Calculus

Problem 31901

Find the slope of the tangent line for the function $y=2^{\cos x}$ at the point where $x=\frac{\pi}{2}$ .

See Solution

Problem 31902

Find the fifth derivative of $f(x)=x^{5}+2 x^{4}$ . What is it? a. else b. 0 c. 120 d. $60 x^{2}+48 x$

See Solution

Problem 31903

Find the derivative $f^{\prime}(2)$ using the limit definition for $f(x)=x^{2}+5x$ .

See Solution

Problem 31904

Find the first-order partial derivatives of $f(x, y, z)=3 x \ln \left(x^{2} y z\right)+x^{y / z}$ .

See Solution

Problem 31905

Find $\frac{d y}{d x}$ if $y=4^{x^{2}}$ and choose the correct option: a, b, c, d, or e.

See Solution

Problem 31906

Find the inflection point of the function $f(x)=x^{3}+6x^{2}+9x-1$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31907

Find $\partial z / \partial s$ and $\partial z / \partial t$ using the Chain Rule for the given functions of $s$ and $t$ .

See Solution

Problem 31908

Find $\partial z / \partial s$ and $\partial z / \partial t$ using the Chain Rule for $z=(x-y)^{5}$ , $x=s^{2} t$ , $y=s t^{2}$ .

See Solution

Problem 31909

Find $\frac{d y}{d x}$ for $y=\frac{e^{x}}{x}$ . Options: a. $\frac{e^{x}}{x^{2}}(1-x)$ b. None c. $\frac{x-1}{x^{2} e^{-x}}$ d. $\frac{x^{2}}{e^{x}}(x-1)$ e. $\frac{x^{2}}{e^{x}}(1-x)$ .

See Solution

Problem 31910

Untersuchen Sie die Funktion $f(x) = x^{4} + 4x + 3$ : Nullstellen, $f'(x)$ , Extrempunkte, $f''(x)$ , Wendepunkte, Verhalten für $x \to \pm\infty$ , Graph skizzieren.

See Solution

Problem 31911

Find the rate of change of $h=2\left(\frac{1}{\sin \theta}-1\right)$ at $\theta=30^{\circ}$ . Options: (A) $-\frac{4}{\sqrt{3}}$ , (B) $-\frac{3}{4}$ , (C) $-4 \sqrt{3}$ , (D) $-\frac{4}{3}$ .

See Solution

Problem 31912

Find the derivative $y^{\prime}$ if $y=e^{\ln x}$ . Options: a. 1, b. $e^{x}$ , c. None, d. $\ln (x)$ , e. 0.

See Solution

Problem 31913

Find the relative minimum point of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Answer as (x,y) or none.

See Solution

Problem 31914

Find $\frac{d y}{d x}$ if $y=\log _{2}(4 x^{2}-8 x)$ . Choices: a. $\frac{4 x^{2}-8 x}{8 x-8}$ b. $\frac{8 x-8}{\ln (2)(4 x^{2}-8 x)}$ c. None d. $\frac{\ln (2)(4 x^{2}-8 x)}{8 x-8}$ e. $\frac{\ln (2)(8 x-8)}{4 x^{2}-8 x}$

See Solution

Problem 31915

Find the inflection point of the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31916

Minimize the energy functional $I(u)=\int_{A} \frac{1}{2} \mu\|\nabla u\|^{2}-(u f) \mathrm{d} A$ for $u \in C^{\infty}(\bar{A})$ .
1. Describe how to find a minimizer.
2. Derive the weak formulation.
3. From the weak form, find the strong form and state the boundary value problem.
4. What Neumann boundary condition is assumed?

See Solution

Problem 31917

Find $\frac{d y}{d x}$ if $y=\ln \left(x^{6}\right)$ . Choices: a. None b. $6 x$ c. $\frac{5}{x}$ d. $\frac{6}{x}$ e. $\frac{1}{x}$

See Solution

Problem 31918

Find $\frac{d y}{d x}$ if $y=4^{x^2}$ . Options: a. $\frac{2 x \times 4^{x^2}}{\ln (4)}$ , b. $4 x^{2}$ , c. $4^{x^2} \ln (4)$ , d. None, e. $\ln (4) \times 4^{x^{2}} \times 2 x$ .

See Solution

Problem 31919

Find the primal basis for the cubic Lagrange element $\mathcal{C G}^{3}(\mu)$ on $\mu=[-1,1]$ using Gauss-Lobatto points. Then, compute the quadratic Lagrangian interpolation $\Pi_{g}^{2} u$ for $u=\log(x)$ in $x \in [1,3]$ using two bases. Which basis is better? Is $x \in [0,2]$ a better interval for approximation?

See Solution

Problem 31920

Bestimme die Ableitung der Funktion $f(x)=\frac{3}{4} \cdot x$ .

See Solution

Problem 31921

In jeder Stunde zerfallen 13\% von $20 \mathrm{~g}$ Plutonium 243. Bestimme die Zerfallsfunktion und die Halbwertszeit.

See Solution

Problem 31922

Bestimme die Zerfallsfunktion für 20 g Plutonium 243, wenn 13\% pro Stunde zerfallen, und berechne die Halbwertszeit.

See Solution

Problem 31923

Bestimmen Sie die Ableitungen der Funktionen: a) $f(x)=e^{-4 x}$ , b) $f(x)=e^{x^{2}}$ , c) $f(x)=e^{2 x+1}$ , d) $f(x)=e^{-\sqrt{x}}$ , e) $f(x)=2 \cdot e^{0.5 x}$ , f) $f(x)=x-e^{x^{3}}$ , g) $f(x)=(1-x) \cdot e^{x}$ , h) $f(x)=x^{2} \cdot e^{-x}$ , i) $f(x)=\sqrt{x} \cdot e^{x}$ , j) $f(x)=\frac{1}{e^{2 x}}$ , k) $f(x)=e^{e^{x}}$ , l) $f(x)=(x^{3}+3 x^{2}) \cdot e^{-x}$ , m) $f(x)=\frac{x^{2}}{e^{x}}$ , n) $f(x)=\sqrt{e^{x}}$ , o) $f(x)=(x^{2}+1) \cdot e^{-x}$ , p) $f(x)=(x^{2}-e^{-2 x})^{2}$ .

See Solution

Problem 31924

Find the end behavior of $f(x)=7 x^{2}+2 x^{3}+16$ by calculating $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

See Solution

Problem 31925

Find the derivative of the function $f(x)=\sqrt{3} \cdot x^{5}+\pi \cdot x^{3}-x+\sin (20)$ . What is $f'(x)$ ?

See Solution

Problem 31926

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\square$ (Simplify your answer.) B. The limit does not exist.

See Solution

Problem 31927

Find the marginal cost from the function $C(x)=178+1.9 x$ . What is $C^{\prime}(x)=\square$ ?

See Solution

Problem 31928

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x-5}{2 x-2}\right)^{4 x^{2}}$ .

See Solution

Problem 31929

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-7 x^{5}$ . What is $f^{\prime}(x)=?$

See Solution

Problem 31930

Find the marginal average cost function for $C(x)=139+6.5 x$ and $R(x)=5 x-0.07 x^{2}$ .

See Solution

Problem 31931

Find the value of the series: $\sum_{n=1}^{\infty} \frac{1}{n^{3}-8}$ .

See Solution

Problem 31932

Find the limit $L=\lim _{x \rightarrow 4} (x^{2}+6)$ and $\delta$ such that $|f(x)-L|<\varepsilon$ for $\varepsilon=0.005$ .

See Solution

Problem 31933

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\square$ or state if it does not exist.

See Solution

Problem 31934

Choose all correct answers (يمكن أن يكون هناك أكثر من إجابة صحيحة) given:
$\int_{1}^{4} f(x) dx=8, \int_{-1}^{4} f(x) dx=-8$
Then, evaluate the following statements.

See Solution

Problem 31935

Find the limits for $f(x)=\frac{x^{2}-9 x+8}{x+8}$ as $x \to -8^{-}, -8^{+}, -8$ .

See Solution

Problem 31936

If $f(x)=\sqrt{\ln x}$ , choose correct statements about its domain and behavior.

See Solution

Problem 31937

Find the derivative $s^{\prime}(x)$ of $s(x)=4x-2$ , then calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , and $s^{\prime}(3)$ .

See Solution

Problem 31938

If $f(x)$ is differentiable with an inflection point at $x=2$ , which statements are true: I. $f^{\prime \prime}(2)=0$ , II. $f^{\prime \prime}(x)$ changes sign at $x=2$ , III. $f(x)$ has an extreme value at $x=2$ ?

See Solution

Problem 31939

Find the area under the curve of $f(x) = \frac{8(\ln x)^{3}}{x}$ from $x=1$ to $x=4$ using the integral $\int_{1}^{4} \frac{8(\ln x)^{3}}{x} \, dx$ .

See Solution

Problem 31940

Find the derivative of the function $5 \sqrt{x^{3}}$ with respect to $x$ .

See Solution

Problem 31941

Calculate the integral from 16 to 25 of $2 \pi(9-x^{\frac{1}{2}})^{2} \sqrt{1+(9-x^{\frac{1}{2}})^{2} x^{-1}} \, dx$ .

See Solution

Problem 31942

Find $F'(x)$ for the following: a) $F(x)=\int_{9}^{x} \frac{1}{t} dt$ , b) $F(x)=\int_{x}^{17} \frac{1}{t} dt$ , c) $F(x)=\int_{12}^{x^{4}} \frac{1}{t} dt$ , d) $F(x)=\int_{2+\cos x}^{x^{2}+1} \frac{1}{t} dt$ .

See Solution

Problem 31943

Find the integral for the area outside $r=3+2 \sin \theta$ and inside $r=2$ .

See Solution

Problem 31944

Find $a$ such that $\int_{a}^{2a} \frac{2x^{3}-5x^{2}+4}{x^{2}} dx=0$ and show $3a^{3}+pa^{2}+q=0$ for constants $p$ and $q$ .

See Solution

Problem 31945

Calculate the sum: $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{n^{2}-2 n+1}$ .

See Solution

Problem 31946

Find the derivative of $y=3^{-x}$ . What is $y'$ ?

See Solution

Problem 31947

Find $f''(x)$ if $f(x)=\int_{0}^{x}(t^{3}+2 t^{2}+3) dt$ .

See Solution

Problem 31948

Approximate the area of region $R$ using a right Riemann sum with the values $r$ at $\theta = 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$ .

See Solution

Problem 31949

Determine the concavity of the function $f(x)=7-6 x^{2}-2 x^{3}$ .

See Solution

Problem 31950

Find the derivative of the function $f(x) = \frac{x^{2} - 2x + 1}{x + 1}$ . What is $f'(x)$ ?

See Solution

Problem 31951

Find the value of $c$ from the Mean Value Theorem for $f(x)=x^{3}-4x$ on $[0,1]$ . Options: a, b, c, d.

See Solution

Problem 31952

Find the length of the curve $x=(y+2)^{2}$ for $1 \leq y \leq 4$ . Choose the correct integral:
1. $\int_{1}^{4}\left(4 y^{2}+16 y+17\right) d y$
2. $\int_{1}^{4}(2 y+4) d y$
3. $\int_{1}^{4} \sqrt{4 y^{2}+16 y+17} d y$

See Solution

Problem 31953

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ for $f(x)=x^{2}-3x+2$ , where $x>1$ .

See Solution

Problem 31954

Evaluate the integral: $\int \frac{4 \sin y}{1-\cos y} d y=$ . Choose the correct answer.

See Solution

Problem 31955

Find the lateral surface area of the cone from revolving $y=\frac{3}{2} x, 0 \leq x \leq 5$ about the $x$ -axis. Set up the integral.

See Solution

Problem 31956

Find the derivative of $y=x^{7} \cdot x^{4}$ using the Product Rule and select the correct answer.

See Solution

Problem 31957

Find the integral of $\frac{\sin x}{\cos x}$ with the substitution $\mu=\cos x$ .

See Solution

Problem 31958

Calculate the integral $\int \sin^{2} x \cos x \, dx$ using the substitution $u=\sin x$ , $du=\cos x \, dx$ .

See Solution

Problem 31959

Evaluate the integral $\int e^{x} \tan \left(e^{x}\right) d x$ .

See Solution

Problem 31960

Given a vibrating guitar string, what statements about displacement $u(x, t)$ are always true? Consider properties of $u(a, t)$ .

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Problem 31961

Find the integral of $\frac{x}{\sqrt{x^{2}+1}}$ with respect to $x$ .

See Solution

Problem 31962

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{100}{x^{3}}$ .

See Solution

Problem 31963

Differentiate the function: $y=4 x^{3}-14 x^{2}+32 x+7$ . Find $\frac{d y}{d x}$ .

See Solution

Problem 31964

Differentiate $y=10 \sqrt{x}$ . What is $\frac{d}{d x}(10 \sqrt{x})$ ?

See Solution

Problem 31965

Find the derivative of the function $y=7 x^{-\frac{3}{2}}+4 x^{-\frac{1}{2}}+x^{3}-5$ . What is $y^{\prime}=?$

See Solution

Problem 31966

Differentiate: $y=\frac{x}{9}-\frac{9}{x}$ , find $\frac{d}{d x}\left(\frac{x}{9}-\frac{9}{x}\right)=\square$ .

See Solution

Problem 31967

Calculate the integral $\int x \sqrt{4 x+1} \, dx$ .

See Solution

Problem 31968

Find the point(s) where the tangent line of $y=-0.025 x^{2}+9 x$ has a slope of 5. The point(s) is/are $\square$ .

See Solution

Problem 31969

Find the integral: $\int \frac{1}{x \ln^{2} x} \, dx$

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Problem 31970

Differentiate $y=\frac{x}{9}-\frac{9}{x}$ and find $\frac{d}{d x}\left(\frac{x}{9}-\frac{9}{x}\right)$ .

See Solution

Problem 31971

Differentiate the function $y=\frac{5 x^{2}-4}{8 x^{3}+9}$ . Find $y^{\prime}=\square$ .

See Solution

Problem 31972

Find the average revenue change rate for $R(x)=89 \sqrt{x}$ when $x=405$ jackets are produced.

See Solution

Problem 31973

Find the derivative $f^{\prime}(1)$ for the function $f(x)=4 x^{2}-6 x+3$ . Simplify your answer.

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Problem 31974

Differentiate the function $F(x)=(x^{3}-8x)^{2}$ . Find $F^{\prime}(x)=$

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Problem 31975

Given the equation $R(v)=\frac{6800}{v}$ , find $R'(v)$ , $R(60)$ , and $R'(60)$ .

See Solution

Problem 31976

Solve the equation $\frac{d^{2} y}{d x^{2}}=e^{x}$ with initial conditions $y(0)=2$ and $y(1)=e$ .

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Problem 31977

Given the equation $R(v)=\frac{6800}{v}$ , find:
a) $R^{\prime}(v)$ , the rate of change of heart rate with respect to $v$ .
b) Heart rate at $v=60 \mathrm{~mL}$ .
c) Rate of change at $v=60 \mathrm{~mL}$ .

See Solution

Problem 31978

Find the integral: $\int x^{2} \sqrt{x+1} \, dx$

See Solution

Problem 31979

Differentiate the function $y=\sqrt{x^{2}+2}$ . Find $\frac{d y}{d x}=\square$ .

See Solution

Problem 31980

Find the derivative of the inverse function $f^{-1}$ at $x=6$ for $f(x)=x^{3}-2$ . Choices: $\frac{1}{12}$ , 12, $\frac{1}{6}$ , $\frac{-1}{12}$ , $-12$ .

See Solution

Problem 31981

Differentiate the function $f(x)=\left(\frac{x-5}{x-2}\right)^{6}$ . Find $f^{\prime}(x)=\square$ .

See Solution

Problem 31982

Calculate the integral $\int_{2}^{16} \frac{d x}{x \sqrt{\ln x}}$ . Choose the correct answer from the options provided.

See Solution

Problem 31983

Calculate the integral: $\int \frac{3 e^{2 x}}{\sin e^{2 x}} d x$ .

See Solution

Problem 31984

Find the rate of change of the particle's position given $s=\sqrt{20+4t}$ at $t=4$ sec. Answer: $\square \mathrm{m/sec}$ .

See Solution

Problem 31985

Find the derivative of $\ln y = \ln [t(t-1)(t-2)]$ . Choose the correct option from the list.

See Solution

Problem 31986

Find the derivative of the consumer credit model $C(x)=9.24 x^{4}-85.45 x^{3}+287.04 x^{2}-309.05 x+2651.7$ .

See Solution

Problem 31987

Differentiate the function $f(x)=(2x^{6}-3x^{5}+5)^{25}$ . Find $f'(x)=$ .

See Solution

Problem 31988

Evaluate the integral: $\int \frac{3 x^{4}}{1+x^{10}} d x$

See Solution

Problem 31989

Model consumer credit with $C(x)=9.24 x^{4}-85.45 x^{3}+287.04 x^{2}-309.05 x+2651.7$ . Find rates of change for 2008 and 2015.

See Solution

Problem 31990

Calculate the integral: $I = \int \frac{4 x^{3}+7 x+5}{x^{2}+1} d x$

See Solution

Problem 31991

Find the local maxima and minima of the function $f(x) = \log x + \sin x + 4$ for positive $x$ .

See Solution

Problem 31992

Check if the series $\sum_{n=1}^{\infty} \frac{5-3 n}{n^{3}}$ converges or diverges. Options: a. divergent b. convergent.

See Solution

Problem 31993

Determine if the series $\sum_{n=1}^{\infty} \frac{6}{n(n+3)}$ converges or diverges and find its value.

See Solution

Problem 31994

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3 n^{2}+2 n}{4+n+n^{2}}$ using tests.

See Solution

Problem 31995

Find the sum of the series $\sum_{n=1}^{\infty} \left( \frac{3}{n^{2}} - \frac{3}{(n+1)^{2}} \right)$ . Options: a. 1 b. 3 c. 0 d. diverges

See Solution

Problem 31996

Determine if the series $\sum_{n=1}^{\infty} \frac{3^{n-1}}{5^{n}}$ converges and find its sum. Options: a. $\frac{5}{7}$ , b. diverges, c. $\frac{5}{6}$ , d. $\frac{14}{29}$ , e. $\frac{1}{2}$ .

See Solution

Problem 31997

1. Find the sum of the series $\sum_{n=1}^{\infty} \frac{n+2}{3^{n}}$ .
2. Find the sum of the series $\sum_{n=1}^{\infty} \frac{2^{n+1}}{n 3^{n-1}}$ .

See Solution

Problem 31998

Calculer l'intégrale suivante : $\int \sqrt{3+2 x} \, dx$

See Solution

Problem 31999

Finn nullpunktene til $f(x)=(2 \sin x+3)^{3}$ for $x \in[0,2 \pi]$ . Regn ut $f\left(\frac{\pi}{6}\right)$ og $f^{\prime}\left(\frac{\pi}{6}\right)$ .

See Solution

Problem 32000

Find the interval(s) where the curve $y=\int_{0}^{x} \frac{t^{2}}{t^{2}+t+4} dt$ is concave downward.

See Solution

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Calculus

Problem 31901

Find the slope of the tangent line for the function y=2cos⁡xy=2^{\cos x}y=2cosx at the point where x=π2x=\frac{\pi}{2}x=2π​.

Problem 31902

Find the fifth derivative of f(x)=x5+2x4f(x)=x^{5}+2 x^{4}f(x)=x5+2x4. What is it? a. else b. 0 c. 120 d. 60x2+48x60 x^{2}+48 x60x2+48x

Problem 31903

Find the derivative f′(2)f^{\prime}(2)f′(2) using the limit definition for f(x)=x2+5xf(x)=x^{2}+5xf(x)=x2+5x.

Problem 31904

Find the first-order partial derivatives of f(x,y,z)=3xln⁡(x2yz)+xy/zf(x, y, z)=3 x \ln \left(x^{2} y z\right)+x^{y / z}f(x,y,z)=3xln(x2yz)+xy/z.

Problem 31905

Find dydx\frac{d y}{d x}dxdy​ if y=4x2y=4^{x^{2}}y=4x2 and choose the correct option: a, b, c, d, or e.

Problem 31906

Find the inflection point of the function f(x)=x3+6x2+9x−1f(x)=x^{3}+6x^{2}+9x-1f(x)=x3+6x2+9x−1 using f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x).

Problem 31907

Find ∂z/∂s\partial z / \partial s∂z/∂s and ∂z/∂t\partial z / \partial t∂z/∂t using the Chain Rule for the given functions of sss and ttt.

Problem 31908

Find ∂z/∂s\partial z / \partial s∂z/∂s and ∂z/∂t\partial z / \partial t∂z/∂t using the Chain Rule for z=(x−y)5z=(x-y)^{5}z=(x−y)5, x=s2tx=s^{2} tx=s2t, y=st2y=s t^{2}y=st2.

Problem 31909

Find dydx\frac{d y}{d x}dxdy​ for y=exxy=\frac{e^{x}}{x}y=xex​. Options: a. exx2(1−x)\frac{e^{x}}{x^{2}}(1-x)x2ex​(1−x) b. None c. x−1x2e−x\frac{x-1}{x^{2} e^{-x}}x2e−xx−1​ d. x2ex(x−1)\frac{x^{2}}{e^{x}}(x-1)exx2​(x−1) e. x2ex(1−x)\frac{x^{2}}{e^{x}}(1-x)exx2​(1−x).

Problem 31910

Untersuchen Sie die Funktion f(x)=x4+4x+3f(x) = x^{4} + 4x + 3f(x)=x4+4x+3: Nullstellen, f′(x)f'(x)f′(x), Extrempunkte, f′′(x)f''(x)f′′(x), Wendepunkte, Verhalten für x→±∞x \to \pm\inftyx→±∞, Graph skizzieren.

Problem 31911

Find the rate of change of h=2(1sin⁡θ−1)h=2\left(\frac{1}{\sin \theta}-1\right)h=2(sinθ1​−1) at θ=30∘\theta=30^{\circ}θ=30∘. Options: (A) −43-\frac{4}{\sqrt{3}}−3​4​, (B) −34-\frac{3}{4}−43​, (C) −43-4 \sqrt{3}−43​, (D) −43-\frac{4}{3}−34​.

Problem 31912

Find the derivative y′y^{\prime}y′ if y=eln⁡xy=e^{\ln x}y=elnx. Options: a. 1, b. exe^{x}ex, c. None, d. ln⁡(x)\ln (x)ln(x), e. 0.

Problem 31913

Find the relative minimum point of f(x)=13x3−2x2+3x+1f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1f(x)=31​x3−2x2+3x+1. Answer as (x,y) or none.

Problem 31914

Problem 31915

Find the inflection point of the function f(x)=x2−2x2−1f(x)=\frac{x^{2}-2}{x^{2}-1}f(x)=x2−1x2−2​ using f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x).

Problem 31916

Problem 31917

Find dydx\frac{d y}{d x}dxdy​ if y=ln⁡(x6)y=\ln \left(x^{6}\right)y=ln(x6). Choices: a. None b. 6x6 x6x c. 5x\frac{5}{x}x5​ d. 6x\frac{6}{x}x6​ e. 1x\frac{1}{x}x1​

Problem 31918

Find dydx\frac{d y}{d x}dxdy​ if y=4x2y=4^{x^2}y=4x2. Options: a. 2x×4x2ln⁡(4)\frac{2 x \times 4^{x^2}}{\ln (4)}ln(4)2x×4x2​, b. 4x24 x^{2}4x2, c. 4x2ln⁡(4)4^{x^2} \ln (4)4x2ln(4), d. None, e. ln⁡(4)×4x2×2x\ln (4) \times 4^{x^{2}} \times 2 xln(4)×4x2×2x.

Problem 31919

Problem 31920

Bestimme die Ableitung der Funktion f(x)=34⋅xf(x)=\frac{3}{4} \cdot xf(x)=43​⋅x.

Problem 31921

In jeder Stunde zerfallen 13\% von 20 g20 \mathrm{~g}20 g Plutonium 243. Bestimme die Zerfallsfunktion und die Halbwertszeit.

Problem 31922

Bestimme die Zerfallsfunktion für 20 g Plutonium 243, wenn 13\% pro Stunde zerfallen, und berechne die Halbwertszeit.

Problem 31923

Problem 31924

Find the end behavior of f(x)=7x2+2x3+16f(x)=7 x^{2}+2 x^{3}+16f(x)=7x2+2x3+16 by calculating lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow-\infty} f(x)limx→−∞​f(x).

Problem 31925

Find the derivative of the function f(x)=3⋅x5+π⋅x3−x+sin⁡(20)f(x)=\sqrt{3} \cdot x^{5}+\pi \cdot x^{3}-x+\sin (20)f(x)=3​⋅x5+π⋅x3−x+sin(20). What is f′(x)f'(x)f′(x)?

Problem 31926

Find the limit: lim⁡x→−3x2−43−x=□\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\squarelimx→−3​3−xx2−4​=□ (Simplify your answer.) B. The limit does not exist.

Problem 31927

Find the marginal cost from the function C(x)=178+1.9xC(x)=178+1.9 xC(x)=178+1.9x. What is C′(x)=□C^{\prime}(x)=\squareC′(x)=□?

Problem 31928

Calculate the limit: lim⁡x→∞(2x−52x−2)4x2\lim _{x \rightarrow \infty}\left(\frac{2 x-5}{2 x-2}\right)^{4 x^{2}}limx→∞​(2x−22x−5​)4x2.

Problem 31929

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=−7x5f(x)=-7 x^{5}f(x)=−7x5. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 31930

Find the marginal average cost function for C(x)=139+6.5xC(x)=139+6.5 xC(x)=139+6.5x and R(x)=5x−0.07x2R(x)=5 x-0.07 x^{2}R(x)=5x−0.07x2.

Problem 31931

Find the value of the series: ∑n=1∞1n3−8\sum_{n=1}^{\infty} \frac{1}{n^{3}-8}n=1∑∞​n3−81​.

Problem 31932

Find the limit L=lim⁡x→4(x2+6)L=\lim _{x \rightarrow 4} (x^{2}+6)L=limx→4​(x2+6) and δ\deltaδ such that ∣f(x)−L∣<ε|f(x)-L|<\varepsilon∣f(x)−L∣<ε for ε=0.005\varepsilon=0.005ε=0.005.

Problem 31933

Find the limit: lim⁡x→−3x2−43−x=□\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\squarelimx→−3​3−xx2−4​=□ or state if it does not exist.

Problem 31934

Choose all correct answers (يمكن أن يكون هناك أكثر من إجابة صحيحة) given: ∫14f(x)dx=8,∫−14f(x)dx=−8 \int_{1}^{4} f(x) dx=8, \int_{-1}^{4} f(x) dx=-8 ∫14​f(x)dx=8,∫−14​f(x)dx=−8 Then, evaluate the following statements.

Problem 31935

Find the limits for f(x)=x2−9x+8x+8f(x)=\frac{x^{2}-9 x+8}{x+8}f(x)=x+8x2−9x+8​ as x→−8−,−8+,−8x \to -8^{-}, -8^{+}, -8x→−8−,−8+,−8.

Problem 31936

If f(x)=ln⁡xf(x)=\sqrt{\ln x}f(x)=lnx​, choose correct statements about its domain and behavior.

Problem 31937

Find the derivative s′(x)s^{\prime}(x)s′(x) of s(x)=4x−2s(x)=4x-2s(x)=4x−2, then calculate s′(1)s^{\prime}(1)s′(1), s′(2)s^{\prime}(2)s′(2), and s′(3)s^{\prime}(3)s′(3).

Problem 31938

If f(x)f(x)f(x) is differentiable with an inflection point at x=2x=2x=2, which statements are true: I. f′′(2)=0f^{\prime \prime}(2)=0f′′(2)=0, II. f′′(x)f^{\prime \prime}(x)f′′(x) changes sign at x=2x=2x=2, III. f(x)f(x)f(x) has an extreme value at x=2x=2x=2?

Problem 31939

Find the area under the curve of f(x)=8(ln⁡x)3xf(x) = \frac{8(\ln x)^{3}}{x}f(x)=x8(lnx)3​ from x=1x=1x=1 to x=4x=4x=4 using the integral ∫148(ln⁡x)3x dx\int_{1}^{4} \frac{8(\ln x)^{3}}{x} \, dx∫14​x8(lnx)3​dx.

Problem 31940

Find the derivative of the function 5x35 \sqrt{x^{3}}5x3​ with respect to xxx.

Problem 31941

Calculate the integral from 16 to 25 of 2π(9−x12)21+(9−x12)2x−1 dx2 \pi(9-x^{\frac{1}{2}})^{2} \sqrt{1+(9-x^{\frac{1}{2}})^{2} x^{-1}} \, dx2π(9−x21​)21+(9−x21​)2x−1​dx.

Problem 31942

Find the slope of the tangent line for the function $y=2^{\cos x}$ at the point where $x=\frac{\pi}{2}$ .

Find the fifth derivative of $f(x)=x^{5}+2 x^{4}$ . What is it? a. else b. 0 c. 120 d. $60 x^{2}+48 x$

Find the derivative $f^{\prime}(2)$ using the limit definition for $f(x)=x^{2}+5x$ .

Find the first-order partial derivatives of $f(x, y, z)=3 x \ln \left(x^{2} y z\right)+x^{y / z}$ .

Find $\frac{d y}{d x}$ if $y=4^{x^{2}}$ and choose the correct option: a, b, c, d, or e.

Find the inflection point of the function $f(x)=x^{3}+6x^{2}+9x-1$ using $f'(x)$ and $f''(x)$ .

Find $\partial z / \partial s$ and $\partial z / \partial t$ using the Chain Rule for the given functions of $s$ and $t$ .

Find $\partial z / \partial s$ and $\partial z / \partial t$ using the Chain Rule for $z=(x-y)^{5}$ , $x=s^{2} t$ , $y=s t^{2}$ .

Find $\frac{d y}{d x}$ for $y=\frac{e^{x}}{x}$ . Options: a. $\frac{e^{x}}{x^{2}}(1-x)$ b. None c. $\frac{x-1}{x^{2} e^{-x}}$ d. $\frac{x^{2}}{e^{x}}(x-1)$ e. $\frac{x^{2}}{e^{x}}(1-x)$ .

Untersuchen Sie die Funktion $f(x) = x^{4} + 4x + 3$ : Nullstellen, $f'(x)$ , Extrempunkte, $f''(x)$ , Wendepunkte, Verhalten für $x \to \pm\infty$ , Graph skizzieren.

Find the rate of change of $h=2\left(\frac{1}{\sin \theta}-1\right)$ at $\theta=30^{\circ}$ . Options: (A) $-\frac{4}{\sqrt{3}}$ , (B) $-\frac{3}{4}$ , (C) $-4 \sqrt{3}$ , (D) $-\frac{4}{3}$ .

Find the derivative $y^{\prime}$ if $y=e^{\ln x}$ . Options: a. 1, b. $e^{x}$ , c. None, d. $\ln (x)$ , e. 0.

Find the relative minimum point of $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ . Answer as (x,y) or none.

Find the inflection point of the function $f(x)=\frac{x^{2}-2}{x^{2}-1}$ using $f'(x)$ and $f''(x)$ .

Find $\frac{d y}{d x}$ if $y=\ln \left(x^{6}\right)$ . Choices: a. None b. $6 x$ c. $\frac{5}{x}$ d. $\frac{6}{x}$ e. $\frac{1}{x}$

Find $\frac{d y}{d x}$ if $y=4^{x^2}$ . Options: a. $\frac{2 x \times 4^{x^2}}{\ln (4)}$ , b. $4 x^{2}$ , c. $4^{x^2} \ln (4)$ , d. None, e. $\ln (4) \times 4^{x^{2}} \times 2 x$ .

Bestimme die Ableitung der Funktion $f(x)=\frac{3}{4} \cdot x$ .

In jeder Stunde zerfallen 13\% von $20 \mathrm{~g}$ Plutonium 243. Bestimme die Zerfallsfunktion und die Halbwertszeit.

Find the end behavior of $f(x)=7 x^{2}+2 x^{3}+16$ by calculating $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ .

Find the derivative of the function $f(x)=\sqrt{3} \cdot x^{5}+\pi \cdot x^{3}-x+\sin (20)$ . What is $f'(x)$ ?

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\square$ (Simplify your answer.) B. The limit does not exist.

Find the marginal cost from the function $C(x)=178+1.9 x$ . What is $C^{\prime}(x)=\square$ ?

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\frac{2 x-5}{2 x-2}\right)^{4 x^{2}}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=-7 x^{5}$ . What is $f^{\prime}(x)=?$

Find the marginal average cost function for $C(x)=139+6.5 x$ and $R(x)=5 x-0.07 x^{2}$ .

Find the value of the series: $\sum_{n=1}^{\infty} \frac{1}{n^{3}-8}$ .

Find the limit $L=\lim _{x \rightarrow 4} (x^{2}+6)$ and $\delta$ such that $|f(x)-L|<\varepsilon$ for $\varepsilon=0.005$ .

Find the limit: $\lim _{x \rightarrow-3} \frac{x^{2}-4}{3-x}=\square$ or state if it does not exist.

Choose all correct answers (يمكن أن يكون هناك أكثر من إجابة صحيحة) given:
$\int_{1}^{4} f(x) dx=8, \int_{-1}^{4} f(x) dx=-8$
Then, evaluate the following statements.

Find the limits for $f(x)=\frac{x^{2}-9 x+8}{x+8}$ as $x \to -8^{-}, -8^{+}, -8$ .

If $f(x)=\sqrt{\ln x}$ , choose correct statements about its domain and behavior.

Find the derivative $s^{\prime}(x)$ of $s(x)=4x-2$ , then calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , and $s^{\prime}(3)$ .

If $f(x)$ is differentiable with an inflection point at $x=2$ , which statements are true: I. $f^{\prime \prime}(2)=0$ , II. $f^{\prime \prime}(x)$ changes sign at $x=2$ , III. $f(x)$ has an extreme value at $x=2$ ?

Find the area under the curve of $f(x) = \frac{8(\ln x)^{3}}{x}$ from $x=1$ to $x=4$ using the integral $\int_{1}^{4} \frac{8(\ln x)^{3}}{x} \, dx$ .

Find the derivative of the function $5 \sqrt{x^{3}}$ with respect to $x$ .

Calculate the integral from 16 to 25 of $2 \pi(9-x^{\frac{1}{2}})^{2} \sqrt{1+(9-x^{\frac{1}{2}})^{2} x^{-1}} \, dx$ .

Find the integral for the area outside $r=3+2 \sin \theta$ and inside $r=2$ .

Find $a$ such that $\int_{a}^{2a} \frac{2x^{3}-5x^{2}+4}{x^{2}} dx=0$ and show $3a^{3}+pa^{2}+q=0$ for constants $p$ and $q$ .

Calculate the sum: $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{n^{2}-2 n+1}$ .

Find the derivative of $y=3^{-x}$ . What is $y'$ ?

Find $f''(x)$ if $f(x)=\int_{0}^{x}(t^{3}+2 t^{2}+3) dt$ .

Approximate the area of region $R$ using a right Riemann sum with the values $r$ at $\theta = 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1$ .

Determine the concavity of the function $f(x)=7-6 x^{2}-2 x^{3}$ .

Find the derivative of the function $f(x) = \frac{x^{2} - 2x + 1}{x + 1}$ . What is $f'(x)$ ?

Find the value of $c$ from the Mean Value Theorem for $f(x)=x^{3}-4x$ on $[0,1]$ . Options: a, b, c, d.

Find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ for $f(x)=x^{2}-3x+2$ , where $x>1$ .

Evaluate the integral: $\int \frac{4 \sin y}{1-\cos y} d y=$ . Choose the correct answer.

Find the lateral surface area of the cone from revolving $y=\frac{3}{2} x, 0 \leq x \leq 5$ about the $x$ -axis. Set up the integral.

Find the derivative of $y=x^{7} \cdot x^{4}$ using the Product Rule and select the correct answer.

Find the integral of $\frac{\sin x}{\cos x}$ with the substitution $\mu=\cos x$ .

Calculate the integral $\int \sin^{2} x \cos x \, dx$ using the substitution $u=\sin x$ , $du=\cos x \, dx$ .

Evaluate the integral $\int e^{x} \tan \left(e^{x}\right) d x$ .

Given a vibrating guitar string, what statements about displacement $u(x, t)$ are always true? Consider properties of $u(a, t)$ .

Find the integral of $\frac{x}{\sqrt{x^{2}+1}}$ with respect to $x$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{100}{x^{3}}$ .

Differentiate the function: $y=4 x^{3}-14 x^{2}+32 x+7$ . Find $\frac{d y}{d x}$ .

Differentiate $y=10 \sqrt{x}$ . What is $\frac{d}{d x}(10 \sqrt{x})$ ?

Find the derivative of the function $y=7 x^{-\frac{3}{2}}+4 x^{-\frac{1}{2}}+x^{3}-5$ . What is $y^{\prime}=?$

Differentiate: $y=\frac{x}{9}-\frac{9}{x}$ , find $\frac{d}{d x}\left(\frac{x}{9}-\frac{9}{x}\right)=\square$ .

Calculate the integral $\int x \sqrt{4 x+1} \, dx$ .

Find the point(s) where the tangent line of $y=-0.025 x^{2}+9 x$ has a slope of 5. The point(s) is/are $\square$ .

Find the integral: $\int \frac{1}{x \ln^{2} x} \, dx$

Differentiate $y=\frac{x}{9}-\frac{9}{x}$ and find $\frac{d}{d x}\left(\frac{x}{9}-\frac{9}{x}\right)$ .

Differentiate the function $y=\frac{5 x^{2}-4}{8 x^{3}+9}$ . Find $y^{\prime}=\square$ .

Find the average revenue change rate for $R(x)=89 \sqrt{x}$ when $x=405$ jackets are produced.

Find the derivative $f^{\prime}(1)$ for the function $f(x)=4 x^{2}-6 x+3$ . Simplify your answer.

Differentiate the function $F(x)=(x^{3}-8x)^{2}$ . Find $F^{\prime}(x)=$