Calculus

Problem 20001

Evaluate $\int_{1}^{0} (9 - 9x^{2}) \, dx$ using $\int_{0}^{1} x^{2} \, dx = \frac{1}{3}$ .

See Solution

Problem 20002

Calculate the average value of $f(x)=\frac{8}{x}$ over the interval $[1,9]$ .

See Solution

Problem 20003

Find $f(-8)$ given that $f^{\prime}(x)$ is continuous, $f(-5)=14$ , and $\int_{-5}^{-8} f^{\prime}(x) dx=-8$ .

See Solution

Problem 20004

Find the derivative of $f(x) = 4x^{2}-8x+9$ and use Rolle's theorem to find $c$ in $(-1,3)$ where $f^{\prime}(c)=0$ .

See Solution

Problem 20005

Bestimmen Sie die Folgenglieder $a_n$ , Häufungspunkte, $\varlimsup$ , $\sup$ , $\varliminf$ und die Konvergenz.

See Solution

Problem 20006

Find the number $c$ in $(-1,3)$ such that $f^{\prime}(c)=0$ for $f(x)=4x^{2}-8x+9$ . Solve $8c-8=0$ .

See Solution

Problem 20007

Find the average value of $f(x)=\frac{1}{x^{2}}$ on the interval $[1,9]$ .

See Solution

Problem 20008

Calculer l'intégrale $\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x$ après avoir factorisé le dénominateur.

See Solution

Problem 20009

Evaluate the integrals given $\int_{0}^{4} f(x) dx=1$ , $\int_{4}^{5} g(x) dx=8$ , $\int_{4}^{5} h(x) dx=-5$ :
(a) $\int_{4}^{0} f(x) dx=$ (b) $\int_{4}^{5} \frac{4 h(x)-g(x)}{2} dx=$ (c) $\int_{5}^{0} r(x) dx=$

See Solution

Problem 20010

Calculez l'intégrale $\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} d x$ en trouvant $A, B, C$ dans $\frac{2 x+2}{x^{3}+6 x}=\frac{A}{x}+\frac{B x+C}{x^{2}+6}$ . Ensuite, donnez l'intégrale avec trois décimales.

See Solution

Problem 20011

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt[4]{x^{4}+1}}{2 x+3}$ .

See Solution

Problem 20012

Calculez les valeurs exactes de $A, B$ et $C$ dans la décomposition $\frac{2 x+2}{x^{3}+6 x}=\frac{A}{x}+\frac{B x+C}{x^{2}+6}$ . Ensuite, trouvez l'intégrale $\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} \mathrm{~d} x$ avec trois décimales.

See Solution

Problem 20013

Calculate the area using the integral: $\int_{1}^{3} (-x+5) \, dx = \text{area}$ . What is the area?

See Solution

Problem 20014

Evaluate the integrals: $\int_{-2}^{2} \frac{11 / 36}{x-3} dx + \int_{-2}^{2} \frac{25 / 3}{x+3} dx + \int_{-2}^{2} \frac{(x+6)}{(x+3)^{2}} dx$ .

See Solution

Problem 20015

Calculez l'intégrale suivante :
$\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x$
(i) Factorisez le dénominateur : $x^{3}+3 x^{2}-9 x-27=(x-a)(x-b)^{2}$ . Trouvez $a$ et $b$ .
(ii) Écrivez la fraction sous la forme :
$\frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{(x-b)^{2}}$
Trouvez $A$ , $B$ , et $C$ .
(iii) Calculez l'intégrale et donnez sa valeur approchée à 3 décimales près.

See Solution

Problem 20016

Find the angular velocity at the $2^{\text{nd}}$ instant when angular acceleration is zero for $\theta=8t^{4}-4t^{3}$ .

See Solution

Problem 20017

Find the derivative of $y=2 x^{2}+5 x-10$ using first principles. Verify with $(K / U=6)$ .

See Solution

Problem 20018

Calculate the integral $\int \cos^{3}(x) e^{\sin(x)} \mathrm{d} x$ using substitution and integration by parts.

See Solution

Problem 20019

Déterminez la fonction $g(t)=\int_{t}^{6t} \frac{x^{2}+1}{x+1} dx$ .

See Solution

Problem 20020

Find the derivatives of: a) $f(x)=\frac{x^{2}+1}{x-1}$ b) $g(x)=\left(x^{2}+3x-5\right) \frac{2}{3}$

See Solution

Problem 20021

Si $g(u)=\int_{u}^{8} \cos (x) d x$ , calculez $\frac{d g}{d u}$ . Pour $g(t)=\int_{e^{2 t}}^{8} \cos (x) d x$ , montrez que $\frac{d g}{d t}=1$ .

See Solution

Problem 20022

Simplifiez $f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}$ en somme de la forme $a x^{b}$ et trouvez la primitive $F(x)$ .

See Solution

Problem 20023

Avec $u=\cos (3 x)$ , trouvez $\int \frac{\sin (3 x)}{\cos ^{5}(3 x)} \mathrm{d} x$ en termes de $u$ .

See Solution

Problem 20024

Trouvez la primitive $F(x)$ de $f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}$ avec $x \neq 0$ . Simplifiez $f(x)$ d'abord.

See Solution

Problem 20025

Compute the integral $\int_{-\pi / 6}^{0} \sin^{4}(3x) \cos^{7}(3x) \, dx$ using substitution $u=$ . Rewrite as $\int_{a}^{b} F(u) \, du$ and find the value of $\int_{-\pi / 6}^{0} \sin^{4}(3x) \cos^{7}(3x) \, dx$ .

See Solution

Problem 20026

Find the max and min of $h(t)=4t-\tan^2 t$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{3}]$ .

See Solution

Problem 20027

Calculate the integrals of the function $f(x)$ defined by points (-2,-5), (-2,-3), (0,2), (1,-1), (3,-1), (5,-2):
(a) $\int_{-5}^{0} f(x) dx=$
(b) $\int_{0}^{5} f(x) dx=$
(c) $\int_{-5}^{5} f(x) dx=$

See Solution

Problem 20028

A 15-ft pole casts a shadow. A 6-ft man walks away at 7 ft/s. Find the shadow's tip speed when he's 30 ft from the pole.

See Solution

Problem 20029

Calculez l'intégrale $\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x$ en utilisant $u=\cos (x)$ .

See Solution

Problem 20030

Find the derivative of $f(x)=2 x^{3}+3 x^{2}$ and where the tangent slope equals 36. $(K / U=3)$

See Solution

Problem 20031

Calculez l'intégrale $\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x$ avec la substitution $u=\cos (x)$ .

See Solution

Problem 20032

Find the tangent line equation to $f(x)=\frac{-x}{(x+1)^{2}}$ at $x=-2$ .

See Solution

Problem 20033

Find points on the curve $y=2-\frac{1}{x}$ where the tangent slope is $\frac{1}{4}$ .

See Solution

Problem 20034

Calculez l'intégrale $\int_{0}^{8} \frac{1}{\sqrt{4+4 x}} d x$ et donnez sa valeur exacte.

See Solution

Problem 20035

How old is a wooden artifact with 21% carbon-14 remaining? (Half-life of carbon-14 is 5730 years.) years

See Solution

Problem 20036

1. What does the derivative of a function represent?
2. How can the product rule replace the quotient rule for $y=\frac{(3+x)}{(2 x-9)}$ ?

See Solution

Problem 20037

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}$ .

See Solution

Problem 20038

Find the normal line equation to the curve $y=\frac{3 x}{\left(x^{2}-1\right)}$ at $x=3$ .

See Solution

Problem 20039

Evaluate these limits: a) $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$ b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$

See Solution

Problem 20040

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3x^{2}+6$ and simplify your answer.

See Solution

Problem 20041

Find the third-order Taylor polynomial for $f$ at 0 using $f(0)=2$ , $f'(0)=2$ , $f''(0)=4$ , $f^{(3)}(0)=6$ and approximate $f(0.1)$ .

See Solution

Problem 20042

Analyze the function $y=\frac{5 x}{x^{2}-9}$ . Find intercepts, extrema, inflection points, and asymptotes.

See Solution

Problem 20043

Evaluate the limits: a) $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$ , b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$ .

See Solution

Problem 20044

Find the third-order Taylor polynomial for $f$ at 9 using $f(9)=3$ , $f'(9)=1$ , $f''(9)=0$ , $f^{(3)}(9)=6$ . Approximate $f(8.8)$ .

See Solution

Problem 20045

Homer throws a bowling ball up; its height $h=-4.9 t^{2}+19.6 t$ . Find: a) landing time, b) impact velocity, c) peak acceleration.

See Solution

Problem 20046

Find the linear and quadratic approximating polynomials for $f(x)=-\frac{1}{x}$ at $a=1$ and use them to estimate $-\frac{1}{0.91}$ .

See Solution

Problem 20047

Find $\left.\frac{d S}{d P}\right|_{t=1}$ using $P(t)=-6 t^{2}+27 t+10$ and $S(t)=-4 t^{2}+4 t+11$ . Round to 1 decimal place.

See Solution

Problem 20048

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=3 \cos (-2 x)$ . What is $p_{1}(x)=\square$ ?

See Solution

Problem 20049

Find the average value $f_{\text {ave }}$ of $f(x)=\sqrt{4-x^{2}}$ from 0 to 2, and a $c$ in $[0,2]$ where $f(c)=f_{\text {ave }}$ .

See Solution

Problem 20050

Alice the Amoeba's displacement is $s=t^{3}-6 t^{2}+9 t+1$ .
a) Find her velocity at $t=3$ seconds. b) When is her acceleration $0 \mathrm{~m} / \mathrm{S}^{2}$ ? c) When is she at rest? d) When is she moving backwards? e) Graph her distance for the first 6 seconds.

See Solution

Problem 20051

Find the third-order Taylor polynomial $p_{3}(x)$ for $f$ at 0 using $f(0)=4, f'(0)=0, f''(0)=8, f^{(3)}(0)=12$ and approximate $f(0.2)$ .

See Solution

Problem 20052

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=3 \cos (4 x)$ . What is $p_{1}(x)=\square$ ?

See Solution

Problem 20053

Calculate $\frac{d y}{d x}$ for $P=x^{0.7} y^{0.3}$ at $x=100$ , $y=300,000$ . Interpret the result.

See Solution

Problem 20054

Find the Taylor polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ at $a=0$ for $f(x)=3 \cos (4 x)$ .

See Solution

Problem 20055

Find the second derivative of $f(x)=\frac{4 x^{2}}{x^{2}-9}$ . Identify intervals of concavity (up and down).

See Solution

Problem 20056

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=\cos(-2x)$ . What is $p_{1}(x)$ ?

See Solution

Problem 20057

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{4}+7 y^{4}=10$ .

See Solution

Problem 20058

Find the half-life of a radioactive substance with a decay rate of $8.4\%$ per day. Round to the nearest hundredth.

See Solution

Problem 20059

Find the function $g(x)$ where $g'(x)=-4096-x^3$ and the maximum value is -5. What is $g(x)$ ?

See Solution

Problem 20060

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=\cos(-2x)$ . Given $p_{1}(x)=1$ , find $p_{2}(x)$ .

See Solution

Problem 20061

Find Taylor polynomials $p_{1}, \ldots, p_{5}$ at $a=0$ for $f(x)=2 e^{-x}$ . What is $p_{1}(x)$ ?

See Solution

Problem 20062

Find the function $g(x)$ such that $g^{\prime}(x)=-1000-x^{3}$ and its maximum value is -7. $g(x)=\square$

See Solution

Problem 20063

Evaluate the limit: $\lim _{x \rightarrow 7}\left(\frac{\frac{1}{x}-\frac{1}{7}}{x-7}\right)=\lim _{x \rightarrow 7} \square=\square$ .

See Solution

Problem 20064

You received 15 mg of dye; after 12 min, 4.5 mg remain. When will it drop below 2 mg? Answer in minutes.

See Solution

Problem 20065

Trouvez la dérivée seconde de la fonction $f(x)=\frac{x}{x^{2}-4}$ .

See Solution

Problem 20066

How long for \$ 9,000 to grow to \$ 45,000 at 5\% continuous compounding? Round to the nearest tenth of a year.

See Solution

Problem 20067

Calculate the integral $L=\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{(\cos x)^{5}} d x$ .

See Solution

Problem 20068

Berechne den Differenzialquotienten von $f(x)=\frac{1}{x}$ bei $x=-2$ .

See Solution

Problem 20069

Evaluate the sum $\sum_{i=1}^{10}(i^{2}-4)$ using summation properties and verify with a graphing utility.

See Solution

Problem 20070

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{x}$ .

See Solution

Problem 20071

Calculate the sum $\sum_{k=1}^{5} \frac{1}{k^{2}+3}$ and round your answer to four decimal places. Use a graphing utility to verify.

See Solution

Problem 20072

A turkey cools from $185^{\circ}$ F to $149^{\circ}$ F in 30 min. Find its temp after 45 min and when it reaches $100^{\circ}$ F.

See Solution

Problem 20073

Find the temperature formula $F(t)$ for an object cooling from $160^{\circ} \mathrm{F}$ in $50^{\circ} \mathrm{F}$ water with $k=0.4$ .

See Solution

Problem 20074

In a community of 4000, a flu spreads as $A=\frac{4000}{1+3999 e^{-0.75 t}}$ . Find: (a) $A$ after 7 days, (b) carrying capacity, (c) days for 300 cases.

See Solution

Problem 20075

Approximate the area under $f(x)=2x+1$ from $x=0$ to $x=2$ using 4 rectangles and right endpoints (Riemann sums).

See Solution

Problem 20076

Find the area under $f(x)=2x+6$ on $[0,2]$ using 4 rectangles.

See Solution

Problem 20077

Find the area between the graph of $f(y)=9y$ and the $y$ -axis for $0 \leq y \leq 2$ . Sketch the region.

See Solution

Problem 20078

Estimate the area under $f(x)=3x+2$ from $x=0$ to $x=3$ using 6 rectangles and right endpoints. Round to 2 decimals.

See Solution

Problem 20079

Find the area under $f(x)=2x+6$ on $[0,2]$ using 4 rectangles.

See Solution

Problem 20080

Graph $f(x)=\frac{105}{1+3 e^{-3 x}}$ for $x=0$ to $x=10$ . Find $f(0)$ , $f(10)$ , its behavior, and limiting value.

See Solution

Problem 20081

Calculate the integral $\int_{2}^{8} (x - 17) \, dx$ using given values: $\int_{2}^{8} x^{3} \, dx = 1020$ , $\int_{2}^{8} x \, dx = 30$ , $\int_{2}^{8} \, dx = 6$ .

See Solution

Problem 20082

A virus spread in a school is modeled by $y=\frac{5600}{1+799 e^{-0.9 x}}$ .
(a) Graph for $0 \leq x \leq 15$ . (b) Find initial infections. (c) Determine the max infections.

See Solution

Problem 20083

Find the rate of area increase for region $A$ when $k=\frac{\pi}{6}$ , given $\frac{dk}{dt}=\frac{\pi}{4}$ . Area: $A=\int_{0}^{k} \frac{\sin x}{x}$ .

See Solution

Problem 20084

A turkey cools from $185^{\circ}$ F to $75^{\circ}$ F. Find its temp after 45 min and when it hits $100^{\circ}$ F using Newton's Law.

See Solution

Problem 20085

Evaluate the integral $\int_{2}^{8}\left(9 x-\frac{1}{10} x^{3}\right) d x$ using given values.

See Solution

Problem 20086

A turkey cools from $185^{\circ}$ F to $75^{\circ}$ F. Find its temp at 45 min if it's $149^{\circ}$ F after 30 min. When is it $100^{\circ}$ F?

See Solution

Problem 20087

Evaluate the following integrals given $\int_{0}^{5} f(x) d x=20$ and $\int_{5}^{7} f(x) d x=12$ : (a) $\int_{0}^{7} f(x) d x$ (b) $\int_{5}^{0} f(x) d x$ (c) $\int_{5}^{5} f(x) d x$ (d) $\int_{0}^{5} 2 f(x) d x$

See Solution

Problem 20088

A turkey cools from $185^{\circ}$ F in a $75^{\circ}$ F room.
(a) Find its temp after 45 min. (b) When will it reach $100^{\circ}$ F? Use $T(t) = T_a + (T_o - T_a) e^{-kt}$ .

See Solution

Problem 20089

Calculate the integral from $\frac{1}{2}$ to $1$ of $\frac{1}{v^{5}}$ with respect to $v$ .

See Solution

Problem 20090

A community of 4000 has a flu spread modeled by $A=\frac{4000}{1+3999 e^{-0.75 t}}$ .
(a) Find $A$ after 7 days. (b) What is the carrying capacity? (c) How many days for 300 people to contract the flu?

See Solution

Problem 20091

A community of 4000 has a flu model $A=\frac{4000}{1+3999 e^{-0.75 t}}$ . Find: (a) $A$ after 7 days, (b) carrying capacity, (c) days for 300 to contract.

See Solution

Problem 20092

A community of 4000 is affected by influenza. Use $A=\frac{4000}{1+3999 e^{-0.75 t}}$ to find:
(a) Flu cases after 7 days. (b) Carrying capacity. (c) Days for 300 cases.

See Solution

Problem 20093

Find the tangent line equation to the curve $5 x^{2}+y^{4}=6 x y$ at the point $(1,1)$ .

See Solution

Problem 20094

Find the tangent line equation to the curve $5 x^{2}+y^{4}=6 x y$ at the point $(1,1)$ .

See Solution

Problem 20095

Use the Limit Comparison Test for convergence/divergence of $\sum_{n=1}^{\infty} \frac{3n^{2}+n+1}{n^{4}+8n^{2}-9}$ . Choose $b_{n}$ and fill in $L$ .

See Solution

Problem 20096

Use the Limit Comparison Test to check if the series converges: $\sum_{k=1}^{\infty} \frac{k^{7}-3}{k^{8}+2}$ .

See Solution

Problem 20097

Find the tangent line equation to $7x^3 + 8y^2 = 15xy$ at $(1,1)$ using its derivative and point-slope form.

See Solution

Problem 20098

Use the Limit Comparison Test for $\sum_{n=1}^{\infty} \frac{0.0012}{n+8}$ . Find $b_{n}$ and limit $L$ .

See Solution

Problem 20099

Determine if the series $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{2}+9}$ converges using the Comparison or Limit Comparison Test.

See Solution

Problem 20100

Find the tangent line equation to the curve $5 x^{2}+4 y^{4}=-9 x y$ at the point $(-1,1)$ .

See Solution

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Calculus

Problem 20001

Evaluate ∫10(9−9x2) dx\int_{1}^{0} (9 - 9x^{2}) \, dx∫10​(9−9x2)dx using ∫01x2 dx=13\int_{0}^{1} x^{2} \, dx = \frac{1}{3}∫01​x2dx=31​.

Problem 20002

Calculate the average value of f(x)=8xf(x)=\frac{8}{x}f(x)=x8​ over the interval [1,9][1,9][1,9].

Problem 20003

Find f(−8)f(-8)f(−8) given that f′(x)f^{\prime}(x)f′(x) is continuous, f(−5)=14f(-5)=14f(−5)=14, and ∫−5−8f′(x)dx=−8\int_{-5}^{-8} f^{\prime}(x) dx=-8∫−5−8​f′(x)dx=−8.

Problem 20004

Find the derivative of f(x)=4x2−8x+9f(x) = 4x^{2}-8x+9f(x)=4x2−8x+9 and use Rolle's theorem to find ccc in (−1,3)(-1,3)(−1,3) where f′(c)=0f^{\prime}(c)=0f′(c)=0.

Problem 20005

Bestimmen Sie die Folgenglieder ana_nan​, Häufungspunkte, lim‾⁡\varlimsuplim, sup⁡\supsup, lim‾⁡\varliminflim​ und die Konvergenz.

Problem 20006

Find the number ccc in (−1,3)(-1,3)(−1,3) such that f′(c)=0f^{\prime}(c)=0f′(c)=0 for f(x)=4x2−8x+9f(x)=4x^{2}-8x+9f(x)=4x2−8x+9. Solve 8c−8=08c-8=08c−8=0.

Problem 20007

Find the average value of f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​ on the interval [1,9][1,9][1,9].

Problem 20008

Calculer l'intégrale ∫−22x2+2x3+3x2−9x−27dx\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x∫−22​x3+3x2−9x−27x2+2​dx après avoir factorisé le dénominateur.

Problem 20009

Problem 20010

Calculez l'intégrale ∫132x+2x3+6xdx\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} d x∫13​x3+6x2x+2​dx en trouvant A,B,CA, B, CA,B,C dans 2x+2x3+6x=Ax+Bx+Cx2+6\frac{2 x+2}{x^{3}+6 x}=\frac{A}{x}+\frac{B x+C}{x^{2}+6}x3+6x2x+2​=xA​+x2+6Bx+C​. Ensuite, donnez l'intégrale avec trois décimales.

Problem 20011

Find the limit: lim⁡x→−∞x4+142x+3\lim _{x \rightarrow-\infty} \frac{\sqrt[4]{x^{4}+1}}{2 x+3}limx→−∞​2x+34x4+1​​.

Problem 20012

Problem 20013

Calculate the area using the integral: ∫13(−x+5) dx=area\int_{1}^{3} (-x+5) \, dx = \text{area}∫13​(−x+5)dx=area. What is the area?

Problem 20014

Evaluate the integrals: ∫−2211/36x−3dx+∫−2225/3x+3dx+∫−22(x+6)(x+3)2dx\int_{-2}^{2} \frac{11 / 36}{x-3} dx + \int_{-2}^{2} \frac{25 / 3}{x+3} dx + \int_{-2}^{2} \frac{(x+6)}{(x+3)^{2}} dx∫−22​x−311/36​dx+∫−22​x+325/3​dx+∫−22​(x+3)2(x+6)​dx.

Problem 20015

Problem 20016

Find the angular velocity at the 2nd2^{\text{nd}}2nd instant when angular acceleration is zero for θ=8t4−4t3\theta=8t^{4}-4t^{3}θ=8t4−4t3.

Problem 20017

Find the derivative of y=2x2+5x−10y=2 x^{2}+5 x-10y=2x2+5x−10 using first principles. Verify with (K/U=6)(K / U=6)(K/U=6).

Problem 20018

Calculate the integral ∫cos⁡3(x)esin⁡(x)dx\int \cos^{3}(x) e^{\sin(x)} \mathrm{d} x∫cos3(x)esin(x)dx using substitution and integration by parts.

Problem 20019

Déterminez la fonction g(t)=∫t6tx2+1x+1dxg(t)=\int_{t}^{6t} \frac{x^{2}+1}{x+1} dxg(t)=∫t6t​x+1x2+1​dx.

Problem 20020

Find the derivatives of: a) f(x)=x2+1x−1f(x)=\frac{x^{2}+1}{x-1}f(x)=x−1x2+1​ b) g(x)=(x2+3x−5)23g(x)=\left(x^{2}+3x-5\right) \frac{2}{3}g(x)=(x2+3x−5)32​

Problem 20021

Si g(u)=∫u8cos⁡(x)dxg(u)=\int_{u}^{8} \cos (x) d xg(u)=∫u8​cos(x)dx, calculez dgdu\frac{d g}{d u}dudg​. Pour g(t)=∫e2t8cos⁡(x)dxg(t)=\int_{e^{2 t}}^{8} \cos (x) d xg(t)=∫e2t8​cos(x)dx, montrez que dgdt=1\frac{d g}{d t}=1dtdg​=1.

Problem 20022

Simplifiez f(x)=3x3+10x−4x3f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}f(x)=x33x3+10x−4​ en somme de la forme axba x^{b}axb et trouvez la primitive F(x)F(x)F(x).

Problem 20023

Avec u=cos⁡(3x)u=\cos (3 x)u=cos(3x), trouvez ∫sin⁡(3x)cos⁡5(3x)dx\int \frac{\sin (3 x)}{\cos ^{5}(3 x)} \mathrm{d} x∫cos5(3x)sin(3x)​dx en termes de uuu.

Problem 20024

Trouvez la primitive F(x)F(x)F(x) de f(x)=3x3+10x−4x3f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}f(x)=x33x3+10x−4​ avec x≠0x \neq 0x=0. Simplifiez f(x)f(x)f(x) d'abord.

Problem 20025

Problem 20026

Find the max and min of h(t)=4t−tan⁡2th(t)=4t-\tan^2 th(t)=4t−tan2t on the interval [−π4,π3][-\frac{\pi}{4}, \frac{\pi}{3}][−4π​,3π​].

Problem 20027

Calculate the integrals of the function f(x)f(x)f(x) defined by points (-2,-5), (-2,-3), (0,2), (1,-1), (3,-1), (5,-2):(a) ∫−50f(x)dx=\int_{-5}^{0} f(x) dx=∫−50​f(x)dx=(b) ∫05f(x)dx=\int_{0}^{5} f(x) dx=∫05​f(x)dx=(c) ∫−55f(x)dx=\int_{-5}^{5} f(x) dx=∫−55​f(x)dx=

Problem 20028

A 15-ft pole casts a shadow. A 6-ft man walks away at 7 ft/s. Find the shadow's tip speed when he's 30 ft from the pole.

Problem 20029

Calculez l'intégrale ∫0−π2sin⁡(x)ecos⁡(x)dx\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x∫0−2π​​sin(x)ecos(x)dx en utilisant u=cos⁡(x)u=\cos (x)u=cos(x).

Problem 20030

Find the derivative of f(x)=2x3+3x2f(x)=2 x^{3}+3 x^{2}f(x)=2x3+3x2 and where the tangent slope equals 36. (K/U=3)(K / U=3)(K/U=3)

Problem 20031

Calculez l'intégrale ∫0−π2sin⁡(x)ecos⁡(x)dx\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x∫0−2π​​sin(x)ecos(x)dx avec la substitution u=cos⁡(x)u=\cos (x)u=cos(x).

Problem 20032

Find the tangent line equation to f(x)=−x(x+1)2f(x)=\frac{-x}{(x+1)^{2}}f(x)=(x+1)2−x​ at x=−2x=-2x=−2.

Problem 20033

Find points on the curve y=2−1xy=2-\frac{1}{x}y=2−x1​ where the tangent slope is 14\frac{1}{4}41​.

Problem 20034

Calculez l'intégrale ∫0814+4xdx\int_{0}^{8} \frac{1}{\sqrt{4+4 x}} d x∫08​4+4x​1​dx et donnez sa valeur exacte.

Problem 20035

How old is a wooden artifact with 21% carbon-14 remaining? (Half-life of carbon-14 is 5730 years.) years

Problem 20036

1. What does the derivative of a function represent? 2. How can the product rule replace the quotient rule for y=(3+x)(2x−9)y=\frac{(3+x)}{(2 x-9)}y=(2x−9)(3+x)​?

Problem 20037

Find the limit as xxx approaches 0 from the right: lim⁡x→0+xesin⁡(π/x)\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}limx→0+​x​esin(π/x).

Problem 20038

Find the normal line equation to the curve y=3x(x2−1)y=\frac{3 x}{\left(x^{2}-1\right)}y=(x2−1)3x​ at x=3x=3x=3.

Problem 20039

Evaluate these limits: a) lim⁡x→−4x2+12x+32x+4\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}limx→−4​x+4x2+12x+32​ b) lim⁡x→44−12+xx−4\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}limx→4​x−44−12+x​​

Problem 20040

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=3x2+6f(x)=3x^{2}+6f(x)=3x2+6 and simplify your answer.

Problem 20041

Find the third-order Taylor polynomial for fff at 0 using f(0)=2f(0)=2f(0)=2, f′(0)=2f'(0)=2f′(0)=2, f′′(0)=4f''(0)=4f′′(0)=4, f(3)(0)=6f^{(3)}(0)=6f(3)(0)=6 and approximate f(0.1)f(0.1)f(0.1).

Problem 20042

Evaluate $\int_{1}^{0} (9 - 9x^{2}) \, dx$ using $\int_{0}^{1} x^{2} \, dx = \frac{1}{3}$ .

Calculate the average value of $f(x)=\frac{8}{x}$ over the interval $[1,9]$ .

Find $f(-8)$ given that $f^{\prime}(x)$ is continuous, $f(-5)=14$ , and $\int_{-5}^{-8} f^{\prime}(x) dx=-8$ .

Find the derivative of $f(x) = 4x^{2}-8x+9$ and use Rolle's theorem to find $c$ in $(-1,3)$ where $f^{\prime}(c)=0$ .

Bestimmen Sie die Folgenglieder $a_n$ , Häufungspunkte, $\varlimsup$ , $\sup$ , $\varliminf$ und die Konvergenz.

Find the number $c$ in $(-1,3)$ such that $f^{\prime}(c)=0$ for $f(x)=4x^{2}-8x+9$ . Solve $8c-8=0$ .

Find the average value of $f(x)=\frac{1}{x^{2}}$ on the interval $[1,9]$ .

Calculer l'intégrale $\int_{-2}^{2} \frac{x^{2}+2}{x^{3}+3 x^{2}-9 x-27} d x$ après avoir factorisé le dénominateur.

Calculez l'intégrale $\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} d x$ en trouvant $A, B, C$ dans $\frac{2 x+2}{x^{3}+6 x}=\frac{A}{x}+\frac{B x+C}{x^{2}+6}$ . Ensuite, donnez l'intégrale avec trois décimales.

Find the limit: $\lim _{x \rightarrow-\infty} \frac{\sqrt[4]{x^{4}+1}}{2 x+3}$ .

Calculate the area using the integral: $\int_{1}^{3} (-x+5) \, dx = \text{area}$ . What is the area?

Evaluate the integrals: $\int_{-2}^{2} \frac{11 / 36}{x-3} dx + \int_{-2}^{2} \frac{25 / 3}{x+3} dx + \int_{-2}^{2} \frac{(x+6)}{(x+3)^{2}} dx$ .

Find the angular velocity at the $2^{\text{nd}}$ instant when angular acceleration is zero for $\theta=8t^{4}-4t^{3}$ .

Find the derivative of $y=2 x^{2}+5 x-10$ using first principles. Verify with $(K / U=6)$ .

Calculate the integral $\int \cos^{3}(x) e^{\sin(x)} \mathrm{d} x$ using substitution and integration by parts.

Déterminez la fonction $g(t)=\int_{t}^{6t} \frac{x^{2}+1}{x+1} dx$ .

Find the derivatives of: a) $f(x)=\frac{x^{2}+1}{x-1}$ b) $g(x)=\left(x^{2}+3x-5\right) \frac{2}{3}$

Si $g(u)=\int_{u}^{8} \cos (x) d x$ , calculez $\frac{d g}{d u}$ . Pour $g(t)=\int_{e^{2 t}}^{8} \cos (x) d x$ , montrez que $\frac{d g}{d t}=1$ .

Simplifiez $f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}$ en somme de la forme $a x^{b}$ et trouvez la primitive $F(x)$ .

Avec $u=\cos (3 x)$ , trouvez $\int \frac{\sin (3 x)}{\cos ^{5}(3 x)} \mathrm{d} x$ en termes de $u$ .

Trouvez la primitive $F(x)$ de $f(x)=\frac{3 x^{3}+10 x^{-4}}{x^{3}}$ avec $x \neq 0$ . Simplifiez $f(x)$ d'abord.

Find the max and min of $h(t)=4t-\tan^2 t$ on the interval $[-\frac{\pi}{4}, \frac{\pi}{3}]$ .

Calculate the integrals of the function $f(x)$ defined by points (-2,-5), (-2,-3), (0,2), (1,-1), (3,-1), (5,-2):
(a) $\int_{-5}^{0} f(x) dx=$
(b) $\int_{0}^{5} f(x) dx=$
(c) $\int_{-5}^{5} f(x) dx=$

Calculez l'intégrale $\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x$ en utilisant $u=\cos (x)$ .

Find the derivative of $f(x)=2 x^{3}+3 x^{2}$ and where the tangent slope equals 36. $(K / U=3)$

Calculez l'intégrale $\int_{0}^{-\frac{\pi}{2}} \sin (x) e^{\cos (x)} \mathrm{d} x$ avec la substitution $u=\cos (x)$ .

Find the tangent line equation to $f(x)=\frac{-x}{(x+1)^{2}}$ at $x=-2$ .

Find points on the curve $y=2-\frac{1}{x}$ where the tangent slope is $\frac{1}{4}$ .

Calculez l'intégrale $\int_{0}^{8} \frac{1}{\sqrt{4+4 x}} d x$ et donnez sa valeur exacte.

1. What does the derivative of a function represent?
2. How can the product rule replace the quotient rule for $y=\frac{(3+x)}{(2 x-9)}$ ?

Find the limit as $x$ approaches 0 from the right: $\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}$ .

Find the normal line equation to the curve $y=\frac{3 x}{\left(x^{2}-1\right)}$ at $x=3$ .

Evaluate these limits: a) $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$ b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=3x^{2}+6$ and simplify your answer.

Find the third-order Taylor polynomial for $f$ at 0 using $f(0)=2$ , $f'(0)=2$ , $f''(0)=4$ , $f^{(3)}(0)=6$ and approximate $f(0.1)$ .

Analyze the function $y=\frac{5 x}{x^{2}-9}$ . Find intercepts, extrema, inflection points, and asymptotes.

Evaluate the limits: a) $\lim _{x \rightarrow-4} \frac{x^{2}+12 x+32}{x+4}$ , b) $\lim _{x \rightarrow 4} \frac{4-\sqrt{12+x}}{x-4}$ .

Find the third-order Taylor polynomial for $f$ at 9 using $f(9)=3$ , $f'(9)=1$ , $f''(9)=0$ , $f^{(3)}(9)=6$ . Approximate $f(8.8)$ .

Homer throws a bowling ball up; its height $h=-4.9 t^{2}+19.6 t$ . Find: a) landing time, b) impact velocity, c) peak acceleration.

Find the linear and quadratic approximating polynomials for $f(x)=-\frac{1}{x}$ at $a=1$ and use them to estimate $-\frac{1}{0.91}$ .

Find $\left.\frac{d S}{d P}\right|_{t=1}$ using $P(t)=-6 t^{2}+27 t+10$ and $S(t)=-4 t^{2}+4 t+11$ . Round to 1 decimal place.

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=3 \cos (-2 x)$ . What is $p_{1}(x)=\square$ ?

Find the average value $f_{\text {ave }}$ of $f(x)=\sqrt{4-x^{2}}$ from 0 to 2, and a $c$ in $[0,2]$ where $f(c)=f_{\text {ave }}$ .

Find the third-order Taylor polynomial $p_{3}(x)$ for $f$ at 0 using $f(0)=4, f'(0)=0, f''(0)=8, f^{(3)}(0)=12$ and approximate $f(0.2)$ .

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=3 \cos (4 x)$ . What is $p_{1}(x)=\square$ ?

Calculate $\frac{d y}{d x}$ for $P=x^{0.7} y^{0.3}$ at $x=100$ , $y=300,000$ . Interpret the result.

Find the Taylor polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ at $a=0$ for $f(x)=3 \cos (4 x)$ .

Find the second derivative of $f(x)=\frac{4 x^{2}}{x^{2}-9}$ . Identify intervals of concavity (up and down).

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=\cos(-2x)$ . What is $p_{1}(x)$ ?

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ for the equation $x^{4}+7 y^{4}=10$ .

Find the half-life of a radioactive substance with a decay rate of $8.4\%$ per day. Round to the nearest hundredth.

Find the function $g(x)$ where $g'(x)=-4096-x^3$ and the maximum value is -5. What is $g(x)$ ?

Find the Taylor polynomials $p_{1}, \ldots, p_{4}$ at $a=0$ for $f(x)=\cos(-2x)$ . Given $p_{1}(x)=1$ , find $p_{2}(x)$ .

Find Taylor polynomials $p_{1}, \ldots, p_{5}$ at $a=0$ for $f(x)=2 e^{-x}$ . What is $p_{1}(x)$ ?

Find the function $g(x)$ such that $g^{\prime}(x)=-1000-x^{3}$ and its maximum value is -7. $g(x)=\square$

Evaluate the limit: $\lim _{x \rightarrow 7}\left(\frac{\frac{1}{x}-\frac{1}{7}}{x-7}\right)=\lim _{x \rightarrow 7} \square=\square$ .

Trouvez la dérivée seconde de la fonction $f(x)=\frac{x}{x^{2}-4}$ .

Calculate the integral $L=\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{(\cos x)^{5}} d x$ .

Berechne den Differenzialquotienten von $f(x)=\frac{1}{x}$ bei $x=-2$ .

Evaluate the sum $\sum_{i=1}^{10}(i^{2}-4)$ using summation properties and verify with a graphing utility.

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{x}$ .

Calculate the sum $\sum_{k=1}^{5} \frac{1}{k^{2}+3}$ and round your answer to four decimal places. Use a graphing utility to verify.

A turkey cools from $185^{\circ}$ F to $149^{\circ}$ F in 30 min. Find its temp after 45 min and when it reaches $100^{\circ}$ F.

Find the temperature formula $F(t)$ for an object cooling from $160^{\circ} \mathrm{F}$ in $50^{\circ} \mathrm{F}$ water with $k=0.4$ .

In a community of 4000, a flu spreads as $A=\frac{4000}{1+3999 e^{-0.75 t}}$ . Find: (a) $A$ after 7 days, (b) carrying capacity, (c) days for 300 cases.