 dtf(x)=\int_{-4}^{x} \sqrt{t^{3}+64} \, dtf(x)=∫−4x​t3+64​dt. What is f′(x)f'(x)f′(x)?

Problem 18721

Evaluate the integral from 1 to 5 of 6x−2−56 x^{-2} - 56x−2−5. What is ∫15(6x−2−5)dx\int_{1}^{5}(6 x^{-2}-5) d x∫15​(6x−2−5)dx?

Problem 18722

Evaluate the integral ∫0220x4 dx\int_{0}^{2} 20 x^{4} \, dx∫02​20x4dx.

Problem 18723

Find the derivative of g(y)=∫1yt15sin⁡t dtg(y)=\int_{1}^{y} t^{15} \sin t \, dtg(y)=∫1y​t15sintdt. What is g′(y)g'(y)g′(y)?

Problem 18724

Evaluate the integral: ∫2124x dx\int_{2}^{12} 4 \sqrt{x} \, dx∫212​4x​dx using the Fundamental Theorem of Calculus.

Problem 18725

Find the partial derivative ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ for f(x,y,z)=xy2−2zy+xf(x, y, z)=x y^{2}-2 z y+xf(x,y,z)=xy2−2zy+x.

Problem 18726

Problem 18727

Find the partial derivatives δfδx\frac{\delta f}{\delta x}δxδf​ and δfδy\frac{\delta f}{\delta y}δyδf​ for f(x,y)=xexyf(x, y)=x e^{x y}f(x,y)=xexy.

Problem 18728

Find the partial derivative ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ for f(x,y,t,a)=txy2+ex+tsin⁡ayf(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a yf(x,y,t,a)=xt​y2+ex+tsinay.

Problem 18729

Evaluate the Laplacian of ψ(x,y,z)=zx2x2+y2+z2\psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}}ψ(x,y,z)=x2+y2+z2zx2​ in Cartesian and spherical coordinates and prove they are equal.

Problem 18730

Find ∂f∂t\frac{\partial f}{\partial t}∂t∂f​ for f(x,y,t,a)=txy2+ex+tsin⁡ayf(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a yf(x,y,t,a)=xt​y2+ex+tsinay. Choose the correct option.

Problem 18731

Find ∂f∂y\frac{\partial f}{\partial y}∂y∂f​ for f(x,y,t,a)=txy2+ex+tsin⁡ayf(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a yf(x,y,t,a)=xt​y2+ex+tsinay. Choose from options (A) to (G).

Problem 18732

Problem 18733

Find the second partial derivative ∂2z∂y2\frac{\partial^{2} z}{\partial y^{2}}∂y2∂2z​ for z(x,y)=sin⁡(xy)z(x, y)=\sin (x y)z(x,y)=sin(xy).

Problem 18734

Find the intervals where the function f(x)=x3−9x2+5f(x)=x^{3}-9 x^{2}+5f(x)=x3−9x2+5 is increasing, decreasing, or constant. Use interval notation.

Problem 18735

Déterminez les dimensions du rectangle d'aire maximale inscrit sous la courbe f(x)=4−x2f(x)=\sqrt{4-x^{2}}f(x)=4−x2​ et au-dessus de l'axe.

Problem 18736

Find the second mixed partial derivative ∂2f∂x∂y\frac{\partial^{2} f}{\partial x \partial y}∂x∂y∂2f​ for f(x,y)=cos⁡(xy)f(x, y)=\cos (x y)f(x,y)=cos(xy).

Problem 18737

Find the time for \$ 4125 to double at a continuous compounding rate of 9.5\%.

Problem 18738

Find the second partial derivative ∂2f∂x2\frac{\partial^{2} f}{\partial x^{2}}∂x2∂2f​ for f(x,y)=cos⁡(xy)f(x, y)=\cos (x y)f(x,y)=cos(xy). Choices: (A) (B) (C) (D) (E) (F) (G)

Problem 18739

Find local maxima, minima, and saddle points of f(x,y)=4+x3+y3−3xyf(x, y)=4+x^{3}+y^{3}-3 x yf(x,y)=4+x3+y3−3xy. Options: (A) (-1,-1) min, (B) (0,0) saddle, (1,1) min, (C) (0,0) max, (D) (0,0) min, (1,1) max, (E) (1,1) saddle, (0,0) min, (F) (0,0) saddle, (1,1) max, (G) Not listed.

Problem 18740

Find the absolute max and min of f(x,y)=x2+4y2−2x2y+4f(x, y)=x^{2}+4 y^{2}-2 x^{2} y+4f(x,y)=x2+4y2−2x2y+4 on −1≤x,y≤1-1 \leq x, y \leq 1−1≤x,y≤1.

Problem 18741

Find the intervals where the function f(x)=x2+4x+4x+1f(x)=\frac{x^{2}+4 x+4}{x+1}f(x)=x+1x2+4x+4​ is increasing, decreasing, or constant. Use interval notation.

Find the limit: $\lim _{x \rightarrow \infty} \frac{5 x-x^{2}}{3 x^{2}+2 x-4}$ .

Find the general antiderivative of $f(x)=3 x^{\frac{1}{3}}+5 x^{-\frac{1}{3}}+7$ . Let $F(x)=\square$ (use C for constant).

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x}{e^{x}}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 5 x}$ .

Find the general antiderivative of $f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}$ .

Find the general antiderivative of $f(x)=\frac{3 x^{6}+10 x^{5}-x^{3}}{x^{4}}$ , denoted as $F(x)=\square$ (use C for constant).

Find the general antiderivative of $f(x)=\frac{3 x^{5}+10 x^{4}-x^{2}}{x^{3}}$ . Let $F(x)=\square$ (use C for the constant).

Bestimme die ersten beiden Ableitungen von $f$ für: a) $f(x)=3+e^{x}$ , b) $f(x)=3 e^{x}$ , c) $f(x)=e^{3 x}$ , d) $f(x)=e^{-x}$ .

Which statement about the function $f(x)=(x+5)(x+1)^{3}(x-2)^{2}$ is true?

Insect population size $P(t)=400 e^{0.04 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P(t)=560$ , (e) when $P(t)$ doubles.

Bestimme PR oder KR für die Ableitung der Funktionen und berechne die erste Ableitung:
a) $f(x)=2 \cdot e^{x+1}$
b) $f(x)=2 x \cdot e^{x}+1$
c) $f(x)=2+e^{3 x}$
d) $f(x)=\frac{x}{e^{x}}$

Problema: Dibuja el gráfico de $T=25+\mathrm{e}^{0.4 t}$ para $0 \leq t \leq 12$ . Encuentra $T$ a la mitad y el tiempo para $T=100^{\circ}C$ .

Insect population $P(t)=500 e^{0.02 t}$ : (a) Find $P(0)$ , (b) growth rate, (c) $P(10)$ , (d) when $P(t)=700$ , (e) when $P(t)$ doubles.

A flu epidemic starts with 5 cases on day $t=0$ . The growth rate is $r(t)=22 e^{0.04 t}$ .
(a) Find the total cases formula $F(t)$ .
(b) Calculate $F(25)$ (round to the nearest whole number).

How long does it take for a bacteria population to double with a growth rate of $8.9\%$ per hour? Round to the nearest hundredth.

Una moneda de un euro cae desde reposo. ¿Cuál es su posición y velocidad tras $1 \mathrm{~s}$ ? Opciones: a) b) c) d) e)

Find the relative rate of change of the Van der Waals equation $P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}$ with respect to $V$ . What is $\frac{\partial P}{\partial V}$ ?

Find the relative rate of change of pressure $P$ in the Van der Waals equation: $P(V, T)=\frac{R T}{V-b}-\frac{a}{V^{2}}$ . What is $\frac{\partial P}{\partial T}$ ? (A) (B) (C) (D) (E) (F) (G)

Find the derivative of $f(x)=\int_{-4}^{x} \sqrt{t^{3}+64} \, dt$ . What is $f'(x)$ ?

Evaluate the integral from 1 to 5 of $6 x^{-2} - 5$ . What is $\int_{1}^{5}(6 x^{-2}-5) d x$ ?

Evaluate the integral $\int_{0}^{2} 20 x^{4} \, dx$ .

Find the derivative of $g(y)=\int_{1}^{y} t^{15} \sin t \, dt$ . What is $g'(y)$ ?

Evaluate the integral: $\int_{2}^{12} 4 \sqrt{x} \, dx$ using the Fundamental Theorem of Calculus.

Find the partial derivative $\frac{\partial f}{\partial x}$ for $f(x, y, z)=x y^{2}-2 z y+x$ .

Find the partial derivatives $\frac{\delta f}{\delta x}$ and $\frac{\delta f}{\delta y}$ for $f(x, y)=x e^{x y}$ .

Find the partial derivative $\frac{\partial f}{\partial x}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ .

Evaluate the Laplacian of $\psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}}$ in Cartesian and spherical coordinates and prove they are equal.

Find $\frac{\partial f}{\partial t}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ . Choose the correct option.

Find $\frac{\partial f}{\partial y}$ for $f(x, y, t, a)=\frac{t}{x} y^{2}+e^{x}+t \sin a y$ . Choose from options (A) to (G).

Find the second partial derivative $\frac{\partial^{2} z}{\partial y^{2}}$ for $z(x, y)=\sin (x y)$ .

Find the intervals where the function $f(x)=x^{3}-9 x^{2}+5$ is increasing, decreasing, or constant. Use interval notation.

Déterminez les dimensions du rectangle d'aire maximale inscrit sous la courbe $f(x)=\sqrt{4-x^{2}}$ et au-dessus de l'axe.

Find the second mixed partial derivative $\frac{\partial^{2} f}{\partial x \partial y}$ for $f(x, y)=\cos (x y)$ .

Find the second partial derivative $\frac{\partial^{2} f}{\partial x^{2}}$ for $f(x, y)=\cos (x y)$ . Choices: (A) (B) (C) (D) (E) (F) (G)

Find local maxima, minima, and saddle points of $f(x, y)=4+x^{3}+y^{3}-3 x y$ . Options: (A) (-1,-1) min, (B) (0,0) saddle, (1,1) min, (C) (0,0) max, (D) (0,0) min, (1,1) max, (E) (1,1) saddle, (0,0) min, (F) (0,0) saddle, (1,1) max, (G) Not listed.

Find the absolute max and min of $f(x, y)=x^{2}+4 y^{2}-2 x^{2} y+4$ on $-1 \leq x, y \leq 1$ .

Find the intervals where the function $f(x)=\frac{x^{2}+4 x+4}{x+1}$ is increasing, decreasing, or constant. Use interval notation.

Identify the true statement about the function $f(x, y)=4y-1+x^{2}-y^{2}$ from the options provided.

Find the volume of the solid formed by rotating the area between $xy=1$ , $y=0$ , $x=1$ , $x=5$ around $x=-1$ using the shell method.

Find $Q$ at $t=0$ in $Q=55 e^{1.03 t}$ , determine if $Q$ is increasing or decreasing, and find the growth rate.

Calculate the volume $V$ of the solid formed by rotating the area between $x=y^{2}$ and $x=1-y^{2}$ around $x=4$ .

Find the asymptotic behavior of $a=\left(\frac{n(n+1)}{n^{2}+4}\right)$ as $n \to \infty$ and check series convergence.

Sea $f(x)=3 \ln (x-2)+1$ a) Encuentra el dominio de $f$ . b) ¿Cuál es la asíntota vertical de $f$ ? c) Halla la intersección con el eje $x$ . d) Encuentra la tangente a $f$ paralela a $y=x$ .

Identify the two true statements about the series $\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n}$ .

Calculate the volume of the region bounded by $y=3x$ , $y=3$ , $y=3\sqrt{x}$ , $x=0$ , $x=1$ rotated about $y=3$ .

How long to double \ $3975 at a continuous$ 6\%$ interest rate?

Solve the equation $(4+x^{4}) \frac{dy}{dx}=\frac{x^{3}}{y}$ using separation of variables. Find $y^{2}=\mathbf{C}$ .

Calculate the volume of the rectangle with length 1 and width $e$ rotated around the axis $O A$ .

Set up the integral for the volume of the solid formed by rotating $y = 2\cos(x^2)$ around the $y$ -axis using cylindrical shells.

Find the volume using cylindrical shells for $y=x^{3}$ , $y=0$ , $x=1$ , and $x=4$ .

Find the integral of the expression: $\int(1-\cot^{2} x) \, dx$ .

Find the positive critical number of the function $f(x)=(x^{2}-2)^{3}$ . Round your answer to three decimal places.

Find the volume using cylindrical shells for the region bounded by $y=\frac{1}{x}$ , $y=0$ , $x=1$ , and $x=3$ .

Evaluate the integral from 4 to 9: $\int_{4}^{9}(6 \sqrt{t}+4 t) d t$ .

Estimate $f(9.2)$ using linear approximation given $f(9)=6.4$ and $f^{\prime}(9)=7.4$ . Round to three decimal places.

Set up an integral for the volume of the region bounded by $y=\ln(x)$ , $y=0$ , and $x=5$ about the $y$ -axis.

Find the volume $V$ of the solid formed by rotating the area between $x=y^{2}$ and $x=1-y^{2}$ around $x=4$ .

Find the function $g(x)$ where $g^{\prime}(x)=6 x^{5}-3 x^{2}+\frac{2}{x}-6$ and $g(1)=14. g(x)=\square$

Calculate the area between $f(x)=x^{2}$ and the $x$ -axis from $x=2$ to $x=9$ using right-endpoint Riemann sums.

Find the derivative of $f(x)=9+3 \tan^{-1}(4.3 \ln(x))$ and evaluate $f'(8)$ . Round to three decimal places.

Solve the initial value problem $(1+x^{20}) y' + 20 x^{19} y = 18 x^{20}$ with $y(0)=2$ .