Calculus

Problem 19501

Find $g^{\prime}(x)$ if $g(x)=\int_{x}^{7} \cos(t^{7}) dt$ . Options: $-\cos(x^{7})$ , $-\sin(x^{7})$ , etc.

See Solution

Problem 19502

Find the value of the integral $\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x$ given $g(0)=0$ and $g(5)=1$ .

See Solution

Problem 19503

Find the derivative of $\int_{\sin x}^{\cos x} t^{2} dt$ . Choices: $\sin^2 x - \cos^2 x$ , $-\cos x \sin^2 x - \sin x \cos^2 x$ , $\cos x \sin^2 x + \sin x \cos^2 x$ , or $\cos^2 x - \sin^2 x$ .

See Solution

Problem 19504

Find the value of the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+9} d t}{x}$ . Choices: -3, 3, $\infty$ , $-\infty$ , 0.

See Solution

Problem 19505

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x)=\{2x^4, -\pi \leq x<0; 5\sin(x), 0 \leq x \leq \pi\}$ .

See Solution

Problem 19506

Find $F'(2)$ for $F(x)=\int_{3x}^{x^3} f(t) dt$ given $f(2)=2, f(4)=-1, f(6)=6, f(8)=4, f(10)=-3$ .

See Solution

Problem 19507

Find $g^{\prime}(2)$ for the function $g(x)=\int_{x^{2}}^{4} 3 t d t$ .

See Solution

Problem 19508

Find the derivative of the integral $\int_{\sin x}^{\cos x} t^{2} dt$ . What is it equal to?

See Solution

Problem 19509

Find the integral of $x(1-x)^{99}$ with respect to $x$ .

See Solution

Problem 19510

Calculate the integral $\int\left(x^{2}+3 x\right)^{5}(2 x+3) d x$ .

See Solution

Problem 19511

Vérifiez si la dérivée seconde de $f(x) = \frac{5x^{2}-36}{x^{2}+12}$ est $\frac{596(2-x)(2+x)}{(x^{2}+12)^{2}}$ .

See Solution

Problem 19512

Find the max and min of $f(x)=x-\frac{8 x}{x+2}$ on $[0,3]$ . Min value = ?, Max value = 0.

See Solution

Problem 19513

Find the volume of the solid formed by revolving the area between $y=3x$ and $y=12\sqrt{x}$ around the $x$ -axis.

See Solution

Problem 19514

Find the volume of the solid formed by revolving the area between $y=3 \sqrt{2 x}$ and $y=8-x$ around the $x$ -axis.

See Solution

Problem 19515

Find the integral of the function: $\int \sqrt{3 x-5} \, dx$ .

See Solution

Problem 19516

Calculate the volume of the solid formed by rotating the area in the first quadrant between $y^{2}=8x$ and $x=2$ around the x-axis.

See Solution

Problem 19517

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

See Solution

Problem 19518

Find $f(x)$ where $\frac{d y}{d x}=80 y x^{15}$ and the $y$ -intercept is 5.

See Solution

Problem 19519

Given $g(x)=f\left(x^{2}-x\right)$ , find $g^{\prime}(3)$ and $f^{\prime \prime}(0)$ if $g^{\prime \prime}(0)=-1$ .

See Solution

Problem 19520

Find the population $P(t)=\frac{700 t}{3 t^{2}+6}$ at $t=0,1,3,8$ . Determine the horizontal asymptote as $t \to \infty$ and its significance.

See Solution

Problem 19521

Use the Limit Comparison Test for convergence/divergence of $\sum_{n=1}^{\infty} \frac{5 n^{2}+n+1}{n^{4}+4 n^{2}-6}$ . Choose $b_{n}$ and find limit $L$ .

See Solution

Problem 19522

Compute $\frac{d}{d x}\left[\int_{1}^{x^{4}} \sin \left(t^{3}\right) d t\right]$ using the Fundamental Theorem.
a) Write $\int_{1}^{x^{4}} \sin \left(t^{3}\right) d t$ as $G(x^{4}) - G(1)$ .
b) Find $\frac{d}{d x}\left[\int_{1}^{x^{4}} \sin \left(t^{3}\right) d t\right]$ .

See Solution

Problem 19523

Given $f$ and $f^{\prime}$ values, find $c$ in $0<c<3$ where $g(c)=5$ and locate $x$ in $[-3, 0]$ where $h^{\prime}(x)=\frac{f(6)-f(-3)}{6-(-3)}$ .

See Solution

Problem 19524

Given $f(x)=\frac{x^{2}}{x^{2}-1}$ , find (a) where $f$ is increasing/decreasing, (b) local max/min $x$ -values, (c) concavity and inflection points.

See Solution

Problem 19525

Estimate the area under $f(x)=1-x^{2}$ using an upper sum with 4 equal-width rectangles over $[a, b]$ .

See Solution

Problem 19526

Find horizontal and vertical asymptotes of $g(x)=|3 x| \cdot \tan ^{-1}\left(\frac{1}{x}\right)$ . Show your work.

See Solution

Problem 19527

Find the max and min of $f(x)=x-\frac{8 x}{x+2}$ on $[0,3]$ . Min value = ?, Max value = 0.

See Solution

Problem 19528

Estabon's drink cools from $204^{\circ} \mathrm{F}$ to $184^{\circ} \mathrm{F}$ in 2 min. How long to reach $99^{\circ} \mathrm{F}$ ?

See Solution

Problem 19529

Find the initial temperature of the soda and its temperature after 18 minutes using $T(x)=-6+24 e^{-0.04 x}$ .

See Solution

Problem 19530

Approximate $\int_{0}^{2} 5 x^{4} d x$ using Simpson's Rule with $n=4$ .

See Solution

Problem 19531

A cup of coffee cools from $205^{\circ} \mathrm{F}$ to $195^{\circ} \mathrm{F}$ in 10 mins. When will it reach $180^{\circ} \mathrm{F}$ ? Use Newton's law of cooling: $T(t)=T_{A}+\left(T_{0}-T_{A}\right) e^{-k t}$ . A. 27 min B. 45 min C. 15 min D. 1 hour

See Solution

Problem 19532

Find the consumer's surplus for the demand function $p=34-x^{2}$ at an equilibrium price of \$9 per unit.

See Solution

Problem 19533

Find the time in hours for a bacteria population to double with a growth rate of $4.6\%$ per hour. Round to the nearest hundredth.

See Solution

Problem 19534

Find the national health expenditures in billions of dollars at the start of 1981, given it was \$500 billion.

See Solution

Problem 19535

Cost function $C(x)=4 x^{2}+12 x+24$ , find marginal cost at $x=5$ , estimate cost for 5.50 units, and break-even point with $R(x)=-x^{2}+29 x+36$ .

See Solution

Problem 19536

Find the time to double \$2700 at a continuous interest rate of 1.5\%. Round to the nearest hundredth.

See Solution

Problem 19537

Find the half-life of a substance decaying at $6.2\%$ per day using the continuous exponential decay model.

See Solution

Problem 19538

Find $\frac{d y}{d x}$ for $y=\int_{a}^{x}(t^{2}-2t) dt$ using the fundamental theorem of calculus.

See Solution

Problem 19539

Find the hourly growth rate of a bacteria population that grows from 3000 to 3121 in 1.5 hours. Express as a percentage.

See Solution

Problem 19540

What does a derivative value of approximately 0 for national health expenditures in 1990 imply about spending that year?

See Solution

Problem 19541

Find the value of $y$ for the integral $y=\int_{x}^{3} \cos t \, dt$ .

See Solution

Problem 19542

Find the value of $y$ for the integral $y=\int_{1}^{\sin x} 3 t^{2} d t$ .

See Solution

Problem 19543

Evaluate the integral from 0 to 3 of the function $x^{2}+4x-1$ .

See Solution

Problem 19544

Find the max and min values of $f$ on the intervals:
10. $f(x)=x^{3}-3 x+1,[0,3]$
11. $f(x)=4 x^{3}+x^{2}-4 x+1,[-1,2]$
12. $f(x)=\sin x+\cos x,\left[0, \frac{\pi}{3}\right]$

See Solution

Problem 19545

Find the maximum acceleration in the interval $0 \leq t \leq 3$ for the particle with velocity $v(t)=t^{3}-3 t^{2}+12 t+4$ .

See Solution

Problem 19546

Evaluate the integral $\int \frac{\sin ^{-1} t \, dt}{\sqrt{1-t^{2}}}$ .

See Solution

Problem 19547

Evaluate the integral: $\int \frac{1+\sin x}{\cos ^{2} x} d x$ .

See Solution

Problem 19548

Find the Maclaurin series for $f(x)=e^{x}+e^{2x}$ .

See Solution

Problem 19549

Trouver le minimum de $f(x)=\frac{x^{3}}{3}-4 x^{2}+12 x+\frac{1}{3}$ sur l'intervalle $[-1,8]$ .

See Solution

Problem 19550

Find the limit: $\lim _{x \rightarrow 0} \frac{x-\ln (1+x)}{x^{2}}$ .

See Solution

Problem 19551

Calculate the area under the curve $y=\log(1+x)$ . Options: (A) $\log(62)$ , (B) $\log(64)-2$ , (C) $\log(32)+2$ , (D) $\log(64)+2$ .

See Solution

Problem 19552

Find the Taylor series of $f(x) = e^x$ around the point $a = 3$ .

See Solution

Problem 19553

Find the Taylor series for $f(x)=e^{x}$ centered at $a=3$ . Do not prove that $R_n(x)$ is 0.

See Solution

Problem 19554

Find the Maclaurin series for $f(x)=\sin(\pi x)$ .

See Solution

Problem 19555

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x - x + \frac{1}{6} x^{3}}{x^{5}}$ .

See Solution

Problem 19556

Juno invests \$ 85,000 at a 4% annual interest rate, compounded continuously. Find the account value in 12 years.

See Solution

Problem 19557

Find the area between the curves $y=x^{2}$ and $y=2-x^{2}$ .

See Solution

Problem 19558

Evaluate the integral: $\int \frac{e^{x}-1}{x} d x$

See Solution

Problem 19559

Bestimme die Grenzwerte: a) $\lim _{x \rightarrow \infty} \frac{x^{2}+1}{2 x^{2}}$ , b) $\lim _{x \rightarrow-\infty} \frac{x^{2}+2 x+1}{4 x^{2}}$ , c) $\lim _{x \rightarrow \infty} \frac{x}{5 x}$ . Verwende Termumformung und die h-Methode.

See Solution

Problem 19560

1. Calculate the area between the line $y=2x+3$ , the x-axis, and the vertical lines $x=-1$ and $x=1$ .
2. Show that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ .

See Solution

Problem 19561

A baseball diamond is a square (sides $90 \, \text{ft}$ ). A player runs to second base at $18 \, \text{ft/sec}$ .
a. Find the rate of distance from third base when $45 \, \text{ft}$ from first base.
b. Determine the rates of angles $\theta_{1}$ and $\theta_{2}$ changing at that moment.
c. When sliding into second base at $15 \, \text{ft/sec}$ , find the rates of angles $\theta_{1}$ and $\theta_{2}$ .

See Solution

Problem 19562

Find the area under the curve $y=3x^{2}$ from $x=1$ to $x=2$ .

See Solution

Problem 19563

Calculate the area under the curve $y=3 x^{2}$ from $x=1$ to $x=2$ .

See Solution

Problem 19564

Berechnen Sie die Ableitung der Funktion $f(x)=2 x^{2}+3$ an den Stellen: a) -2, b) $-1,25$ , c) 2,5.

See Solution

Problem 19565

Find the Taylor series for $f(x) = \cos x$ centered at $a = \pi$ .

See Solution

Problem 19566

Finde die Stellen $a$ und $b$ für $f(x)=x^{3}-1,5 x^{2}-20 x$ mit Steigung 40 und berechne $\{b-a ;-(a \cdot b+1) ; b^{2}+a-b^{0}\}$ .

See Solution

Problem 19567

Bestimme die Extremstellen der Funktion $f(x)=x^{3}-21 x^{2}+135 x+19$ und finde ein dreibuchstabiges Wort basierend auf den Hinweisen.

See Solution

Problem 19568

A particle moves on $y=x^{2}$ with $dx/dt=4 \mathrm{~m/s}$ . Find the rate of change of angle $\theta$ at $x=1 \mathrm{~m}$ .

See Solution

Problem 19569

A rectangle's length $\ell$ increases at $2 \mathrm{~cm/sec}$ and width $w$ decreases at $2 \mathrm{~cm/sec}$ . At $\ell=24 \mathrm{~cm}$ and $w=7 \mathrm{~cm}$ , find the rates of change for area, perimeter, and diagonals. Identify which are increasing, decreasing, or constant.

See Solution

Problem 19570

Kosten-Gewinn-Rechnung:
Gegeben ist die Kostenfunktion $K(x)=0,01 x^{3}-x^{2}+50 x+720$ .
a) Bestimme die Gewinnfunktion $G(x)$ für den Verkaufspreis von $53€$ .
b) Berechne den Gewinn für bestimmte Stückzahlen und skizziere $G$ .
c) Benenne markante Punkte und deren Bedeutung.
d) Finde die Nullstellen von $G$ und das Gewinnintervall.
e) Bestimme die erste Ableitung von $G$ und berechne den maximalen Gewinn.
f) Finde die Stückzahlen für ca. $500€$ Gewinn.
g) Erkläre die Bedeutung des Hochpunkts bei $x=33 \frac{1}{3}$ .

See Solution

Problem 19571

Find the body's velocity, speed, acceleration, and jerk at $t=\frac{\pi}{4}$ sec for $s=3 \sin t + 12 \cos t$ .

See Solution

Problem 19572

Finde die Bedingungen für Extrema der Funktion $f_{a}(x)=a x^{4}-4 x^{3}+a^{2} x$ durch Berechnung der ersten und zweiten Ableitung.

See Solution

Problem 19573

Calculate the area between the curve $y=2 x+x^{2}-x^{3}$ , the $x$ -axis, and lines $x=-1$ and $x=1$ .

See Solution

Problem 19574

Une compagnie achète un appareil à 2500\ $. Trouvez le profit maximal avec$ R $et$ C $donnés par$ \frac{d R}{d t}=100(18-3 \sqrt{t}) $et$ \frac{d C}{d t}=100(2+\sqrt{t})$.

See Solution

Problem 19575

Find the dimensions and area of a rectangle on the $x$ -axis with vertices on the parabola $y=64-x^{2}$ that maximizes area.

See Solution

Problem 19576

Find the hourly growth rate for a bacteria population that grows from 1000 to 1124 in 2.5 hours.

See Solution

Problem 19577

Find the derivative $f^{\prime}(x)$ and the critical numbers of $f(x)=x^{3}-3x-1$ .

See Solution

Problem 19578

Determine the convergence or divergence of the series $a_{n}=\frac{n^{n^{2}}}{(n+1)^{n^{2}}}$ .

See Solution

Problem 19579

Find the area between the curves $y=2-x^{2}$ and $y=-x$ using the integral $A=\int_{a}^{b}[f(x)-g(x)] dx$ .

See Solution

Problem 19580

Find the Taylor Series for $\int x^{2} e^{-x^{2}} dx$ using the Taylor Series of $e^{x}$ and state the radius of convergence.

See Solution

Problem 19581

Find the mass of a radioactive substance after 4 days, starting at 3542 kg and decaying at 13% per day. Round to the nearest tenth. $\square \mathrm{kg}$

See Solution

Problem 19582

Find the Taylor series of the function $f(x) = x^{5}-2 x^{4}+x^{3}-x^{2}+2 x-1$ at the point $x_0=1$ .

See Solution

Problem 19583

Calculate the future value of an investment of \ $6,200 at a 9% continuous compounding rate over 10 years using$ A = Pe^{rt}$.

See Solution

Problem 19584

Determine convergence type (absolute, conditional, or divergent) of the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{n+2}{n(n+1)}$ .

See Solution

Problem 19585

Calculate the sum $\sum_{k=1}^{\infty} \frac{4^{k-1}+3^{k-1}-1}{12^{k-1}}$ .

See Solution

Problem 19586

Find the limit of $f(x) = 2 \sqrt{x+1} + x + 1$ as $x \to \infty$ .

See Solution

Problem 19587

Find the rate of change of revenue when selling widgets at \ $270, given$ x=\frac{90000}{\sqrt{3p+1}}$.

See Solution

Problem 19588

Is it true that $\int_{11}^{30} f(x) dx = -\int_{30}^{11} f(x) dx$ for $f$ on $(11,30)$ ? a) True b) False

See Solution

Problem 19589

Find the value of $\int_{0}^{4} f(x) dx$ where $f(x)=x-1$ for $0 \leq x<2$ and $f(x)=1-x$ for $2 \leq x \leq 4$ . Options: a) -4 b) 31 c) 12 d) 242 e) None.

See Solution

Problem 19590

Find the derivative: $\frac{d}{d x} \int_{\frac{\pi}{2}}^{x^{2}} \sin (u) d u$ . Choose the correct option.

See Solution

Problem 19591

Calculate the total energy requirement for a female Collie between her third and fifth birthday using $E(t)=5 t^{-0.2}$ . Options: a) 1022.616 b) 540.648 c) 494.790 d) 852.180

See Solution

Problem 19592

Find the total reaction from hour 2 to hour 3 for $R'(t)=\frac{6}{t}+\frac{4}{t^{3}}+e^{t}$ . Round up your answer. Choices: a) 15 b) 16 c) 18 d) 19

See Solution

Problem 19593

Check if the Mean Value Theorem applies to $f(x)=9 \ln x-10$ on $[1,14]$ . If yes, find $c$ in $[1,14]$ . DNE if not.

See Solution

Problem 19594

Calculate the limit: $\lim _{x \rightarrow+\infty} 2 \sqrt{x+1}+x+1$ .

See Solution

Problem 19595

Find the inflection points of $f(x)=8 x^{4}+110 x^{3}-42 x^{2}+11$ . Answer as a list, e.g., 3,-2.

See Solution

Problem 19596

Find the inflection points of $f(x)=8 x^{4}+110 x^{3}-42 x^{2}+11$ . List them as $x_1,x_2$ .

See Solution

Problem 19597

Solve for $y(x)$ given $\frac{d y}{d x}=\frac{x}{x^{2}-5}$ and $y(0)=\ln(5)$ .

See Solution

Problem 19598

Is it true that $\int_{5}^{8} f(x) d x = \int_{5}^{25} f(x) d x - \int_{8}^{25} f(x) d x$ ? a) True b) False

See Solution

Problem 19599

Which symbol represents "left" in the context of limits: $x \rightarrow \infty$ or $x \rightarrow -\infty$ ?

See Solution

Problem 19600

Find $dy/dx$ using logarithmic differentiation for $y=(9+x)^{4/x}$ .

See Solution

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Calculus

Problem 19501

Find g′(x)g^{\prime}(x)g′(x) if g(x)=∫x7cos⁡(t7)dtg(x)=\int_{x}^{7} \cos(t^{7}) dtg(x)=∫x7​cos(t7)dt. Options: −cos⁡(x7)-\cos(x^{7})−cos(x7), −sin⁡(x7)-\sin(x^{7})−sin(x7), etc.

Problem 19502

Find the value of the integral ∫05g′(x)1+(g(x))2dx\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x∫05​1+(g(x))2g′(x)​dx given g(0)=0g(0)=0g(0)=0 and g(5)=1g(5)=1g(5)=1.

Problem 19503

Problem 19504

Find the value of the limit: lim⁡x→0∫x0t2+9dtx\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+9} d t}{x}x→0lim​x∫x0​t2+9​dt​. Choices: -3, 3, ∞\infty∞, −∞-\infty−∞, 0.

Problem 19505

Evaluate the integral ∫−ππf(x)dx\int_{-\pi}^{\pi} f(x) dx∫−ππ​f(x)dx where f(x)={2x4,−π≤x<0;5sin⁡(x),0≤x≤π}f(x)=\{2x^4, -\pi \leq x<0; 5\sin(x), 0 \leq x \leq \pi\}f(x)={2x4,−π≤x<0;5sin(x),0≤x≤π}.

Problem 19506

Find F′(2)F'(2)F′(2) for F(x)=∫3xx3f(t)dtF(x)=\int_{3x}^{x^3} f(t) dtF(x)=∫3xx3​f(t)dt given f(2)=2,f(4)=−1,f(6)=6,f(8)=4,f(10)=−3f(2)=2, f(4)=-1, f(6)=6, f(8)=4, f(10)=-3f(2)=2,f(4)=−1,f(6)=6,f(8)=4,f(10)=−3.

Problem 19507

Find g′(2)g^{\prime}(2)g′(2) for the function g(x)=∫x243tdtg(x)=\int_{x^{2}}^{4} 3 t d tg(x)=∫x24​3tdt.

Problem 19508

Find the derivative of the integral ∫sin⁡xcos⁡xt2dt\int_{\sin x}^{\cos x} t^{2} dt∫sinxcosx​t2dt. What is it equal to?

Problem 19509

Find the integral of x(1−x)99x(1-x)^{99}x(1−x)99 with respect to xxx.

Problem 19510

Calculate the integral ∫(x2+3x)5(2x+3)dx\int\left(x^{2}+3 x\right)^{5}(2 x+3) d x∫(x2+3x)5(2x+3)dx.

Problem 19511

Vérifiez si la dérivée seconde de f(x)=5x2−36x2+12f(x) = \frac{5x^{2}-36}{x^{2}+12}f(x)=x2+125x2−36​ est 596(2−x)(2+x)(x2+12)2\frac{596(2-x)(2+x)}{(x^{2}+12)^{2}}(x2+12)2596(2−x)(2+x)​.

Problem 19512

Find the max and min of f(x)=x−8xx+2f(x)=x-\frac{8 x}{x+2}f(x)=x−x+28x​ on [0,3][0,3][0,3]. Min value = ?, Max value = 0.

Problem 19513

Find the volume of the solid formed by revolving the area between y=3xy=3xy=3x and y=12xy=12\sqrt{x}y=12x​ around the xxx-axis.

Problem 19514

Find the volume of the solid formed by revolving the area between y=32xy=3 \sqrt{2 x}y=32x​ and y=8−xy=8-xy=8−x around the xxx-axis.

Problem 19515

Find the integral of the function: ∫3x−5 dx\int \sqrt{3 x-5} \, dx∫3x−5​dx.

Problem 19516

Calculate the volume of the solid formed by rotating the area in the first quadrant between y2=8xy^{2}=8xy2=8x and x=2x=2x=2 around the x-axis.

Problem 19517

Find the volume of the solid formed by rotating the area between y=−x2+2xy=-x^{2}+2xy=−x2+2x and the xxx-axis around the xxx-axis.

Problem 19518

Find f(x)f(x)f(x) where dydx=80yx15\frac{d y}{d x}=80 y x^{15}dxdy​=80yx15 and the yyy-intercept is 5.

Problem 19519

Given g(x)=f(x2−x)g(x)=f\left(x^{2}-x\right)g(x)=f(x2−x), find g′(3)g^{\prime}(3)g′(3) and f′′(0)f^{\prime \prime}(0)f′′(0) if g′′(0)=−1g^{\prime \prime}(0)=-1g′′(0)=−1.

Problem 19520

Find the population P(t)=700t3t2+6P(t)=\frac{700 t}{3 t^{2}+6}P(t)=3t2+6700t​ at t=0,1,3,8t=0,1,3,8t=0,1,3,8. Determine the horizontal asymptote as t→∞t \to \inftyt→∞ and its significance.

Problem 19521

Use the Limit Comparison Test for convergence/divergence of ∑n=1∞5n2+n+1n4+4n2−6\sum_{n=1}^{\infty} \frac{5 n^{2}+n+1}{n^{4}+4 n^{2}-6}n=1∑∞​n4+4n2−65n2+n+1​. Choose bnb_{n}bn​ and find limit LLL.

Problem 19522

Problem 19523

Given fff and f′f^{\prime}f′ values, find ccc in 0<c<30<c<30<c<3 where g(c)=5g(c)=5g(c)=5 and locate xxx in [−3,0][-3, 0][−3,0] where h′(x)=f(6)−f(−3)6−(−3)h^{\prime}(x)=\frac{f(6)-f(-3)}{6-(-3)}h′(x)=6−(−3)f(6)−f(−3)​.

Problem 19524

Given f(x)=x2x2−1f(x)=\frac{x^{2}}{x^{2}-1}f(x)=x2−1x2​, find (a) where fff is increasing/decreasing, (b) local max/min xxx-values, (c) concavity and inflection points.

Problem 19525

Estimate the area under f(x)=1−x2f(x)=1-x^{2}f(x)=1−x2 using an upper sum with 4 equal-width rectangles over [a,b][a, b][a,b].

Problem 19526

Find horizontal and vertical asymptotes of g(x)=∣3x∣⋅tan⁡−1(1x)g(x)=|3 x| \cdot \tan ^{-1}\left(\frac{1}{x}\right)g(x)=∣3x∣⋅tan−1(x1​). Show your work.

Problem 19527

Find the max and min of f(x)=x−8xx+2f(x)=x-\frac{8 x}{x+2}f(x)=x−x+28x​ on [0,3][0,3][0,3]. Min value = ?, Max value = 0.

Problem 19528

Estabon's drink cools from 204∘F204^{\circ} \mathrm{F}204∘F to 184∘F184^{\circ} \mathrm{F}184∘F in 2 min. How long to reach 99∘F99^{\circ} \mathrm{F}99∘F?

Problem 19529

Find the initial temperature of the soda and its temperature after 18 minutes using T(x)=−6+24e−0.04xT(x)=-6+24 e^{-0.04 x}T(x)=−6+24e−0.04x.

Problem 19530

Approximate ∫025x4dx\int_{0}^{2} 5 x^{4} d x∫02​5x4dx using Simpson's Rule with n=4n=4n=4.

Problem 19531

Problem 19532

Find the consumer's surplus for the demand function p=34−x2p=34-x^{2}p=34−x2 at an equilibrium price of \$9 per unit.

Problem 19533

Find the time in hours for a bacteria population to double with a growth rate of 4.6%4.6\%4.6% per hour. Round to the nearest hundredth.

Problem 19534

Find the national health expenditures in billions of dollars at the start of 1981, given it was \$500 billion.

Problem 19535

Cost function C(x)=4x2+12x+24C(x)=4 x^{2}+12 x+24C(x)=4x2+12x+24, find marginal cost at x=5x=5x=5, estimate cost for 5.50 units, and break-even point with R(x)=−x2+29x+36R(x)=-x^{2}+29 x+36R(x)=−x2+29x+36.

Problem 19536

Find the time to double \$2700 at a continuous interest rate of 1.5\%. Round to the nearest hundredth.

Problem 19537

Find the half-life of a substance decaying at 6.2%6.2\%6.2% per day using the continuous exponential decay model.

Problem 19538

Find dydx\frac{d y}{d x}dxdy​ for y=∫ax(t2−2t)dty=\int_{a}^{x}(t^{2}-2t) dty=∫ax​(t2−2t)dt using the fundamental theorem of calculus.

Problem 19539

Find the hourly growth rate of a bacteria population that grows from 3000 to 3121 in 1.5 hours. Express as a percentage.

Problem 19540

What does a derivative value of approximately 0 for national health expenditures in 1990 imply about spending that year?

Problem 19541

Find the value of yyy for the integral y=∫x3cos⁡t dty=\int_{x}^{3} \cos t \, dty=∫x3​costdt.

Find $g^{\prime}(x)$ if $g(x)=\int_{x}^{7} \cos(t^{7}) dt$ . Options: $-\cos(x^{7})$ , $-\sin(x^{7})$ , etc.

Find the value of the integral $\int_{0}^{5} \frac{g^{\prime}(x)}{1+(g(x))^{2}} d x$ given $g(0)=0$ and $g(5)=1$ .

Find the value of the limit: $\lim _{x \rightarrow 0} \frac{\int_{x}^{0} \sqrt{t^{2}+9} d t}{x}$ . Choices: -3, 3, $\infty$ , $-\infty$ , 0.

Evaluate the integral $\int_{-\pi}^{\pi} f(x) dx$ where $f(x)=\{2x^4, -\pi \leq x<0; 5\sin(x), 0 \leq x \leq \pi\}$ .

Find $F'(2)$ for $F(x)=\int_{3x}^{x^3} f(t) dt$ given $f(2)=2, f(4)=-1, f(6)=6, f(8)=4, f(10)=-3$ .

Find $g^{\prime}(2)$ for the function $g(x)=\int_{x^{2}}^{4} 3 t d t$ .

Find the derivative of the integral $\int_{\sin x}^{\cos x} t^{2} dt$ . What is it equal to?

Find the integral of $x(1-x)^{99}$ with respect to $x$ .

Calculate the integral $\int\left(x^{2}+3 x\right)^{5}(2 x+3) d x$ .

Vérifiez si la dérivée seconde de $f(x) = \frac{5x^{2}-36}{x^{2}+12}$ est $\frac{596(2-x)(2+x)}{(x^{2}+12)^{2}}$ .

Find the max and min of $f(x)=x-\frac{8 x}{x+2}$ on $[0,3]$ . Min value = ?, Max value = 0.

Find the volume of the solid formed by revolving the area between $y=3x$ and $y=12\sqrt{x}$ around the $x$ -axis.

Find the volume of the solid formed by revolving the area between $y=3 \sqrt{2 x}$ and $y=8-x$ around the $x$ -axis.

Find the integral of the function: $\int \sqrt{3 x-5} \, dx$ .

Calculate the volume of the solid formed by rotating the area in the first quadrant between $y^{2}=8x$ and $x=2$ around the x-axis.

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

Find $f(x)$ where $\frac{d y}{d x}=80 y x^{15}$ and the $y$ -intercept is 5.

Given $g(x)=f\left(x^{2}-x\right)$ , find $g^{\prime}(3)$ and $f^{\prime \prime}(0)$ if $g^{\prime \prime}(0)=-1$ .

Find the population $P(t)=\frac{700 t}{3 t^{2}+6}$ at $t=0,1,3,8$ . Determine the horizontal asymptote as $t \to \infty$ and its significance.

Use the Limit Comparison Test for convergence/divergence of $\sum_{n=1}^{\infty} \frac{5 n^{2}+n+1}{n^{4}+4 n^{2}-6}$ . Choose $b_{n}$ and find limit $L$ .

Given $f$ and $f^{\prime}$ values, find $c$ in $0<c<3$ where $g(c)=5$ and locate $x$ in $[-3, 0]$ where $h^{\prime}(x)=\frac{f(6)-f(-3)}{6-(-3)}$ .

Given $f(x)=\frac{x^{2}}{x^{2}-1}$ , find (a) where $f$ is increasing/decreasing, (b) local max/min $x$ -values, (c) concavity and inflection points.

Estimate the area under $f(x)=1-x^{2}$ using an upper sum with 4 equal-width rectangles over $[a, b]$ .

Find horizontal and vertical asymptotes of $g(x)=|3 x| \cdot \tan ^{-1}\left(\frac{1}{x}\right)$ . Show your work.

Find the max and min of $f(x)=x-\frac{8 x}{x+2}$ on $[0,3]$ . Min value = ?, Max value = 0.

Estabon's drink cools from $204^{\circ} \mathrm{F}$ to $184^{\circ} \mathrm{F}$ in 2 min. How long to reach $99^{\circ} \mathrm{F}$ ?

Find the initial temperature of the soda and its temperature after 18 minutes using $T(x)=-6+24 e^{-0.04 x}$ .

Approximate $\int_{0}^{2} 5 x^{4} d x$ using Simpson's Rule with $n=4$ .

Find the consumer's surplus for the demand function $p=34-x^{2}$ at an equilibrium price of \$9 per unit.

Find the time in hours for a bacteria population to double with a growth rate of $4.6\%$ per hour. Round to the nearest hundredth.

Cost function $C(x)=4 x^{2}+12 x+24$ , find marginal cost at $x=5$ , estimate cost for 5.50 units, and break-even point with $R(x)=-x^{2}+29 x+36$ .

Find the half-life of a substance decaying at $6.2\%$ per day using the continuous exponential decay model.

Find $\frac{d y}{d x}$ for $y=\int_{a}^{x}(t^{2}-2t) dt$ using the fundamental theorem of calculus.

Find the value of $y$ for the integral $y=\int_{x}^{3} \cos t \, dt$ .

Find the value of $y$ for the integral $y=\int_{1}^{\sin x} 3 t^{2} d t$ .

Evaluate the integral from 0 to 3 of the function $x^{2}+4x-1$ .

Find the maximum acceleration in the interval $0 \leq t \leq 3$ for the particle with velocity $v(t)=t^{3}-3 t^{2}+12 t+4$ .

Evaluate the integral $\int \frac{\sin ^{-1} t \, dt}{\sqrt{1-t^{2}}}$ .

Evaluate the integral: $\int \frac{1+\sin x}{\cos ^{2} x} d x$ .

Find the Maclaurin series for $f(x)=e^{x}+e^{2x}$ .

Trouver le minimum de $f(x)=\frac{x^{3}}{3}-4 x^{2}+12 x+\frac{1}{3}$ sur l'intervalle $[-1,8]$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{x-\ln (1+x)}{x^{2}}$ .

Calculate the area under the curve $y=\log(1+x)$ . Options: (A) $\log(62)$ , (B) $\log(64)-2$ , (C) $\log(32)+2$ , (D) $\log(64)+2$ .

Find the Taylor series of $f(x) = e^x$ around the point $a = 3$ .

Find the Taylor series for $f(x)=e^{x}$ centered at $a=3$ . Do not prove that $R_n(x)$ is 0.

Find the Maclaurin series for $f(x)=\sin(\pi x)$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin x - x + \frac{1}{6} x^{3}}{x^{5}}$ .

Find the area between the curves $y=x^{2}$ and $y=2-x^{2}$ .

Evaluate the integral: $\int \frac{e^{x}-1}{x} d x$

1. Calculate the area between the line $y=2x+3$ , the x-axis, and the vertical lines $x=-1$ and $x=1$ .
2. Show that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ .

Find the area under the curve $y=3x^{2}$ from $x=1$ to $x=2$ .

Calculate the area under the curve $y=3 x^{2}$ from $x=1$ to $x=2$ .

Berechnen Sie die Ableitung der Funktion $f(x)=2 x^{2}+3$ an den Stellen: a) -2, b) $-1,25$ , c) 2,5.

Find the Taylor series for $f(x) = \cos x$ centered at $a = \pi$ .

Finde die Stellen $a$ und $b$ für $f(x)=x^{3}-1,5 x^{2}-20 x$ mit Steigung 40 und berechne $\{b-a ;-(a \cdot b+1) ; b^{2}+a-b^{0}\}$ .

Bestimme die Extremstellen der Funktion $f(x)=x^{3}-21 x^{2}+135 x+19$ und finde ein dreibuchstabiges Wort basierend auf den Hinweisen.

A particle moves on $y=x^{2}$ with $dx/dt=4 \mathrm{~m/s}$ . Find the rate of change of angle $\theta$ at $x=1 \mathrm{~m}$ .

Find the body's velocity, speed, acceleration, and jerk at $t=\frac{\pi}{4}$ sec for $s=3 \sin t + 12 \cos t$ .

Finde die Bedingungen für Extrema der Funktion $f_{a}(x)=a x^{4}-4 x^{3}+a^{2} x$ durch Berechnung der ersten und zweiten Ableitung.

Calculate the area between the curve $y=2 x+x^{2}-x^{3}$ , the $x$ -axis, and lines $x=-1$ and $x=1$ .

Une compagnie achète un appareil à 2500\ $. Trouvez le profit maximal avec$ R $et$ C $donnés par$ \frac{d R}{d t}=100(18-3 \sqrt{t}) $et$ \frac{d C}{d t}=100(2+\sqrt{t})$.