Calculus

Problem 23601

Find the value of $\lim _{x \rightarrow 4^{-}} \frac{1}{(x-4)^{3}}$ : 0, 1, $\infty$ , or $-\infty$ ?

See Solution

Problem 23602

Find the derivative of $f(x)=4x^3+2x^2+1$ using $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ .

See Solution

Problem 23603

Find the remaining amount of a $669 \mathrm{mg}$ radioactive substance after 87 years using $A(t)=669 e^{-0.033 t}$ .

See Solution

Problem 23604

Find the value of $k$ in the integral $\int \frac{x^{4}+2}{x^{5}+10 x} d x=k \ln \left(x^{5}+10 x\right)+C$ .

See Solution

Problem 23605

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

See Solution

Problem 23606

Find the instantaneous rate of change of $f(x)=e^{x+1}$ at $x=-3$ using the limit below. No need to simplify.

See Solution

Problem 23607

Find the average rate of change of $f(x)=125(0.9)^{x}$ between years 11-15 and compare it to 1-5.

See Solution

Problem 23608

Untersuchen Sie die Form einer Satellitenschüssel (Querschnitt: $40 \mathrm{~cm}$ breit, $6,8 \mathrm{~cm}$ tief).
a) Schätzen Sie, wie viel Bier (ca. 5,6 Liter) passt hinein. b) Legen Sie den Querschnitt für die Rotation um die $x$ -Achse fest. c) Bestimmen Sie die Funktion $f(x)$ für die Schüssel. d) Skizzieren Sie $f$ , zeichnen Sie Rechtecke für die Untersumme $\underline{S}_{4}$ und Obersumme $\bar{S}_{4}$ ein. Berechnen Sie den Rauminhalt der Rotationskörper und die Biermenge.

See Solution

Problem 23609

Calculate the average population change of 3,000 bass growing at $2\%$ per year from Year 1-4 and Year 5-8.

See Solution

Problem 23610

A bolt fell from $400 \mathrm{ft}$ . What was its speed upon hitting the ground using $V^{2}=64 s$ ?

See Solution

Problem 23611

Ein Tauchroboter bewegt sich vertikal. Gegeben ist $h(t) = \frac{9}{40} t^{3}-\frac{27}{2} t^{2}+\frac{405}{2} t$ für $0 \leq t \leq 30$ .
a) Zeigen Sie, dass der Abstand nach 10 Minuten 900 Meter beträgt. b) Berechnen Sie die zurückgelegte Strecke in den ersten 15 Minuten. c) Erklären Sie die Bedeutung des Wendepunkts von $h$ .

See Solution

Problem 23612

A cylinder's radius decreases at 9 m/min with a constant volume of 728 m³. Find the height change rate when height is 3 m. Use $V=\pi r^{2} h$ and round to three decimals.

See Solution

Problem 23613

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x+7$ , where $h \neq 0$ .

See Solution

Problem 23614

Find relative extrema of $f(x)=2 x^{3}-3 x^{2}-4$ using the Second Derivative Test. Justify your answers.

See Solution

Problem 23615

Find the value of the relative minimum for the function $g(x)=x^{3}-3x^{2}+1$ .

See Solution

Problem 23616

Find $\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x}$ for $f(x)=x^{3}$ .

See Solution

Problem 23617

Find the $x$ -values of relative extrema for $g$ where $g^{\prime}(x)=x^{2}+2x-8$ using the First Derivative Test.

See Solution

Problem 23618

What is the value of $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ if $f(x)=-3$ ?

See Solution

Problem 23619

Calculate the orbital speed $v$ and altitude $h$ of POES, which orbits Earth 14.1 times daily. Use $T$ , $R_{\oplus}$ , $G$ , and $M_{\oplus}$ .

See Solution

Problem 23620

If $f(x)=\frac{1}{x}$ , find $f^{\prime}(-2)$ .

See Solution

Problem 23621

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{4}}{3^{n}}$ converges or diverges. State the tests used.

See Solution

Problem 23622

Evaluate the integral $\int_{-1}^{2} 3 x(2 x^{2}-1)^{3} dx$ .

See Solution

Problem 23623

求 $t \geq 0$ 时，粒子沿 $x$ 轴的速度 $v(t)=t^{3}-6 t^{2}+10 t-4$ 何时方向由右变左？

See Solution

Problem 23624

Given functions $C(x) = -0.02x^2 + 100x + 100$ and $p(x) = 300 - 0.1x$ , find profit $P$ , average profit, and marginal profit for $x = 900$ .

See Solution

Problem 23625

Evaluate the integral from 1 to 2: $\int_{1}^{2}\left(1+3 z-z^{-2}\right) d z$ .

See Solution

Problem 23626

Find the profit function $P$ , average profit, and marginal profit for $C(x)=-0.02 x^2+100 x+100$ , $p(x)=300-0.1 x$ , $a=900$ .

See Solution

Problem 23627

Evaluate the integral $\int_{0}^{1}(x+\sqrt{x}) \, dx$ using the Fundamental Theorem of Calculus Part 2.

See Solution

Problem 23628

Find $F^{\prime}(x)$ using the Fundamental Theorem given $F(x)=\int_{3}^{x}(t^{2}+t+1) dt$ .

See Solution

Problem 23629

Find Jean's angular velocity at $t=3$ from $\theta(t)=\frac{\pi t^{2}}{8}$ and when it is greatest.

See Solution

Problem 23630

How far does an object travel in 10 seconds if $v(t) = (12t - 4)$ m/s?

See Solution

Problem 23631

Find Jean's angular velocity from $\theta(t) = \frac{\pi t^{2}}{8}$ at $t = 3$ and when it's maximum. Answer in terms of $\pi$ .

See Solution

Problem 23632

Find the function that passes through $(1,1)$ with $\frac{d y}{d x}=4 x^{2}$ .

See Solution

Problem 23633

Find the function passing through $(0,-5)$ with slope $\frac{d y}{d x}=2$ .

See Solution

Problem 23634

Find Jean's angular velocity from $\theta(t)=\frac{\pi t^{2}}{72}$ at $t=3$ and when it's at its maximum.

See Solution

Problem 23635

Find the indefinite integral: $\int \frac{x^{2}-2 \sqrt{x}}{x} dx$ . Show all work and simplify if possible.

See Solution

Problem 23636

A sandal dropped from a 15.0 m mast on a ship moving at 1.75 m/s south. Find its velocity when it hits the deck: (a) relative to the ship, (b) relative to a stationary observer on shore. (c) Discuss how the answers are consistent for the position where it hits the deck.

See Solution

Problem 23637

A ball rolls down a hill at $5 \mathrm{ft/s}$ . What is its velocity after 15 seconds with $a = -32 \mathrm{ft/s}^{2}$ ?

See Solution

Problem 23638

Find Jean's angular velocity at $t=3$ for $\theta(t)=\frac{\pi t^{2}}{72}$ and when is it greatest?

See Solution

Problem 23639

Find how many times the velocity of $y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)$ equals 0 in 4 seconds. (A) 0 (B) 3 (C) 5 (D) 6 (E) 7

See Solution

Problem 23640

Velocity function is $v(t)=-t^{2}+5 t-4$ on $[0,5]$ .
5a) Sketch $v(t)$ , find where $v(t)=0$ , and shade the region. 5b) Calculate displacement on $[0,5]$ . 5c) Calculate distance traveled on $[0,5]$ .

See Solution

Problem 23641

Find Juan's angular velocity from $\phi(t)=\frac{\pi t(24-t)}{72}$ at $t=3$ and when it is maximized.

See Solution

Problem 23642

Find the position of an object after 10s with acceleration $a(t)=3t$ , initial position $1m$ , and initial velocity $2m/s$ .

See Solution

Problem 23643

Calculate the indefinite integral: $\int e^{-4 x} dx$ and include the constant $C$ .

See Solution

Problem 23644

Find critical points of $f(x)=-4 x^{3}+36 x$ on $[-4,5]$ , then determine local and absolute extrema.

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Problem 23645

Find the Hessian matrix of $f(x_{1}, x_{2})=5 x_{1}^{2}-6 x_{1}+2 x_{2}^{2}-2 x_{2}(1+x_{1})+\frac{2}{9}$ . Determine its definiteness and extreme points.

See Solution

Problem 23646

Find critical points of $f(x)=-2x^3+42x$ on $[-5,4]$ and determine local/absolute max/min values.

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Problem 23647

Calculate the indefinite integral: $\int\left(15 e^{3 x}-\frac{15}{x}\right) d x$ and include the constant $C$ .

See Solution

Problem 23648

Calculate the indefinite integral and include the constant $C$ . Use absolute values when needed: $\int(36 x^{2}+3 x^{-1}+24 e^{3 x}) dx$

See Solution

Problem 23649

Determine where the function $f(x)=x^{2} \sin x-2 \sin x+2 x \cos x$ is increasing or decreasing on $(0,2 \pi)$ .

See Solution

Problem 23650

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{2-\operatorname{sen} x}{x}$ .

See Solution

Problem 23651

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{2-\sin x}{x}$ .

See Solution

Problem 23652

Analyze the Leonard-Jones potential $V(r) = \frac{1}{r^{6}} - \frac{A}{r^{3}}$ as $r \to 0$ . What happens to $V(r)$ ?

See Solution

Problem 23653

Find the indefinite integral using substitution: $\int \frac{\cos x}{(\sin x-1)^{2}} d x$ . Show all work.

See Solution

Problem 23654

Calculate the fetal growth rate $\frac{\mathrm{dH}}{\mathrm{dt}}$ using $H=-30.11+1.301 t^{2}-0.6554 t^{2} \log t$ . Compare growth at $t=8$ and $t=36$ weeks, then for $\frac{1}{H} \frac{\mathrm{dH}}{\mathrm{dt}}$ .

See Solution

Problem 23655

球体的体积以每秒3立方厘米的速度减少。当半径以每秒0.25厘米的速度减少时，半径是多少？

See Solution

Problem 23656

Find the average value of $f(x)=12 \sqrt{x}$ on the interval $[1,36]$ .

See Solution

Problem 23657

Evaluate the integral from 1 to $e^{9}$ of $\frac{1}{x} \, dx$ .

See Solution

Problem 23658

Find the tangent line $y=m x+b$ to $f(x)=12 x+9-15 e^{x}$ at $(0,-6)$ . Determine $m$ and $b$ .

See Solution

Problem 23659

Evaluate the integral $\int_{0}^{4} \frac{x}{\sqrt{9+x^{2}}} d x$ using substitution. Show all work.

See Solution

Problem 23660

Find the expression for electric charge $q$ in coulombs as a function of time $t$ given $i=3 t^{2}+2$ and $q=0$ when $t=0$ .

See Solution

Problem 23661

Find the critical numbers $A$ and $B$ for the function $f(x)=x^{2} e^{20 x}$ . Determine if $f(x)$ is INC or DEC in the intervals $(-\infty, A]$ , $[A, B]$ , and $[B, \infty)$ .

See Solution

Problem 23662

Calculez les dérivées des fonctions suivantes : a) $f(x)=(4 x+1)^{3}-\frac{2}{\sqrt[4]{x}}+\frac{x^{2}+1}{x^{2}-1}$ , b) $f(x)=2^{\ln \left(\arcsin x^{3}\right)}$ , c) $f(x)=\cos ^{3} x \cdot \sin e^{x}$ , d) $f(x)=\left[\arctan x+\ln \left(1+x^{2}\right)\right]^{4}$ , e) $f(x)=\frac{1}{\cos \left(2^{x}\right)} \cdot \log _{2} e^{2 x}$ , f) $f(x)=x^{2} \cdot e^{x^{3}}$ .

See Solution

Problem 23663

Evaluate the integrals given $\int_{0}^{4} f(x) d x=10$ , $\int_{0}^{4} g(x) d x=20$ , with $f$ even and $g$ odd.
8a) $\int_{-4}^{4} f(x) d x$
8b) $\int_{-4}^{4} 3 g(x) d x$
8c) $\int_{-4}^{4}(4 f(x)-3 g(x)) d x$
8d) $\int_{0}^{1} 8 x f(4 x^{2}) d x$ (Hint: Use $w=4 x^{2}$ )
8e) $\int_{-2}^{2} 3 x f(x) d x$ (Hint: Check the symmetry of $3 x f(x)$ )

See Solution

Problem 23664

Find the critical number of the function $f(x)=(7 x+8) e^{2 x}$ .
$x=$

See Solution

Problem 23665

Find the inflection points of $f(x)=x^{2} e^{4 x}$ at $x=C$ and $x=D$ , then determine concavity on $(-\infty, C)$ , $(C, D)$ , $(D, \infty)$ .

See Solution

Problem 23666

Find the electric charge $q$ in coulombs as a function of time $t$ (seconds) given $i=4 t^{4}-4$ and $q=5$ at $t=0$ .

See Solution

Problem 23667

Find the value of $Y$ that maximizes utility for $U=X^{0.5} Y^{0.5}$ with prices $4$ for $X$ , $2$ for $Y$ , and budget $20$ .

See Solution

Problem 23668

Find the voltage across a $4 \mathrm{~F}$ capacitor after 4 seconds if current $i=2 t+1$ charges it from $9 \mathrm{~V}$ .

See Solution

Problem 23669

Differentiate the head circumference formula $H = -30.11 + 1.301 t^{2} - 0.6554 t^{2} \log t$ with respect to $t$ .

See Solution

Problem 23670

Evaluate the integral using symmetry: $\int_{-\pi / 6}^{\pi / 6} 2 \sec ^{2} x \, dx$ .

See Solution

Problem 23671

Bestimme die allgemeine Stammfunktion von $f(x)=6x^{3}-8x^{3}$ .

See Solution

Problem 23672

Bestimme die Stammfunktion von $f(x)=x^{7}+x^{3}$ .

See Solution

Problem 23673

Bestimme die allgemeine Stammfunktion von $f(x)=6 x^{6}+9 x^{9}$ .

See Solution

Problem 23674

Find the average velocity over $[1,2]$ for $s(t)=-16 t^{2}+100 t$ , where $t$ is in hours and $s$ in feet.

See Solution

Problem 23675

Check if the series $\sum_{k=1}^{\infty} \frac{k(k+7)}{(k+6)^{2}}$ converges or diverges. If convergent, find the sum.

See Solution

Problem 23676

Evaluate the integral using symmetry: $\int_{-2}^{2}\left(7-|x|^{3}\right) d x$ .

See Solution

Problem 23677

Calculez les dérivées des fonctions suivantes : a) $f(x)=(4 x+1)^{3}-\frac{2}{\sqrt[4]{x}}+\frac{x^{2}+1}{x^{2}-1}$ b) $f(x)=2^{\ln(\arcsin x^{3})}$ c) $f(x)=\cos^{3} x \cdot \sin e^{x}$ d) $f(x)=[\arctan x+\ln(1+x^{2})]^{4}$ e) $f(x)=\frac{1}{\cos(2^{x})} \cdot \log_{2} e^{2 x}$ f) $f(x)=x^{2} \cdot e^{x^{3}}$ Domaine de $f(x)=2 x^{2} \cdot \ln x-2 x^{2}$ .

See Solution

Problem 23678

Evaluate the integral or state "IMPOSSIBLE": $\int x^{2} \sqrt[3]{x^{3}-2} d x$ . Use $C$ for the constant.

See Solution

Problem 23679

Bestimme die Stammfunktion von $f(x)=2 x^{7}(4-x)$ .

See Solution

Problem 23680

Find the second derivative $y''$ for the equation $x^{4}+y^{4}=16$ .

See Solution

Problem 23681

Find the average value of $f(x)=\frac{5}{x^{2}+1}$ over the interval [-1,1] and graph the function with the average value.

See Solution

Problem 23682

Bestimme die allgemeine Stammfunktion von $f(x)=8 x^{7}-4 x^{9}$ .

See Solution

Problem 23683

Calculate the average value of $f(x)=-\cos x$ from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ and graph it.

See Solution

Problem 23684

Bestimme die allgemeine Stammfunktion von $f(x)=8 \sin (x)+8 \sqrt{x}$ .

See Solution

Problem 23685

Find the derivative of the function $h(x)=\int_{1}^{e^{x}} 9 \ln (t) \, dt$ . What is $h^{\prime}(x)$ ?

See Solution

Problem 23686

Bestimme die Stammfunktion von $f(x)=\frac{6}{\sqrt{x}}+4 \sin (x)$ .

See Solution

Problem 23687

Evaluate the integral or state if it's impossible: $\int x^{4} \sqrt{x^{5}-4} d x$ (Use $C$ for the constant.)

See Solution

Problem 23688

Evaluate the integral from 1 to 3: $\int_{1}^{3}\left(\frac{1}{x^{2}}-\frac{6}{x^{3}}\right) d x$

See Solution

Problem 23689

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{5}{7 \sqrt{x}}$ .

See Solution

Problem 23690

Berechne das Integral $\int_{0}^{2} s(t) \, dt$ mit $s(t) = 16e^{-0.5t} - 14e^{-t} - 2$ und bestimme die prozentuale Abweichung vom Näherungswert.

See Solution

Problem 23691

Find the average value of $f(x)=\frac{1}{x^{2}+1}$ over the interval [-1,1] and graph it with the average value indicated.

See Solution

Problem 23692

Bestimme die Stammfunktion von $f(x)=\frac{x^{6}}{3}+\frac{7}{3} \sqrt{x}$ .

See Solution

Problem 23693

Calculate the indefinite integral and include the constant $C$ : $\int \frac{3 x^{3}+2 x^{2}+5 x}{x} d x$

See Solution

Problem 23694

Find the tangent line of $y=f(x)$ at $x=1$ for $f(x)=\frac{1}{x}-\frac{4}{\sqrt{x}}+\frac{2}{x^{2}}$ . The tangent line is $y=\square$ .

See Solution

Problem 23695

Evaluate the integral from 1 to 3 of $\left(\frac{1}{x^{2}}-\frac{6}{x^{3}}\right)$ . Result: $\frac{10}{3}$ .

See Solution

Problem 23696

Evaluate the integral or state "IMPOSSIBLE": $\int x \sqrt[3]{x^{5}-3} d x$ . Use $C$ for the constant of integration.

See Solution

Problem 23697

Calculate the integral $\int x(x-5)^{2.7} d x$ .

See Solution

Problem 23698

Hill's function models oxygen binding in hemoglobin: $f(P)=\frac{P^{n}}{k^{n}+P^{n}}$ . Find $f^{\prime}(P)$ and show $f^{\prime}(P)>0$ for $P>0$ .

See Solution

Problem 23699

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{3}}{4}-\frac{4}{5 x^{3}}$ .

See Solution

Problem 23700

Bestimme die allgemeine Stammfunktion von $f(x)=\frac{x^{6}-9 x^{5}}{8 x-72}$ .

See Solution

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Calculus

Problem 23601

Find the value of lim⁡x→4−1(x−4)3\lim _{x \rightarrow 4^{-}} \frac{1}{(x-4)^{3}}limx→4−​(x−4)31​: 0, 1, ∞\infty∞, or −∞-\infty−∞?

Problem 23602

Find the derivative of f(x)=4x3+2x2+1f(x)=4x^3+2x^2+1f(x)=4x3+2x2+1 using lim⁡Δx→0f(x+Δx)−f(x)Δx\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}limΔx→0​Δxf(x+Δx)−f(x)​.

Problem 23603

Find the remaining amount of a 669mg669 \mathrm{mg}669mg radioactive substance after 87 years using A(t)=669e−0.033tA(t)=669 e^{-0.033 t}A(t)=669e−0.033t.

Problem 23604

Find the value of kkk in the integral ∫x4+2x5+10xdx=kln⁡(x5+10x)+C\int \frac{x^{4}+2}{x^{5}+10 x} d x=k \ln \left(x^{5}+10 x\right)+C∫x5+10xx4+2​dx=kln(x5+10x)+C.

Problem 23605

Find the volume of the solid formed by rotating the area between x=2−y2x=2-y^{2}x=2−y2 and x=y4x=y^{4}x=y4 around the yyy-axis.

Problem 23606

Find the instantaneous rate of change of f(x)=ex+1f(x)=e^{x+1}f(x)=ex+1 at x=−3x=-3x=−3 using the limit below. No need to simplify.

Problem 23607

Find the average rate of change of f(x)=125(0.9)xf(x)=125(0.9)^{x}f(x)=125(0.9)x between years 11-15 and compare it to 1-5.

Problem 23608

Problem 23609

Calculate the average population change of 3,000 bass growing at 2%2\%2% per year from Year 1-4 and Year 5-8.

Problem 23610

A bolt fell from 400ft400 \mathrm{ft}400ft. What was its speed upon hitting the ground using V2=64sV^{2}=64 sV2=64s?

Problem 23611

Problem 23612

A cylinder's radius decreases at 9 m/min with a constant volume of 728 m³. Find the height change rate when height is 3 m. Use V=πr2hV=\pi r^{2} hV=πr2h and round to three decimals.

Problem 23613

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=9x+7f(x)=9x+7f(x)=9x+7, where h≠0h \neq 0h=0.

Problem 23614

Find relative extrema of f(x)=2x3−3x2−4f(x)=2 x^{3}-3 x^{2}-4f(x)=2x3−3x2−4 using the Second Derivative Test. Justify your answers.

Problem 23615

Find the value of the relative minimum for the function g(x)=x3−3x2+1g(x)=x^{3}-3x^{2}+1g(x)=x3−3x2+1.

Problem 23616

Find lim⁡Δx→0(x+Δx)3−x3Δx\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x}limΔx→0​Δx(x+Δx)3−x3​ for f(x)=x3f(x)=x^{3}f(x)=x3.

Problem 23617

Find the xxx-values of relative extrema for ggg where g′(x)=x2+2x−8g^{\prime}(x)=x^{2}+2x-8g′(x)=x2+2x−8 using the First Derivative Test.

Problem 23618

What is the value of lim⁡Δx→0f(x+Δx)−f(x)Δx\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}limΔx→0​Δxf(x+Δx)−f(x)​ if f(x)=−3f(x)=-3f(x)=−3?

Problem 23619

Calculate the orbital speed vvv and altitude hhh of POES, which orbits Earth 14.1 times daily. Use TTT, R⊕R_{\oplus}R⊕​, GGG, and M⊕M_{\oplus}M⊕​.

Problem 23620

If f(x)=1xf(x)=\frac{1}{x}f(x)=x1​, find f′(−2)f^{\prime}(-2)f′(−2).

Problem 23621

Determine if the series ∑n=1∞n43n\sum_{n=1}^{\infty} \frac{n^{4}}{3^{n}}∑n=1∞​3nn4​ converges or diverges. State the tests used.

Problem 23622

Evaluate the integral ∫−123x(2x2−1)3dx\int_{-1}^{2} 3 x(2 x^{2}-1)^{3} dx∫−12​3x(2x2−1)3dx.

Problem 23623

求 t≥0t \geq 0t≥0 时，粒子沿 xxx 轴的速度 v(t)=t3−6t2+10t−4v(t)=t^{3}-6 t^{2}+10 t-4v(t)=t3−6t2+10t−4 何时方向由右变左？

Problem 23624

Given functions C(x)=−0.02x2+100x+100C(x) = -0.02x^2 + 100x + 100C(x)=−0.02x2+100x+100 and p(x)=300−0.1xp(x) = 300 - 0.1xp(x)=300−0.1x, find profit PPP, average profit, and marginal profit for x=900x = 900x=900.

Problem 23625

Evaluate the integral from 1 to 2: ∫12(1+3z−z−2)dz\int_{1}^{2}\left(1+3 z-z^{-2}\right) d z∫12​(1+3z−z−2)dz.

Problem 23626

Find the profit function PPP, average profit, and marginal profit for C(x)=−0.02x2+100x+100C(x)=-0.02 x^2+100 x+100C(x)=−0.02x2+100x+100, p(x)=300−0.1xp(x)=300-0.1 xp(x)=300−0.1x, a=900a=900a=900.

Problem 23627

Evaluate the integral ∫01(x+x) dx\int_{0}^{1}(x+\sqrt{x}) \, dx∫01​(x+x​)dx using the Fundamental Theorem of Calculus Part 2.

Problem 23628

Find F′(x)F^{\prime}(x)F′(x) using the Fundamental Theorem given F(x)=∫3x(t2+t+1)dtF(x)=\int_{3}^{x}(t^{2}+t+1) dtF(x)=∫3x​(t2+t+1)dt.

Problem 23629

Find Jean's angular velocity at t=3t=3t=3 from θ(t)=πt28\theta(t)=\frac{\pi t^{2}}{8}θ(t)=8πt2​ and when it is greatest.

Problem 23630

How far does an object travel in 10 seconds if v(t)=(12t−4)v(t) = (12t - 4)v(t)=(12t−4) m/s?

Problem 23631

Find Jean's angular velocity from θ(t)=πt28\theta(t) = \frac{\pi t^{2}}{8}θ(t)=8πt2​ at t=3t = 3t=3 and when it's maximum. Answer in terms of π\piπ.

Problem 23632

Find the function that passes through (1,1)(1,1)(1,1) with dydx=4x2\frac{d y}{d x}=4 x^{2}dxdy​=4x2.

Problem 23633

Find the function passing through (0,−5)(0,-5)(0,−5) with slope dydx=2\frac{d y}{d x}=2dxdy​=2.

Problem 23634

Find Jean's angular velocity from θ(t)=πt272\theta(t)=\frac{\pi t^{2}}{72}θ(t)=72πt2​ at t=3t=3t=3 and when it's at its maximum.

Problem 23635

Find the indefinite integral: ∫x2−2xxdx\int \frac{x^{2}-2 \sqrt{x}}{x} dx∫xx2−2x​​dx. Show all work and simplify if possible.

Problem 23636

A sandal dropped from a 15.0 m mast on a ship moving at 1.75 m/s south. Find its velocity when it hits the deck: (a) relative to the ship, (b) relative to a stationary observer on shore. (c) Discuss how the answers are consistent for the position where it hits the deck.

Problem 23637

A ball rolls down a hill at 5ft/s5 \mathrm{ft/s}5ft/s. What is its velocity after 15 seconds with a=−32ft/s2a = -32 \mathrm{ft/s}^{2}a=−32ft/s2?

Problem 23638

Find Jean's angular velocity at t=3t=3t=3 for θ(t)=πt272\theta(t)=\frac{\pi t^{2}}{72}θ(t)=72πt2​ and when is it greatest?

Problem 23639

Find how many times the velocity of y(t)=16cos⁡(5t)−14sin⁡(5t)y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)y(t)=61​cos(5t)−41​sin(5t) equals 0 in 4 seconds. (A) 0 (B) 3 (C) 5 (D) 6 (E) 7

Problem 23640

Velocity function is v(t)=−t2+5t−4v(t)=-t^{2}+5 t-4v(t)=−t2+5t−4 on [0,5][0,5][0,5]. 5a) Sketch v(t)v(t)v(t), find where v(t)=0v(t)=0v(t)=0, and shade the region. 5b) Calculate displacement on [0,5][0,5][0,5]. 5c) Calculate distance traveled on [0,5][0,5][0,5].

Problem 23641

Find the value of $\lim _{x \rightarrow 4^{-}} \frac{1}{(x-4)^{3}}$ : 0, 1, $\infty$ , or $-\infty$ ?

Find the derivative of $f(x)=4x^3+2x^2+1$ using $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ .

Find the remaining amount of a $669 \mathrm{mg}$ radioactive substance after 87 years using $A(t)=669 e^{-0.033 t}$ .

Find the value of $k$ in the integral $\int \frac{x^{4}+2}{x^{5}+10 x} d x=k \ln \left(x^{5}+10 x\right)+C$ .

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

Find the instantaneous rate of change of $f(x)=e^{x+1}$ at $x=-3$ using the limit below. No need to simplify.

Find the average rate of change of $f(x)=125(0.9)^{x}$ between years 11-15 and compare it to 1-5.

Calculate the average population change of 3,000 bass growing at $2\%$ per year from Year 1-4 and Year 5-8.

A bolt fell from $400 \mathrm{ft}$ . What was its speed upon hitting the ground using $V^{2}=64 s$ ?

A cylinder's radius decreases at 9 m/min with a constant volume of 728 m³. Find the height change rate when height is 3 m. Use $V=\pi r^{2} h$ and round to three decimals.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x+7$ , where $h \neq 0$ .

Find relative extrema of $f(x)=2 x^{3}-3 x^{2}-4$ using the Second Derivative Test. Justify your answers.

Find the value of the relative minimum for the function $g(x)=x^{3}-3x^{2}+1$ .

Find $\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x}$ for $f(x)=x^{3}$ .

Find the $x$ -values of relative extrema for $g$ where $g^{\prime}(x)=x^{2}+2x-8$ using the First Derivative Test.

What is the value of $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ if $f(x)=-3$ ?

Calculate the orbital speed $v$ and altitude $h$ of POES, which orbits Earth 14.1 times daily. Use $T$ , $R_{\oplus}$ , $G$ , and $M_{\oplus}$ .

If $f(x)=\frac{1}{x}$ , find $f^{\prime}(-2)$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{4}}{3^{n}}$ converges or diverges. State the tests used.

Evaluate the integral $\int_{-1}^{2} 3 x(2 x^{2}-1)^{3} dx$ .

求 $t \geq 0$ 时，粒子沿 $x$ 轴的速度 $v(t)=t^{3}-6 t^{2}+10 t-4$ 何时方向由右变左？

Given functions $C(x) = -0.02x^2 + 100x + 100$ and $p(x) = 300 - 0.1x$ , find profit $P$ , average profit, and marginal profit for $x = 900$ .

Evaluate the integral from 1 to 2: $\int_{1}^{2}\left(1+3 z-z^{-2}\right) d z$ .

Find the profit function $P$ , average profit, and marginal profit for $C(x)=-0.02 x^2+100 x+100$ , $p(x)=300-0.1 x$ , $a=900$ .

Evaluate the integral $\int_{0}^{1}(x+\sqrt{x}) \, dx$ using the Fundamental Theorem of Calculus Part 2.

Find $F^{\prime}(x)$ using the Fundamental Theorem given $F(x)=\int_{3}^{x}(t^{2}+t+1) dt$ .

Find Jean's angular velocity at $t=3$ from $\theta(t)=\frac{\pi t^{2}}{8}$ and when it is greatest.

How far does an object travel in 10 seconds if $v(t) = (12t - 4)$ m/s?

Find Jean's angular velocity from $\theta(t) = \frac{\pi t^{2}}{8}$ at $t = 3$ and when it's maximum. Answer in terms of $\pi$ .

Find the function that passes through $(1,1)$ with $\frac{d y}{d x}=4 x^{2}$ .

Find the function passing through $(0,-5)$ with slope $\frac{d y}{d x}=2$ .

Find Jean's angular velocity from $\theta(t)=\frac{\pi t^{2}}{72}$ at $t=3$ and when it's at its maximum.

Find the indefinite integral: $\int \frac{x^{2}-2 \sqrt{x}}{x} dx$ . Show all work and simplify if possible.

A ball rolls down a hill at $5 \mathrm{ft/s}$ . What is its velocity after 15 seconds with $a = -32 \mathrm{ft/s}^{2}$ ?

Find Jean's angular velocity at $t=3$ for $\theta(t)=\frac{\pi t^{2}}{72}$ and when is it greatest?

Find how many times the velocity of $y(t)=\frac{1}{6} \cos (5 t)-\frac{1}{4} \sin (5 t)$ equals 0 in 4 seconds. (A) 0 (B) 3 (C) 5 (D) 6 (E) 7

Velocity function is $v(t)=-t^{2}+5 t-4$ on $[0,5]$ .
5a) Sketch $v(t)$ , find where $v(t)=0$ , and shade the region. 5b) Calculate displacement on $[0,5]$ . 5c) Calculate distance traveled on $[0,5]$ .

Find Juan's angular velocity from $\phi(t)=\frac{\pi t(24-t)}{72}$ at $t=3$ and when it is maximized.

Find the position of an object after 10s with acceleration $a(t)=3t$ , initial position $1m$ , and initial velocity $2m/s$ .

Calculate the indefinite integral: $\int e^{-4 x} dx$ and include the constant $C$ .

Find critical points of $f(x)=-4 x^{3}+36 x$ on $[-4,5]$ , then determine local and absolute extrema.

Find the Hessian matrix of $f(x_{1}, x_{2})=5 x_{1}^{2}-6 x_{1}+2 x_{2}^{2}-2 x_{2}(1+x_{1})+\frac{2}{9}$ . Determine its definiteness and extreme points.

Find critical points of $f(x)=-2x^3+42x$ on $[-5,4]$ and determine local/absolute max/min values.

Calculate the indefinite integral: $\int\left(15 e^{3 x}-\frac{15}{x}\right) d x$ and include the constant $C$ .

Calculate the indefinite integral and include the constant $C$ . Use absolute values when needed: $\int(36 x^{2}+3 x^{-1}+24 e^{3 x}) dx$

Determine where the function $f(x)=x^{2} \sin x-2 \sin x+2 x \cos x$ is increasing or decreasing on $(0,2 \pi)$ .

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{2-\operatorname{sen} x}{x}$ .

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{2-\sin x}{x}$ .

Analyze the Leonard-Jones potential $V(r) = \frac{1}{r^{6}} - \frac{A}{r^{3}}$ as $r \to 0$ . What happens to $V(r)$ ?

Find the indefinite integral using substitution: $\int \frac{\cos x}{(\sin x-1)^{2}} d x$ . Show all work.

Find the average value of $f(x)=12 \sqrt{x}$ on the interval $[1,36]$ .

Evaluate the integral from 1 to $e^{9}$ of $\frac{1}{x} \, dx$ .

Find the tangent line $y=m x+b$ to $f(x)=12 x+9-15 e^{x}$ at $(0,-6)$ . Determine $m$ and $b$ .

Evaluate the integral $\int_{0}^{4} \frac{x}{\sqrt{9+x^{2}}} d x$ using substitution. Show all work.

Find the expression for electric charge $q$ in coulombs as a function of time $t$ given $i=3 t^{2}+2$ and $q=0$ when $t=0$ .

Find the critical numbers $A$ and $B$ for the function $f(x)=x^{2} e^{20 x}$ . Determine if $f(x)$ is INC or DEC in the intervals $(-\infty, A]$ , $[A, B]$ , and $[B, \infty)$ .

Find the critical number of the function $f(x)=(7 x+8) e^{2 x}$ .
$x=$

Find the inflection points of $f(x)=x^{2} e^{4 x}$ at $x=C$ and $x=D$ , then determine concavity on $(-\infty, C)$ , $(C, D)$ , $(D, \infty)$ .

Find the electric charge $q$ in coulombs as a function of time $t$ (seconds) given $i=4 t^{4}-4$ and $q=5$ at $t=0$ .

Find the value of $Y$ that maximizes utility for $U=X^{0.5} Y^{0.5}$ with prices $4$ for $X$ , $2$ for $Y$ , and budget $20$ .

Find the voltage across a $4 \mathrm{~F}$ capacitor after 4 seconds if current $i=2 t+1$ charges it from $9 \mathrm{~V}$ .

Differentiate the head circumference formula $H = -30.11 + 1.301 t^{2} - 0.6554 t^{2} \log t$ with respect to $t$ .

Evaluate the integral using symmetry: $\int_{-\pi / 6}^{\pi / 6} 2 \sec ^{2} x \, dx$ .

Bestimme die allgemeine Stammfunktion von $f(x)=6x^{3}-8x^{3}$ .

Bestimme die Stammfunktion von $f(x)=x^{7}+x^{3}$ .

Bestimme die allgemeine Stammfunktion von $f(x)=6 x^{6}+9 x^{9}$ .

Find the average velocity over $[1,2]$ for $s(t)=-16 t^{2}+100 t$ , where $t$ is in hours and $s$ in feet.