Calculus

Problem 21601

Find the integral of the function: $\int 8 e^{4 x} d x$

See Solution

Problem 21602

Find $f(x)$ given $f^{\prime}(x)=\frac{3}{x^{5}}$ and $f\left(\frac{1}{2}\right)=1$ .

See Solution

Problem 21603

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2x+2$ , where $h \neq 0$ .

See Solution

Problem 21604

Function $f(x)=9-x^{2}$ is positive/negative on $[0,4]$ . Sketch it, then find net area using Riemann sums with $n=4$ .

See Solution

Problem 21605

Evaluate the integral $\int_{-1.32}^{-1.32} 1957 e^{x^{4}} \, dx$ .

See Solution

Problem 21606

Find a suitable substitution $u$ for the integral $\int x^{2} e^{x^{3}} d x$ . Is $u=x^{2}$ a good choice?

See Solution

Problem 21607

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} \cdot n}(x-1)^{n}$ converges.

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Problem 21608

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{a} \cdot x}(x-1)^{n}$ converges.

See Solution

Problem 21609

Find $\int_{1}^{4} f(x) \, dx$ given that $F(1) = 19$ and $F(4) = 13$ .

See Solution

Problem 21610

Evaluate $\int_{a}^{c} f(x) d x$ given $\int_{a}^{b} f(x) d x=7$ and $\int_{b}^{c} f(x) d x=9$ .

See Solution

Problem 21611

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} n^{2} x^{n}$ converges.

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Problem 21612

Which option correctly states the Fundamental Theorem of Calculus for a continuous function $f(x)$ ?

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Problem 21613

Given that $f$ is an even function and $\int_{-3}^{3} f(x) d x=14$ , find: a. $\int_{0}^{3} f(x) d x$ b. $\int_{-3}^{3} x f(x) d x$ .

See Solution

Problem 21614

Find the function $f$ from the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{7} \Delta x_{k}$ on $[8,16]$ .

See Solution

Problem 21615

Evaluate $\int_{a}^{c} f(x) d x$ given $\int_{a}^{b} f(x) d x=7$ and $\int_{b}^{c} f(x) d x=9$ .

See Solution

Problem 21616

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{n !(x+2)^{n}}{n^{n}}$ .

See Solution

Problem 21617

Graph the integrand, calculate $\Delta x$ , grid points, left and right Riemann sums for $\int^{9}\left(\frac{1}{x}+2\right) d x ; n=4$ .

See Solution

Problem 21618

Set up $R_{4}$ for $f(x)=x^{2}-3$ over the interval $[6,14]$ .

See Solution

Problem 21619

Find the average value of $f(x)=\frac{5}{x}$ on $[3, 3e]$ and graph the function with the average value indicated.

See Solution

Problem 21620

Calculate the average value of the function $f(x)=2 x^{2}$ over the interval $[3,5]$ .

See Solution

Problem 21621

Find $\Delta x$ for the Riemann sum $L_{4}$ of $f(x)=e^{(3 x^{2})}+\ln (x)$ over $[3,11]$ .

See Solution

Problem 21622

Find the average height of the arch modeled by $y=615\left[1-\left(\frac{x}{290}\right)^{2}\right]$ over its base.

See Solution

Problem 21623

Evaluate the integral: $\int \frac{3 x+2}{x^{2}+1} dx = \square + C$

See Solution

Problem 21624

Invest \$18,864 at 5.7% interest, compounded continuously. Find the function, balances for 1, 2, 5, 10 years, and doubling time.

See Solution

Problem 21625

Find the tangent line's parametric equations at $t=1$ for the particle with $x(t)=4 t^{2}$ and $y(t)=4 t^{3}$ .

See Solution

Problem 21626

Evaluate the integral: $\int 3 \sin^{3}(x) \cos(x) \, dx = \square + C$

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Problem 21627

A particle's position is $x=\cos(t)$ and $y=\sin\left(\frac{t}{2}\right)$ . Find when it first stops moving.

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Problem 21628

Find the weight of a person who is 5 feet, 4 inches tall using the formula $\frac{\mathrm{dW}}{\mathrm{dh}}=0.0012 \mathrm{~h}^{2}+0.01 \mathrm{~h}$ and $W(80)=216.8$ pounds.

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Problem 21629

Evaluate the integral $\int x^{2}(x^{3}-9)^{40} dx$ using the substitution $u=x^{3}-9$ . Find the result in terms of $x$ .

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Problem 21630

Find the weight of a person who is 5 feet 4 inches tall using the formula $\frac{d W}{d h}=0.0012 h^{2}+0.01 h$ and $W(80)=216.8$ .

See Solution

Problem 21631

Find the velocity and position functions for a particle with acceleration $\overrightarrow{\mathbf{a}}(t)=\sin t \hat{\mathbf{i}}+2 \cos t \hat{\mathbf{j}}+6 t \hat{\mathbf{k}}$ , initial velocity $\overrightarrow{\mathbf{v}}(0)=-\hat{\mathbf{k}}$ , and initial position $\overrightarrow{\mathbf{r}}(0)=\hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ .

See Solution

Problem 21632

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{k^{4}+7}$ . Radius: $R=\square$

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Problem 21633

Find where the function $f(x)=x^{4}-50 x^{2}+625$ is increasing and decreasing, and identify local extreme values.

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Problem 21634

Transform the integral $\int \frac{\cos (y)}{\sin (y)+1} d y$ using $u=\square$ and $d u=\square d y$ to get $\int d u$ .

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Problem 21635

Find the indefinite integral $\int(\ln (z))^{7} \frac{1}{z} d z$ using the substitution $u=\square$ and $d u=\square d z$ .

See Solution

Problem 21636

Find the radius and interval of convergence for the series $\sum_{k=1}^{\infty} \frac{x^{2 k+1}}{15^{k-1}}$ . What is $R=$ ?

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Problem 21637

Find the indefinite integral $\int x^{2} \sqrt[4]{x^{3}-4} d x$ using the substitution $u=\square$ and $d u=\square d x$ .

See Solution

Problem 21638

Find intervals where the function $g(x)=x \sqrt{50-x^{2}}$ is increasing or decreasing, and identify local extremes.

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Problem 21639

Find local max and min of $f$ using First and Second Derivative Tests for $f(x)=x+\sqrt{1-x}$ and $f(x)=\frac{x}{x^{2}+4}$ . Which method do you prefer?

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Problem 21640

Find the radius of convergence for the series $\sum_{k=1}^{\infty} \frac{(3 k) ! x^{k}}{(k !)^{3}}$ . What is $R$ ?

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Problem 21641

Find the inflection points, local maxima/minima, and intervals of differentiability for $y=-x+\sin 2x$ on $[-\frac{5\pi}{6}, \frac{5\pi}{6}]$ .

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Problem 21642

Determine the radius of convergence for $\sum_{k=1}^{\infty} \frac{k ! x^{5 k}}{(6 k)^{k}}$ in terms of $e$ .

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Problem 21643

Find the indefinite integral $\int(\ln (z))^{6} \frac{1}{z} d z$ using the substitution $u=\ln(z)$ .

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Problem 21644

Invest \ $17,180 at 6.1% interest, compounded continuously. Find the function$ P(t)$ and balances for 1, 2, 5, 10 years, plus doubling time.

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Problem 21645

Graph the function $y=x^{2}-2x-3$ : find domain, symmetries, derivatives, critical points, inflection points, and extremes. Domain: $\square$ .

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Problem 21646

Gegeben sind die Funktionen $f(x)=-\frac{1}{2} x^{2}+4 x-6$ und $g(x)=-2 x+7,5$ . Beantworte folgende Fragen: a) Fußpunkte des Hügels? b) Länge der Untergehung? c) Steilheit am westlichen Fußpunkt? d) Steigungswinkel dort? e) Höhe des Hügels und Hochpunkt? f) Westlicher Schnittpunkt von $f(x)$ und $g(x)$ ? g) Tangente in diesem Schnittpunkt? h) Schnittwinkel zwischen $f$ und $g$ ?

See Solution

Problem 21647

Find the first four nonzero terms of the Taylor series for $f(x)=\frac{5}{x^{4}}$ at $a=1$ , and the interval of convergence.

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Problem 21648

Find the interval where the function $f(x)=\frac{3 x^{2}-1}{x^{2}+3}$ is concave downward. A. $(-\infty,-1)$ and $(1, \infty)$ B. $(-1,1)$ C. $(0, \infty)$ D. $(-\infty,-1)$ E. $(1, \infty)$

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Problem 21649

Find the second derivative of $y=f(x)$ given $y^{\prime}=4 \sin x$ for $-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}$ .

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Problem 21650

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y=\sqrt{\frac{(x+1)^{8}}{(3 x-5)^{9}}}$ . $\frac{d y}{d x}=\square$

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Problem 21651

Calculate the slope of the function $f(x)=x^{2}+2$ at the point $(-1,3)$ .

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Problem 21652

Bestimmen Sie die positive Zahl $z$ aus den Integralen: a) $\int_{0}^{2} x \, dx = 18$ und b) $\int_{1}^{2} 4x \, dx = 30$ .

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Problem 21653

Find the second derivative $y^{\prime \prime}$ of $y=f(x)$ where $y^{\prime}=(10 x^{2}-20 x)(x-5)^{2}$ and sketch the graph of $f$ . $y^{\prime \prime}=\square$

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Problem 21654

Find the second derivative $y^{\prime \prime}$ if $y^{\prime}=-\pi \sec ^{2} x$ , then sketch $f$ using critical points.

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Problem 21655

Finde a für die Stammfunktionen: a) $f(x)=3 x^{2}; F(x)=x^{a}$ b) $f(x)=2 x; F(x)=x^{2}-a$ c) $f(x)=2 x; F(x)=x^{2}+1+a$ d) $f(x)=(a+1) \cdot x; F(x)=x^{a+1}$ .

See Solution

Problem 21656

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}$ using algebra.

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Problem 21657

What is the present value needed to reach \$1800 in 60 years at a continuous compounding rate of 5.5%?

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Problem 21658

Find the second derivative $y^{\prime \prime}$ of $y=f(x)$ given $y^{\prime}=x(x-9)^{2}$ and sketch the graph of $\mathrm{f}$ . $y^{\prime \prime}=\square$

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Problem 21659

Finde die Extrem- und Wendepunkte der Funktionen und bestimme die Art der Extrema. a) $f(x)=x^{5}-4 x^{2}$ b) $f(x)=\frac{1}{3} x^{3}+x^{2}+4 x$ c) $f(x)=x^{3}-x-1$

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Problem 21660

Find $\lim _{x \rightarrow 4} f(x)$ if $\lim _{x \rightarrow 4} g(x)=2$ and $\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}=\pi$ .

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Problem 21661

Find $\lim_{x \rightarrow 1} f(x)$ for $f(x)=\frac{\frac{1}{x}-1}{x-1}$ and compare with options (4), (i1), (C), (D).

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Problem 21662

Bestimme die Stammfunktion für folgende Funktionen: a) $f(x)=0,5 x^{3}$ , b) $f(x)=\frac{1}{4} x^{-2}$ , c) $f(x)=\frac{2}{5 x^{2}}$ , d) $f(x)=(2 x+2)^{3}$ , e) $f(x)=\frac{1}{3} x^{3}$ , f) $f(x)=x^{2} \cdot x^{3}$ , g) $f(x)=\frac{1}{x^{2}}+x$ , h) $f(x)=(2 x+1)^{-2}$ .

See Solution

Problem 21663

Calculate the following integrals: a) $\int_{0}^{2}(2+x)^{3} dx$ , b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) dx$ , c) $\int_{0}^{2} \frac{(7}{(1}$ .

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Problem 21664

Given the function $f(x)=\frac{2 x}{x^{2}-49}$ , find critical numbers, intervals of decrease, local max/min, inflection points, concavity, and asymptotes.

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Problem 21665

Berechnen Sie die Integrale: a) $\int_{0}^{2}(2+x)^{3} d x$ , b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) d x$ , e) $\int_{-0,5}^{0} e^{2 x+1} d x$ , f) $\int_{-1}^{0} e^{-x} d x$ .

See Solution

Problem 21666

A substance grows at $12\%$ daily. If it starts at 614 grams, find its mass after 4 days. Round to the nearest tenth.

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Problem 21667

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{1}{n 3^{n}}$ .

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Problem 21668

Finde die $x$ -Werte der Funktion $f(x)=-x^{4}+3 x^{3}$ mit waagerechter Tangente unter Verwendung des Vorzeichenwechselkriteriums.

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Problem 21669

Berechne die Stammfunktion von $f(x)=0,1\left(x^{2}-1\right)(x-3)(x+4)$ .

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Problem 21670

Bestimme die Extrempunkte und den Wendepunkt der Funktion $f(x)=-\frac{1}{4} x^{3}+\frac{3}{2} x^{2}$ und zeige, dass die Gerade durch die Extrempunkte den Wendepunkt schneidet.

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Problem 21671

Find the limit: $\lim _{x \rightarrow 0+} x^{x}$ .

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Problem 21672

Bestimmen Sie die Intervalle für strenges Wachstum/Fall von $f(x)=x^{3}-\frac{9}{4} x^{2}-3 x$ , die Extrempunkte und deren Art. Berechnen Sie Wendepunkte und Nullstellen.

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Problem 21673

Gegeben die Funktion $f(x)=\frac{1}{4} x^{3}-2 x$ (a) Finde die Nullstellen von $f$ . (b) Zeige $f^{\prime}(-2)=1$ und berechne die Tangente an $x=-2$ .

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Problem 21674

Find the limit as $x$ approaches 1 for the expression $\frac{x^{2}-\sqrt{x}}{\sqrt{x}-1}$ .

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Problem 21675

Gegeben ist der Graph der Funktion $f$ . Bestimmen Sie die Steigung an einigen Punkten und skizzieren Sie $f'$ im Koordinatensystem.

See Solution

Problem 21676

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$ as $x$ approaches 1.

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Problem 21677

Erdbeerertrag $f(x)=-\frac{1}{8} x^{3}+\frac{3}{4} x^{2}+8$ für $0 \leq x \leq 7$ . (a) Graph skizzieren. (b) Ertrag ohne Dünger? Maximaler Ertrag und größter Ertragszuwachs?

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Problem 21678

Berechne die mittlere Änderungsrate für die Funktionen in den angegebenen Intervallen. a) $f(x)=x^{3}+1 ; \quad I=[0 ; 2]$ b) $f(x)=-x^{2}+1 ; \quad I=[1 ; 3]$ c) $f(x)=x^{2}+x+2 ; \quad I=[-1 ; 2]$

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Problem 21679

Bestimme die Werte von $x$ und $y$ , sodass die Fläche $A = x(400 - \frac{1}{2} x)$ maximal wird. Maximalwert ist $A_{max} = 80000 \, m^2$ .

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Problem 21680

Erdbeerertrag $f(x)=-\frac{1}{8} x^{3}+\frac{3}{4} x^{2}+8$ für Düngemenge $x$ in 100 kg.
(a) Skizziere $f$ für $0 \leq x \leq 7$ . (b) Ertrag ohne Düngung. (c) Düngemenge für maximalen Ertrag. (d) Düngemenge für größten Ertragszuwachs.

See Solution

Problem 21681

Gegeben ist die Funktion $f(x)=-x^{2}+4$ . Finde eine Stammfunktion $F$ und berechne die Fläche zwischen $f$ und der X-Achse.

See Solution

Problem 21682

Find the derivative of $f(x) = \sqrt{2x + 1} \cdot (2x + 1)^2$ .

See Solution

Problem 21683

Berechne die Stammfunktion von $f(x)=-(x-2)^{2}+1$ .

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Problem 21684

Indiana Jones fährt mit einer Lore. An Höhe A (40 m) hat sie $v_{0}=162 \text{ km/h}$ . Bestimme $v_{B}$ (0 m) und $v_{C}$ (20 m).

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Problem 21685

Berechne die Fläche A zwischen den Funktionen $f(x)=6-\frac{3}{\pi} x$ und $g(x)=3 \cdot \sin \left(\frac{x}{2}\right)$ an der y-Achse.

See Solution

Problem 21686

Find the difference quotient for $f(x) = \sqrt{x^{2}+9}$ , which is $\frac{f(x+h) - f(x)}{h}$ .

See Solution

Problem 21687

Bestimme die Fläche unter $f(x) = x^3 - 2x^2 + x$ im Intervall $[-1; 2]$ zwischen den Nullstellen $x = 0$ und $x = 1$ .

See Solution

Problem 21688

Find the biomass at $t=9$ hours if it starts at 13 mg and changes at $\frac{1}{7} \mathrm{mg}/$ hour from $t=2$ to $t=9$ . a) 12 b) 13 c) 14 d) 15 e) None

See Solution

Problem 21689

Find the difference quotient of the function $f(x)=\sqrt{2 x+2}$ . Which option is correct? a) b) c) d) e) None.

See Solution

Problem 21690

Find the difference quotient for $f(x)=\sqrt{3x+2}$ . Which option is correct? a) b) c) d) e) None.

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Problem 21691

Is the function $f(x)$ continuous at $x=0$ where $f(x)=\lfloor x+5\rfloor$ for $x<0$ , $2 e^{-2 x}$ for $0 \leq x<7$ , and $3 \ln (x-2)$ for $x \geq 7$ ?

See Solution

Problem 21692

Find the integral of the function: $\int\left(7 e^{3 t}+5 t+2\right) d t$

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Problem 21693

Find the critical points of the function $y=f(x)=4x^{3}+18x^{2}-48x+6$ .

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Problem 21694

Find $\lim _{x \rightarrow 0} f(x)$ for the piecewise function: $f(x)=\left\{\begin{array}{cc}\lfloor x+5\rfloor & x<0 \\ 2 e^{-2 x} & 0 \leq x<7 \\ 3 \ln (x-2) & x \geq 7\end{array}\right.$ . Choices: a) 3 b) 2 c) 4 d) 6 e) D.N.E.

See Solution

Problem 21695

Find the time when the drug concentration $K(x)=\frac{6x}{x^{2}+25}$ is maximum. Choose from: a) 25, b) 5, c) 5 and -5, d) 30, e) None.

See Solution

Problem 21696

Approximate the change in drug concentration $C(x)=\frac{10 x}{(9+x^{3})}$ for $x$ from 0.5 to 1.5. Choices: a) 0.4 b) 1.02 c) 0.35 d) 1.05 e) 1.88

See Solution

Problem 21697

Approximate the change in drug concentration $C(x)=\frac{10 x}{9+x^{2}}$ for $x$ changing from 1.5 to 3.

See Solution

Problem 21698

Find the critical numbers of $f(x)=\frac{1}{3} \sqrt{48-3 x^{2}}$ . Options: a) 0 b) $-4,0,4$ c) $-3,0,3$ d) $-2,0,2$ e) None.

See Solution

Problem 21699

Find the monkey's acceleration at time $t=5$ from the distance $S(t)=t \sin (2 t)+t^{2}$ . Round up your answer. Options: a) 27 b) 10 c) 3 d) 15

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Problem 21700

Find the monkey's acceleration at time $t=2$ for the distance function $S(t)=t \sin (3 t)+t^{2}$ . Round up the result.

See Solution

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Calculus

Problem 21601

Find the integral of the function: ∫8e4xdx\int 8 e^{4 x} d x∫8e4xdx

Problem 21602

Find f(x)f(x)f(x) given f′(x)=3x5f^{\prime}(x)=\frac{3}{x^{5}}f′(x)=x53​ and f(12)=1f\left(\frac{1}{2}\right)=1f(21​)=1.

Problem 21603

Calculate the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−2x+2f(x)=-2x+2f(x)=−2x+2, where h≠0h \neq 0h=0.

Problem 21604

Function f(x)=9−x2f(x)=9-x^{2}f(x)=9−x2 is positive/negative on [0,4][0,4][0,4]. Sketch it, then find net area using Riemann sums with n=4n=4n=4.

Problem 21605

Evaluate the integral ∫−1.32−1.321957ex4 dx\int_{-1.32}^{-1.32} 1957 e^{x^{4}} \, dx∫−1.32−1.32​1957ex4dx.

Problem 21606

Find a suitable substitution uuu for the integral ∫x2ex3dx\int x^{2} e^{x^{3}} d x∫x2ex3dx. Is u=x2u=x^{2}u=x2 a good choice?

Problem 21607

Find the values of xxx for which the series ∑n=1∞(−1)n3n⋅n(x−1)n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} \cdot n}(x-1)^{n}∑n=1∞​3n⋅n(−1)n​(x−1)n converges.

Problem 21608

Find the values of xxx for which the series ∑n=1∞(−1)n3a⋅x(x−1)n\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{a} \cdot x}(x-1)^{n}∑n=1∞​3a⋅x(−1)n​(x−1)n converges.

Problem 21609

Find ∫14f(x) dx\int_{1}^{4} f(x) \, dx∫14​f(x)dx given that F(1)=19F(1) = 19F(1)=19 and F(4)=13F(4) = 13F(4)=13.

Problem 21610

Evaluate ∫acf(x)dx\int_{a}^{c} f(x) d x∫ac​f(x)dx given ∫abf(x)dx=7\int_{a}^{b} f(x) d x=7∫ab​f(x)dx=7 and ∫bcf(x)dx=9\int_{b}^{c} f(x) d x=9∫bc​f(x)dx=9.

Problem 21611

Find the values of xxx for which the series ∑n=1∞n2xn\sum_{n=1}^{\infty} n^{2} x^{n}∑n=1∞​n2xn converges.

Problem 21612

Which option correctly states the Fundamental Theorem of Calculus for a continuous function f(x)f(x)f(x)?

Problem 21613

Given that fff is an even function and ∫−33f(x)dx=14\int_{-3}^{3} f(x) d x=14∫−33​f(x)dx=14, find: a. ∫03f(x)dx\int_{0}^{3} f(x) d x∫03​f(x)dx b. ∫−33xf(x)dx\int_{-3}^{3} x f(x) d x∫−33​xf(x)dx.

Problem 21614

Find the function fff from the limit of Riemann sums: lim⁡Δ→0∑k=1n(xk∗)7Δxk\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{7} \Delta x_{k}Δ→0lim​k=1∑n​(xk∗​)7Δxk​ on [8,16][8,16][8,16].

Problem 21615

Evaluate ∫acf(x)dx\int_{a}^{c} f(x) d x∫ac​f(x)dx given ∫abf(x)dx=7\int_{a}^{b} f(x) d x=7∫ab​f(x)dx=7 and ∫bcf(x)dx=9\int_{b}^{c} f(x) d x=9∫bc​f(x)dx=9.

Problem 21616

Find the radius of convergence for the series ∑n=1∞n!(x+2)nnn\sum_{n=1}^{\infty} \frac{n !(x+2)^{n}}{n^{n}}∑n=1∞​nnn!(x+2)n​.

Problem 21617

Graph the integrand, calculate Δx\Delta xΔx, grid points, left and right Riemann sums for ∫9(1x+2)dx;n=4\int^{9}\left(\frac{1}{x}+2\right) d x ; n=4∫9(x1​+2)dx;n=4.

Problem 21618

Set up R4R_{4}R4​ for f(x)=x2−3f(x)=x^{2}-3f(x)=x2−3 over the interval [6,14][6,14][6,14].

Problem 21619

Find the average value of f(x)=5xf(x)=\frac{5}{x}f(x)=x5​ on [3,3e][3, 3e][3,3e] and graph the function with the average value indicated.

Problem 21620

Calculate the average value of the function f(x)=2x2f(x)=2 x^{2}f(x)=2x2 over the interval [3,5][3,5][3,5].

Problem 21621

Find Δx\Delta xΔx for the Riemann sum L4L_{4}L4​ of f(x)=e(3x2)+ln⁡(x)f(x)=e^{(3 x^{2})}+\ln (x)f(x)=e(3x2)+ln(x) over [3,11][3,11][3,11].

Problem 21622

Find the average height of the arch modeled by y=615[1−(x290)2]y=615\left[1-\left(\frac{x}{290}\right)^{2}\right]y=615[1−(290x​)2] over its base.

Problem 21623

Evaluate the integral: ∫3x+2x2+1dx=□+C\int \frac{3 x+2}{x^{2}+1} dx = \square + C∫x2+13x+2​dx=□+C

Problem 21624

Invest \$18,864 at 5.7% interest, compounded continuously. Find the function, balances for 1, 2, 5, 10 years, and doubling time.

Problem 21625

Find the tangent line's parametric equations at t=1t=1t=1 for the particle with x(t)=4t2x(t)=4 t^{2}x(t)=4t2 and y(t)=4t3y(t)=4 t^{3}y(t)=4t3.

Problem 21626

Evaluate the integral: ∫3sin⁡3(x)cos⁡(x) dx=□+C\int 3 \sin^{3}(x) \cos(x) \, dx = \square + C∫3sin3(x)cos(x)dx=□+C

Problem 21627

A particle's position is x=cos⁡(t)x=\cos(t)x=cos(t) and y=sin⁡(t2)y=\sin\left(\frac{t}{2}\right)y=sin(2t​). Find when it first stops moving.

Problem 21628

Find the weight of a person who is 5 feet, 4 inches tall using the formula dWdh=0.0012 h2+0.01 h\frac{\mathrm{dW}}{\mathrm{dh}}=0.0012 \mathrm{~h}^{2}+0.01 \mathrm{~h}dhdW​=0.0012 h2+0.01 h and W(80)=216.8W(80)=216.8W(80)=216.8 pounds.

Problem 21629

Evaluate the integral ∫x2(x3−9)40dx\int x^{2}(x^{3}-9)^{40} dx∫x2(x3−9)40dx using the substitution u=x3−9u=x^{3}-9u=x3−9. Find the result in terms of xxx.

Problem 21630

Find the weight of a person who is 5 feet 4 inches tall using the formula dWdh=0.0012h2+0.01h\frac{d W}{d h}=0.0012 h^{2}+0.01 hdhdW​=0.0012h2+0.01h and W(80)=216.8W(80)=216.8W(80)=216.8.

Problem 21631

Problem 21632

Find the radius and interval of convergence for the series ∑k=0∞(2x−3)kk4+7\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{k^{4}+7}∑k=0∞​k4+7(2x−3)k​. Radius: R=□R=\squareR=□

Problem 21633

Find where the function f(x)=x4−50x2+625f(x)=x^{4}-50 x^{2}+625f(x)=x4−50x2+625 is increasing and decreasing, and identify local extreme values.

Problem 21634

Transform the integral ∫cos⁡(y)sin⁡(y)+1dy\int \frac{\cos (y)}{\sin (y)+1} d y∫sin(y)+1cos(y)​dy using u=□u=\squareu=□ and du=□dyd u=\square d ydu=□dy to get ∫du\int d u∫du.

Problem 21635

Find the indefinite integral ∫(ln⁡(z))71zdz\int(\ln (z))^{7} \frac{1}{z} d z∫(ln(z))7z1​dz using the substitution u=□u=\squareu=□ and du=□dzd u=\square d zdu=□dz.

Problem 21636

Find the radius and interval of convergence for the series ∑k=1∞x2k+115k−1\sum_{k=1}^{\infty} \frac{x^{2 k+1}}{15^{k-1}}k=1∑∞​15k−1x2k+1​. What is R=R=R=?

Problem 21637

Find the indefinite integral ∫x2x3−44dx\int x^{2} \sqrt[4]{x^{3}-4} d x∫x24x3−4​dx using the substitution u=□u=\squareu=□ and du=□dxd u=\square d xdu=□dx.

Problem 21638

Find intervals where the function g(x)=x50−x2g(x)=x \sqrt{50-x^{2}}g(x)=x50−x2​ is increasing or decreasing, and identify local extremes.

Problem 21639

Find local max and min of fff using First and Second Derivative Tests for f(x)=x+1−xf(x)=x+\sqrt{1-x}f(x)=x+1−x​ and f(x)=xx2+4f(x)=\frac{x}{x^{2}+4}f(x)=x2+4x​. Which method do you prefer?

Problem 21640

Find the radius of convergence for the series ∑k=1∞(3k)!xk(k!)3\sum_{k=1}^{\infty} \frac{(3 k) ! x^{k}}{(k !)^{3}}∑k=1∞​(k!)3(3k)!xk​. What is RRR?

Find the integral of the function: $\int 8 e^{4 x} d x$

Find $f(x)$ given $f^{\prime}(x)=\frac{3}{x^{5}}$ and $f\left(\frac{1}{2}\right)=1$ .

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2x+2$ , where $h \neq 0$ .

Function $f(x)=9-x^{2}$ is positive/negative on $[0,4]$ . Sketch it, then find net area using Riemann sums with $n=4$ .

Evaluate the integral $\int_{-1.32}^{-1.32} 1957 e^{x^{4}} \, dx$ .

Find a suitable substitution $u$ for the integral $\int x^{2} e^{x^{3}} d x$ . Is $u=x^{2}$ a good choice?

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} \cdot n}(x-1)^{n}$ converges.

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{a} \cdot x}(x-1)^{n}$ converges.

Find $\int_{1}^{4} f(x) \, dx$ given that $F(1) = 19$ and $F(4) = 13$ .

Evaluate $\int_{a}^{c} f(x) d x$ given $\int_{a}^{b} f(x) d x=7$ and $\int_{b}^{c} f(x) d x=9$ .

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} n^{2} x^{n}$ converges.

Which option correctly states the Fundamental Theorem of Calculus for a continuous function $f(x)$ ?

Given that $f$ is an even function and $\int_{-3}^{3} f(x) d x=14$ , find: a. $\int_{0}^{3} f(x) d x$ b. $\int_{-3}^{3} x f(x) d x$ .

Find the function $f$ from the limit of Riemann sums: $\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(x_{k}^{*}\right)^{7} \Delta x_{k}$ on $[8,16]$ .

Evaluate $\int_{a}^{c} f(x) d x$ given $\int_{a}^{b} f(x) d x=7$ and $\int_{b}^{c} f(x) d x=9$ .

Find the radius of convergence for the series $\sum_{n=1}^{\infty} \frac{n !(x+2)^{n}}{n^{n}}$ .

Graph the integrand, calculate $\Delta x$ , grid points, left and right Riemann sums for $\int^{9}\left(\frac{1}{x}+2\right) d x ; n=4$ .

Set up $R_{4}$ for $f(x)=x^{2}-3$ over the interval $[6,14]$ .

Find the average value of $f(x)=\frac{5}{x}$ on $[3, 3e]$ and graph the function with the average value indicated.

Calculate the average value of the function $f(x)=2 x^{2}$ over the interval $[3,5]$ .

Find $\Delta x$ for the Riemann sum $L_{4}$ of $f(x)=e^{(3 x^{2})}+\ln (x)$ over $[3,11]$ .

Find the average height of the arch modeled by $y=615\left[1-\left(\frac{x}{290}\right)^{2}\right]$ over its base.

Evaluate the integral: $\int \frac{3 x+2}{x^{2}+1} dx = \square + C$

Find the tangent line's parametric equations at $t=1$ for the particle with $x(t)=4 t^{2}$ and $y(t)=4 t^{3}$ .

Evaluate the integral: $\int 3 \sin^{3}(x) \cos(x) \, dx = \square + C$

A particle's position is $x=\cos(t)$ and $y=\sin\left(\frac{t}{2}\right)$ . Find when it first stops moving.

Find the weight of a person who is 5 feet, 4 inches tall using the formula $\frac{\mathrm{dW}}{\mathrm{dh}}=0.0012 \mathrm{~h}^{2}+0.01 \mathrm{~h}$ and $W(80)=216.8$ pounds.

Evaluate the integral $\int x^{2}(x^{3}-9)^{40} dx$ using the substitution $u=x^{3}-9$ . Find the result in terms of $x$ .

Find the weight of a person who is 5 feet 4 inches tall using the formula $\frac{d W}{d h}=0.0012 h^{2}+0.01 h$ and $W(80)=216.8$ .

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty} \frac{(2 x-3)^{k}}{k^{4}+7}$ . Radius: $R=\square$

Find where the function $f(x)=x^{4}-50 x^{2}+625$ is increasing and decreasing, and identify local extreme values.

Transform the integral $\int \frac{\cos (y)}{\sin (y)+1} d y$ using $u=\square$ and $d u=\square d y$ to get $\int d u$ .

Find the indefinite integral $\int(\ln (z))^{7} \frac{1}{z} d z$ using the substitution $u=\square$ and $d u=\square d z$ .

Find the radius and interval of convergence for the series $\sum_{k=1}^{\infty} \frac{x^{2 k+1}}{15^{k-1}}$ . What is $R=$ ?

Find the indefinite integral $\int x^{2} \sqrt[4]{x^{3}-4} d x$ using the substitution $u=\square$ and $d u=\square d x$ .

Find intervals where the function $g(x)=x \sqrt{50-x^{2}}$ is increasing or decreasing, and identify local extremes.

Find local max and min of $f$ using First and Second Derivative Tests for $f(x)=x+\sqrt{1-x}$ and $f(x)=\frac{x}{x^{2}+4}$ . Which method do you prefer?

Find the radius of convergence for the series $\sum_{k=1}^{\infty} \frac{(3 k) ! x^{k}}{(k !)^{3}}$ . What is $R$ ?

Find the inflection points, local maxima/minima, and intervals of differentiability for $y=-x+\sin 2x$ on $[-\frac{5\pi}{6}, \frac{5\pi}{6}]$ .

Determine the radius of convergence for $\sum_{k=1}^{\infty} \frac{k ! x^{5 k}}{(6 k)^{k}}$ in terms of $e$ .

Find the indefinite integral $\int(\ln (z))^{6} \frac{1}{z} d z$ using the substitution $u=\ln(z)$ .

Invest \ $17,180 at 6.1% interest, compounded continuously. Find the function$ P(t)$ and balances for 1, 2, 5, 10 years, plus doubling time.

Graph the function $y=x^{2}-2x-3$ : find domain, symmetries, derivatives, critical points, inflection points, and extremes. Domain: $\square$ .

Find the first four nonzero terms of the Taylor series for $f(x)=\frac{5}{x^{4}}$ at $a=1$ , and the interval of convergence.

Find the second derivative of $y=f(x)$ given $y^{\prime}=4 \sin x$ for $-\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}$ .

Find the derivative of $y$ with respect to $x$ using logarithmic differentiation for $y=\sqrt{\frac{(x+1)^{8}}{(3 x-5)^{9}}}$ . $\frac{d y}{d x}=\square$

Calculate the slope of the function $f(x)=x^{2}+2$ at the point $(-1,3)$ .

Bestimmen Sie die positive Zahl $z$ aus den Integralen: a) $\int_{0}^{2} x \, dx = 18$ und b) $\int_{1}^{2} 4x \, dx = 30$ .

Find the second derivative $y^{\prime \prime}$ of $y=f(x)$ where $y^{\prime}=(10 x^{2}-20 x)(x-5)^{2}$ and sketch the graph of $f$ . $y^{\prime \prime}=\square$

Find the second derivative $y^{\prime \prime}$ if $y^{\prime}=-\pi \sec ^{2} x$ , then sketch $f$ using critical points.

Find the limit: $\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}$ using algebra.

Find the second derivative $y^{\prime \prime}$ of $y=f(x)$ given $y^{\prime}=x(x-9)^{2}$ and sketch the graph of $\mathrm{f}$ . $y^{\prime \prime}=\square$

Finde die Extrem- und Wendepunkte der Funktionen und bestimme die Art der Extrema. a) $f(x)=x^{5}-4 x^{2}$ b) $f(x)=\frac{1}{3} x^{3}+x^{2}+4 x$ c) $f(x)=x^{3}-x-1$

Find $\lim _{x \rightarrow 4} f(x)$ if $\lim _{x \rightarrow 4} g(x)=2$ and $\lim _{x \rightarrow 4} \frac{f(x)}{g(x)}=\pi$ .

Find $\lim_{x \rightarrow 1} f(x)$ for $f(x)=\frac{\frac{1}{x}-1}{x-1}$ and compare with options (4), (i1), (C), (D).

Calculate the following integrals: a) $\int_{0}^{2}(2+x)^{3} dx$ , b) $\int_{2}^{3}\left(1+\frac{1}{x^{2}}\right) dx$ , c) $\int_{0}^{2} \frac{(7}{(1}$ .

Given the function $f(x)=\frac{2 x}{x^{2}-49}$ , find critical numbers, intervals of decrease, local max/min, inflection points, concavity, and asymptotes.

A substance grows at $12\%$ daily. If it starts at 614 grams, find its mass after 4 days. Round to the nearest tenth.

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{1}{n 3^{n}}$ .

Finde die $x$ -Werte der Funktion $f(x)=-x^{4}+3 x^{3}$ mit waagerechter Tangente unter Verwendung des Vorzeichenwechselkriteriums.

Berechne die Stammfunktion von $f(x)=0,1\left(x^{2}-1\right)(x-3)(x+4)$ .

Bestimme die Extrempunkte und den Wendepunkt der Funktion $f(x)=-\frac{1}{4} x^{3}+\frac{3}{2} x^{2}$ und zeige, dass die Gerade durch die Extrempunkte den Wendepunkt schneidet.

Find the limit: $\lim _{x \rightarrow 0+} x^{x}$ .

Bestimmen Sie die Intervalle für strenges Wachstum/Fall von $f(x)=x^{3}-\frac{9}{4} x^{2}-3 x$ , die Extrempunkte und deren Art. Berechnen Sie Wendepunkte und Nullstellen.

Gegeben die Funktion $f(x)=\frac{1}{4} x^{3}-2 x$ (a) Finde die Nullstellen von $f$ . (b) Zeige $f^{\prime}(-2)=1$ und berechne die Tangente an $x=-2$ .

Find the limit as $x$ approaches 1 for the expression $\frac{x^{2}-\sqrt{x}}{\sqrt{x}-1}$ .