Calculus

Problem 31801

Find the limits: a. $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{\left|4 y^{3}-y^{2}\right|}$ b. $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}$

See Solution

Problem 31802

Find the limit: $\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}$ .

See Solution

Problem 31803

Find the antiderivative $F(x)$ of $f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}$ such that $F(1)=0$ .

See Solution

Problem 31804

Evaluate the integral from 5 to 25 of $\frac{1}{x(\ln x)^{2}}$ dx. What is the result?

See Solution

Problem 31805

Find the limit: $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{|4 y^{3}-y^{2}|}$

See Solution

Problem 31806

Find the antiderivative $F(x)$ of $f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}$ with $F(1)=0$ . What is $F(x)$ ?

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Problem 31807

For the function $f(x)=-4 x^{2}$ , find slopes of secant lines and predict the tangent slope at $x=3$ .

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Problem 31808

Find $f(2)$ if $f^{\prime \prime}(x)=10 x+4 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=3$ .

See Solution

Problem 31809

Find the slope of the secant line for $f(x) = 17 \cos x$ on $[\frac{\pi}{2}, \pi]$ and conjecture the tangent slope.

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Problem 31810

A 10 m wire with radius changing from $5 \times 10^{-4}$ m to $9.8 \times 10^{-4}$ m holds a $3.14$ kg load. Find its length increase using $Y=2 \times 10^{11} \mathrm{Nm}^{-2}$ .

See Solution

Problem 31811

For the function $f(x)=-4 x^{2}$ , find slopes of secant lines for intervals near $x=3$ and conjecture the tangent slope.

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Problem 31812

For $f(x)=-3x^{2}$ , find slopes of secant lines for intervals and predict the tangent slope at $x=3$ .

See Solution

Problem 31813

A ball is thrown with an initial velocity of $52 \mathrm{ft/s}$ . Its height is $y=52t-16t^{2}$ .
(a) Find average velocity from $t=2$ for: (i) 0.5 s (ii) 0.1 s (iii) 0.05 s (iv) 0.01 s
(b) Estimate instantaneous velocity at $t=2$ .

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Problem 31814

Given $f^{\prime \prime}(x)=-2 \cos (x)$ , find $f^{\prime}(x)$ and $f(x)$ using constants C and D.

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Problem 31815

Find the function $f(t)$ given that $f^{\prime \prime}(t)=2 e^{t}+3 \sin (t)$ , $f(0)=-9$ , and $f(\pi)=9$ .

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Problem 31816

Calculate the integral $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ and select the correct answer from the options.

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Problem 31817

Find the integral of $\frac{1}{1+e^{x}}$ . Choose the correct answer from the options given.

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Problem 31818

Find the slope of the secant line for $f(x) = 17 \cos x$ between $x = \frac{\pi}{2}$ and $x = \pi$ .

See Solution

Problem 31819

Find the derivative of $y=2^{x}$ . Options: $y' = 2^{x} \ln 2$ , $y' = 2^{x} \ln x$ , $y' = \frac{2^{x}}{\ln 2}$ , $y' = 2^{-} x \ln 2$ , None.

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Problem 31820

Find the derivative of $y=\frac{t}{\ln t}$ . Choose the correct option from the list.

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Problem 31821

Find the derivative of $y=t \cdot \ln t$ . Choose the correct answer from the options provided.

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Problem 31822

Find the derivative of $x e^{x} + e^{x}$ . What is the result?

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Problem 31823

Find the derivative of $y=\frac{\ln t}{t}$ . Choose the correct option from the list.

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Problem 31824

Find the derivative of $x e^{x} - e^{x}$ . What is it?

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Problem 31825

Find the derivative of $-x e^{x}+e^{x}$ . What is it?

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Problem 31826

Evaluate the integral from 5 to 25: $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

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Problem 31827

Find the derivative of $y=\frac{\ln t}{t}$ .

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Problem 31828

Simplify the energy functional, find the weak form, derive the boundary value problem, and interpret the PDE and conditions.

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Problem 31829

Find the integral of $\frac{2}{1+e^{x}}$ . Choose the correct answer from the options provided.

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Problem 31830

Evaluate the integral $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ and choose the correct answer from the options.

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Problem 31831

Find the derivative of $-x e^{x}+e^{x}$ . What is it?

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Problem 31832

Find the integral of $\frac{2 e^{x}}{1+e^{x}}$ . Choose the correct answer from the options given.

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Problem 31833

Find the derivative of $y=3^{x}$ . Options: $y'=\frac{3^{x}}{\ln 3}$ , $y'=3^{x} \ln$ , $y'=3^{x} \ln x$ , None, $y'=3^{-x} \ln 3$ .

See Solution

Problem 31834

Find the integral of $\frac{1}{1+e^{x}}$ . Choose the correct answer from the options provided.

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Problem 31835

Evaluate the integral from 3 to 9 of $\frac{1}{x(\ln x)^{2}} \, dx$ . What is the result?

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Problem 31836

Find the derivative of $y=3^{-x}$ . Choices: $y'=-3^{-x} \ln 3$ , $y'=\frac{3^{x}}{\ln 3}$ , etc.

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Problem 31837

Evaluate the integral $\int \frac{1}{1+e^{x}}$ and choose the correct answer from the options given.

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Problem 31838

Find the integral of $\frac{2 e^{x}}{1+e^{x}}$ . What is the result?

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Problem 31839

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}$ .

See Solution

Problem 31840

Find the limit as $x$ approaches -4 for $3x^3 - 2x^2 + 3x + 8$ or state if it doesn't exist.

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Problem 31841

Find the derivative of $y=2^{-x}$ . What is $y^{\prime}$ ? Options: $-2^{-x} \ln 2$ , $\frac{2^{x}}{\ln 2}$ , $2^{x} \ln x$ , None, $2^{-x} \ln 2$ .

See Solution

Problem 31842

Calculate the integral from 5 to 25 of $\frac{1}{x(\ln x)^{2}} \, dx$ . Options include $\frac{1}{\ln 2}$ , $\ln 25$ , etc.

See Solution

Problem 31843

Find the derivative of $y=3^{-x}$ . Choose the correct option.

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Problem 31844

Find the limit: $\lim _{x \rightarrow 4} \frac{-4 x}{\sqrt{3 x-8}}$ . Choose A for the limit value or B if it does not exist.

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Problem 31845

Evaluate the integral from 5 to 25: $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

See Solution

Problem 31846

Find the integral of $\frac{e^{x}}{1+e^{x}}$ . Choose the correct answer from the options provided.

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Problem 31847

Calculez les intégrales $I=\int_{0}^{\ln 2}\left(\frac{\mathrm{e}^{2 x}-\mathrm{e}^{-x}}{2}\right) \mathrm{d} x$ et $J=\int_{0}^{1}\left(\sin (5 x)-\frac{1}{x+1}\right) \mathrm{d} x$ . Trouvez une primitive $F$ de $f(x)=\frac{1+\sqrt{x}+x}{x}$ .

See Solution

Problem 31848

Evaluate the integral from 2 to 4: $\int_{2}^{4} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

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Problem 31849

Find the derivative of $y = t \cdot \ln(t)$ . Choose the correct option from the list.

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Problem 31850

Find the derivative of $x e^{x} - e^{x}$ . What is the result?

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Problem 31851

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{7+h}-\frac{1}{7}}{h}$ and simplify your answer.

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Problem 31852

Find the limit: $\lim _{x \rightarrow 2} \frac{4 x^{2}+7 x+6}{8 x-4}$ or state if it doesn't exist.

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Problem 31853

Find the limit: $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{|4 y^{3}-y^{2}|}$ .

See Solution

Problem 31854

Find the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$ . Simplify it to $\lim _{x \rightarrow-5} \square$ .

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Problem 31855

Find the limit: $\lim _{h \rightarrow 0} \frac{(12+h)^{2}-144}{h}$ and simplify your answer.

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Problem 31856

Find the limit: $\lim _{h \rightarrow 0} \frac{4}{\sqrt{36+5 h}+9}$ . A: Simplify your answer; B: Limit does not exist.

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Problem 31857

Find the limit: $\lim _{x \rightarrow 169} \frac{\sqrt{x}-13}{x-169}$ or state if it doesn't exist.

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Problem 31858

Find the limit: $\lim _{t \rightarrow 5}\left(t^{2}+8 t+4\right)$ . A. Simplify your answer. B. It doesn't exist.

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Problem 31859

Find the limit as $x$ approaches 5 for $2x - 8$ . Is it A. $2x - 8 = \square$ or B. limit does not exist?

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Problem 31860

Evaluate the integral $\int_{3}^{9} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

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Problem 31861

Find the limit as $x$ approaches 6 for the constant function 8. What is the result? A. $\lim _{x \rightarrow 6} 8=\square$ B. Limit does not exist.

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Problem 31862

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-3 x-4}{x-4}$ . Simplify it to $\lim _{x \rightarrow 4}(\square)$ .

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Problem 31863

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{256+h}-16}{h}$ or state if it doesn't exist.

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Problem 31864

Given the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+9, & x<-9 \\ \sqrt{x+9}, & x \geq-9\end{array}\right.$ , find the limits as $x$ approaches -9 from the left, right, and overall.

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Problem 31865

Find the average velocity of the object given by $s(t)=-7 t^{2}+28 t$ over these intervals: (a) $[1,6]$ , (b) $[1,5]$ , (c) $[1,4]$ , (d) $[1,1+h]$ where $h>0$ .

See Solution

Problem 31866

Given $f(x)=\left\{\begin{array}{ll}x^{2}+9, & x<-9 \\ \sqrt{x+9}, & x \geq-9\end{array}\right.$ , find the limits as $x$ approaches -9.

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Problem 31867

Find average velocities for $s(t)=-16 t^{2}+106 t$ over intervals $[1,2]$ , $[1,1.5]$ , $[1,1.1]$ , $[1,1.01]$ , $[1,1.001]$ . What is the instantaneous velocity at $t=1$ ?

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Problem 31868

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

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Problem 31869

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

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Problem 31870

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

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Problem 31871

Evaluate the integral from 2 to 16: $\int_{2}^{16} \frac{d x}{2 x \sqrt{\ln x}}$ . Choose the correct answer.

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Problem 31872

5 (4.5pts) Plate equations 5.1 (2.5pts) Airy stress function
Given plane stress $\sigma_{i 3}=0$ and $f=0$ , solve the biharmonic equation $\Delta \Delta \lambda=0$ for $\lambda$ in $A \subseteq \mathbb{R}^{2}$ .
1. (1.5pts) Why is $\lambda$ and the biharmonic equation a suitable simplification?
2. (1pts) Identify valid solutions for the biharmonic problem from the options: - 15 - $\ln(x^{2}+y^{2})$ - $\exp(x) \cosh(y)$ - $\cos(x+y) \sinh(y) \cosh(x)$ - $\exp(5) \sin(5x)$ - $y^{6}-x^{4}$ - $y \sin(2x) \exp(2y)$

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Problem 31873

Find the gradient $\nabla f$ of $f(x, y, z)=x^{2}+2xy-z^{3}$ at the point $(1,1,1)$ . Choices: a. $4 i+j$ , b. $4 i+2 j-3 k$ , c. $4 i-j$ , d. None.

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Problem 31874

A hemispherical tank has water depth $h$ . Volume is $V=\pi\left(30-\frac{2}{3} h\right)(h+1)^{2}$ . At $h=10$ , find $\frac{dh}{dt}$ when water flows in at $6\pi \mathrm{~m}^{3}/\mathrm{s}$ .

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Problem 31875

Evaluate the integral from 2 to 16: $\int_{2}^{16} \frac{d x}{2 x \sqrt{\ln x}}$ . Choose the correct answer.

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Problem 31876

Calculate the integral from 5 to 25 of $\frac{1}{x(\ln x)^{2}}$ and choose the correct answer.

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Problem 31877

Solve the Laplace equation $\Delta w = 0$ in volume $V$ with Dirichlet $w = g$ and Neumann $\frac{\partial w}{\partial n} = h$ conditions. Derive the weak formulation.

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Problem 31878

Find the derivative of $y=3^{-x}$ . Choose the correct option from the following: $y'=-3^{-x} \ln 3$ or others.

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Problem 31879

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

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Problem 31880

Find $\frac{d f^{-1}}{d x}$ for $f(x)=x^{3}-2$ at $x=-1$ (where $f(1)=-1$ ). Choose from the options.

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Problem 31881

Given a rectangle $A=[2a, h]$ in the $x-y$ -plane with thickness $t=1$ , find stress boundary conditions for the biharmonic equation.
1. What are the stress boundary conditions?
2. Sketch the system with boundary conditions.
3. Does $\lambda=2 m_{z}(y/h)^{3}$ satisfy the boundary value problem? Justify.

See Solution

Problem 31882

Find average velocities for $s(t)=-16 t^{2}+106 t$ over intervals $[1,2]$ , $[1,1.5]$ , $[1,1.1]$ , $[1,1.01]$ , $[1,1.001]$ . Conjecture instantaneous velocity at $t=1$ .

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Problem 31883

Find the limits: b) $\lim _{x \rightarrow \infty} \frac{2 x^{3}+x^{2}-3}{x^{3}+x+2}$ c) $\lim _{x \rightarrow \infty} \frac{x+1}{2 x^{2}+5 x}$ .

See Solution

Problem 31884

Solve the Laplace equation $\Delta w=0$ in $V \subseteq \mathbb{R}^{3}$ .
1. Find the weak formulation with Neumann and Dirichlet conditions.
2. Is $\mathbf{v}=\operatorname{curl} \phi$ appropriate for test functions? Justify.
3. Can this PDE be linked to an energy functional? Explain.
4. Identify exact solution candidates from the list provided.

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Problem 31885

Find the limit: $\lim _{x \rightarrow-1}(7-x)^{\frac{7}{3}}$ or state if it does not exist.

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Problem 31886

Find $\frac{dy}{dx}$ for the function $y = x^2 - 1$ .

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Problem 31887

Approximate $f(x, y)=x^{2}+y^{2}$ at $(1.05,1.05)$ using linearization. Choose: a. 2.25 b. 2.15 c. 2.20 d. None e. 2.25

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Problem 31888

Find the derivative of $f(x) = 3x - 1$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .

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Problem 31889

Simplify the energy functional: $\int_{V} \frac{1}{2} \varepsilon: \mathbb{C}: \varepsilon - \mathbf{u} \cdot \mathbf{f} dV - \int_{A_{N}} \mathbf{u} \cdot \mathrm{t} dA$ with given kinematics.

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Problem 31890

Find the nature of the critical point $(-2,1)$ for the function $f(x, y)=-x^{2}-2 y^{2}-4 x+4 y+6$ .

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Problem 31891

Euler-Bernoulli beam problem:
1. Find weak formulation of $E I \frac{\mathrm{d}^{4}}{\mathrm{~d} x^{4}} w=q$ with boundary terms.
2. State Dirichlet and Neumann conditions and their physical meaning.
3. Discuss continuity requirements for approximations and Lagrange elements usage.

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Problem 31892

Soit la fonction $f$ définie par
$f(x) = \begin{cases} \frac{\sqrt{x^{2}+x+2}-2}{x-1} & \text{si } x \neq 1 \\ \frac{3}{4} & \text{pour } x=1 \end{cases}$
1) Est-ce que $f$ est continue en 1 ? 2) Est-ce que $f$ est dérivable en 1 ? 3) Justifiez que $f$ est dérivable pour $x \neq 1$ . 4) Calculez $f^{\prime}(x)$ pour $x \neq 1$ . 5) Calculez $\lim_{x \to -\infty} f(x)$ et $\lim_{x \to +\infty} f(x)$ .

See Solution

Problem 31893

Soit la fonction $f$ définie par :
$f(x) = \frac{\sqrt{x^{2}+x+2}-2}{x-1}$ pour $x \neq 1$ et $f(1) = \frac{3}{4}$ .
1) $f$ est-elle continue en 1 ? 2) $f$ est-elle dérivable en 1 ? 3) Justifie que $f$ est dérivable pour $x \neq 1$ . 4) Calcule $f'(x)$ pour $x \neq 1$ . 5) Calcule $\lim_{x \to -\infty} f(x)$ et $\lim_{x \to +\infty} f(x)$ .

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Problem 31894

1) Soit la fonction $f(x)=\frac{3 x-1}{3 x^{2}-2 x-1}$ . Quelle est une primitive de $f$ parmi les options suivantes ?
a) $F(x)=\frac{-1}{-6 x^{2}+4 x-2}$ ; b) $F(x)=\frac{1}{6 x^{2}+4 x+2}$ ; c) $F(x)=\frac{x}{6 x-3}$ ; d) $F(x)=3 x-1$ .
2) Dans le plan complexe, les points $A$ , $B$ , et $C$ forment-ils un triangle équilatéral, rectangle, isocèle et rectangle, ou aucune des réponses ?
3) Dans l'espace avec le repère $(0, I, J, K)$ , les points $A(-1, 2, 1)$ , $B(1, -6, -1)$ , et $C(2, 2, 2)$ sont sur la droite $(D)$ de vecteur directeur $\vec{u}(1, 1, -3)$ et le plan $(P)$ .

See Solution

Problem 31895

Find the first-order partial derivatives of $f(x, y) = x^{3} - 4xy^{2} + y^{4}$ .

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Problem 31896

Find the slope of the tangent line for $y=2^{\cos x}$ at $x=\frac{\pi}{2}$ . Choices: $1 / \ln 2$ , $\ln \frac{1}{2}$ , $\ln 2$ , $1 / \ln 3$ .

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Problem 31897

Differentiate the function: $\frac{(2 e^{2 x}+e^{3})^{2}}{e^{3 x}}$ with respect to $x$ .

See Solution

Problem 31898

Find the derivative of the function $y = \frac{(2e^{2x} + e^3)^2}{e^{3x}}$ .

See Solution

Problem 31899

Find the first-order partial derivatives of $f(x, y)=4 e^{\frac{x}{y}}+\tan ^{-1}\left(\frac{y}{x}\right)$ .

See Solution

Problem 31900

Find: $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{n}{n^{2}+k^{2}}$

See Solution

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Calculus

Problem 31801

Find the limits: a. lim⁡y→0.25−4y−1∣4y3−y2∣\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{\left|4 y^{3}-y^{2}\right|}limy→0.25−​∣4y3−y2∣4y−1​ b. lim⁡x→0tan⁡(5x)5x\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}limx→0​5xtan(5x)​

Problem 31802

Find the limit: lim⁡x→23x2−7x+2x−4\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}limx→2​x−43x2−7x+2​.

Problem 31803

Find the antiderivative F(x)F(x)F(x) of f(x)=8x3−8x5f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}f(x)=x38​−x58​ such that F(1)=0F(1)=0F(1)=0.

Problem 31804

Evaluate the integral from 5 to 25 of 1x(ln⁡x)2\frac{1}{x(\ln x)^{2}}x(lnx)21​ dx. What is the result?

Problem 31805

Find the limit: lim⁡y→0.25−4y−1∣4y3−y2∣\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{|4 y^{3}-y^{2}|}limy→0.25−​∣4y3−y2∣4y−1​

Problem 31806

Find the antiderivative F(x)F(x)F(x) of f(x)=8x3−8x5f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}f(x)=x38​−x58​ with F(1)=0F(1)=0F(1)=0. What is F(x)F(x)F(x)?

Problem 31807

For the function f(x)=−4x2f(x)=-4 x^{2}f(x)=−4x2, find slopes of secant lines and predict the tangent slope at x=3x=3x=3.

Problem 31808

Find f(2)f(2)f(2) if f′′(x)=10x+4sin⁡(x)f^{\prime \prime}(x)=10 x+4 \sin (x)f′′(x)=10x+4sin(x), f(0)=3f(0)=3f(0)=3, and f′(0)=3f^{\prime}(0)=3f′(0)=3.

Problem 31809

Find the slope of the secant line for f(x)=17cos⁡xf(x) = 17 \cos xf(x)=17cosx on [π2,π][\frac{\pi}{2}, \pi][2π​,π] and conjecture the tangent slope.

Problem 31810

A 10 m wire with radius changing from 5×10−45 \times 10^{-4}5×10−4 m to 9.8×10−49.8 \times 10^{-4}9.8×10−4 m holds a 3.143.143.14 kg load. Find its length increase using Y=2×1011Nm−2Y=2 \times 10^{11} \mathrm{Nm}^{-2}Y=2×1011Nm−2.

Problem 31811

For the function f(x)=−4x2f(x)=-4 x^{2}f(x)=−4x2, find slopes of secant lines for intervals near x=3x=3x=3 and conjecture the tangent slope.

Problem 31812

For f(x)=−3x2f(x)=-3x^{2}f(x)=−3x2, find slopes of secant lines for intervals and predict the tangent slope at x=3x=3x=3.

Problem 31813

A ball is thrown with an initial velocity of 52ft/s52 \mathrm{ft/s}52ft/s. Its height is y=52t−16t2y=52t-16t^{2}y=52t−16t2. (a) Find average velocity from t=2t=2t=2 for: (i) 0.5 s (ii) 0.1 s (iii) 0.05 s (iv) 0.01 s(b) Estimate instantaneous velocity at t=2t=2t=2.

Problem 31814

Given f′′(x)=−2cos⁡(x)f^{\prime \prime}(x)=-2 \cos (x)f′′(x)=−2cos(x), find f′(x)f^{\prime}(x)f′(x) and f(x)f(x)f(x) using constants C and D.

Problem 31815

Find the function f(t)f(t)f(t) given that f′′(t)=2et+3sin⁡(t)f^{\prime \prime}(t)=2 e^{t}+3 \sin (t)f′′(t)=2et+3sin(t), f(0)=−9f(0)=-9f(0)=−9, and f(π)=9f(\pi)=9f(π)=9.

Problem 31816

Calculate the integral ∫12ln⁡x3xdx\int_{1}^{2} \frac{\ln x^{3}}{x} d x∫12​xlnx3​dx and select the correct answer from the options.

Problem 31817

Find the integral of 11+ex\frac{1}{1+e^{x}}1+ex1​. Choose the correct answer from the options given.

Problem 31818

Find the slope of the secant line for f(x)=17cos⁡xf(x) = 17 \cos xf(x)=17cosx between x=π2x = \frac{\pi}{2}x=2π​ and x=πx = \pix=π.

Problem 31819

Find the derivative of y=2xy=2^{x}y=2x. Options: y′=2xln⁡2y' = 2^{x} \ln 2y′=2xln2, y′=2xln⁡xy' = 2^{x} \ln xy′=2xlnx, y′=2xln⁡2y' = \frac{2^{x}}{\ln 2}y′=ln22x​, y′=2−xln⁡2y' = 2^{-} x \ln 2y′=2−xln2, None.

Problem 31820

Find the derivative of y=tln⁡ty=\frac{t}{\ln t}y=lntt​. Choose the correct option from the list.

Problem 31821

Find the derivative of y=t⋅ln⁡ty=t \cdot \ln ty=t⋅lnt. Choose the correct answer from the options provided.

Problem 31822

Find the derivative of xex+exx e^{x} + e^{x}xex+ex. What is the result?

Problem 31823

Find the derivative of y=ln⁡tty=\frac{\ln t}{t}y=tlnt​. Choose the correct option from the list.

Problem 31824

Find the derivative of xex−exx e^{x} - e^{x}xex−ex. What is it?

Problem 31825

Find the derivative of −xex+ex-x e^{x}+e^{x}−xex+ex. What is it?

Problem 31826

Evaluate the integral from 5 to 25: ∫525dxx(ln⁡x)2\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}∫525​x(lnx)2dx​. What is the result?

Problem 31827

Find the derivative of y=ln⁡tty=\frac{\ln t}{t}y=tlnt​.

Problem 31828

Simplify the energy functional, find the weak form, derive the boundary value problem, and interpret the PDE and conditions.

Problem 31829

Find the integral of 21+ex\frac{2}{1+e^{x}}1+ex2​. Choose the correct answer from the options provided.

Problem 31830

Evaluate the integral ∫525dxx(ln⁡x)2\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}∫525​x(lnx)2dx​ and choose the correct answer from the options.

Problem 31831

Find the derivative of −xex+ex-x e^{x}+e^{x}−xex+ex. What is it?

Problem 31832

Find the integral of 2ex1+ex\frac{2 e^{x}}{1+e^{x}}1+ex2ex​. Choose the correct answer from the options given.

Problem 31833

Find the derivative of y=3xy=3^{x}y=3x. Options: y′=3xln⁡3y'=\frac{3^{x}}{\ln 3}y′=ln33x​, y′=3xln⁡y'=3^{x} \lny′=3xln, y′=3xln⁡xy'=3^{x} \ln xy′=3xlnx, None, y′=3−xln⁡3y'=3^{-x} \ln 3y′=3−xln3.

Problem 31834

Find the integral of 11+ex\frac{1}{1+e^{x}}1+ex1​. Choose the correct answer from the options provided.

Problem 31835

Evaluate the integral from 3 to 9 of 1x(ln⁡x)2 dx\frac{1}{x(\ln x)^{2}} \, dxx(lnx)21​dx. What is the result?

Problem 31836

Find the derivative of y=3−xy=3^{-x}y=3−x. Choices: y′=−3−xln⁡3y'=-3^{-x} \ln 3y′=−3−xln3, y′=3xln⁡3y'=\frac{3^{x}}{\ln 3}y′=ln33x​, etc.

Problem 31837

Evaluate the integral ∫11+ex\int \frac{1}{1+e^{x}}∫1+ex1​ and choose the correct answer from the options given.

Problem 31838

Find the integral of 2ex1+ex\frac{2 e^{x}}{1+e^{x}}1+ex2ex​. What is the result?

Problem 31839

Find the limit: lim⁡x→0tan⁡(5x)5x\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}limx→0​5xtan(5x)​.

Problem 31840

Find the limits: a. $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{\left|4 y^{3}-y^{2}\right|}$ b. $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}$

Find the limit: $\lim _{x \rightarrow 2} \frac{3 x^{2}-7 x+2}{x-4}$ .

Find the antiderivative $F(x)$ of $f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}$ such that $F(1)=0$ .

Evaluate the integral from 5 to 25 of $\frac{1}{x(\ln x)^{2}}$ dx. What is the result?

Find the limit: $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{|4 y^{3}-y^{2}|}$

Find the antiderivative $F(x)$ of $f(x)=\frac{8}{x^{3}}-\frac{8}{x^{5}}$ with $F(1)=0$ . What is $F(x)$ ?

For the function $f(x)=-4 x^{2}$ , find slopes of secant lines and predict the tangent slope at $x=3$ .

Find $f(2)$ if $f^{\prime \prime}(x)=10 x+4 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=3$ .

Find the slope of the secant line for $f(x) = 17 \cos x$ on $[\frac{\pi}{2}, \pi]$ and conjecture the tangent slope.

A 10 m wire with radius changing from $5 \times 10^{-4}$ m to $9.8 \times 10^{-4}$ m holds a $3.14$ kg load. Find its length increase using $Y=2 \times 10^{11} \mathrm{Nm}^{-2}$ .

For the function $f(x)=-4 x^{2}$ , find slopes of secant lines for intervals near $x=3$ and conjecture the tangent slope.

For $f(x)=-3x^{2}$ , find slopes of secant lines for intervals and predict the tangent slope at $x=3$ .

A ball is thrown with an initial velocity of $52 \mathrm{ft/s}$ . Its height is $y=52t-16t^{2}$ .
(a) Find average velocity from $t=2$ for: (i) 0.5 s (ii) 0.1 s (iii) 0.05 s (iv) 0.01 s
(b) Estimate instantaneous velocity at $t=2$ .

Given $f^{\prime \prime}(x)=-2 \cos (x)$ , find $f^{\prime}(x)$ and $f(x)$ using constants C and D.

Find the function $f(t)$ given that $f^{\prime \prime}(t)=2 e^{t}+3 \sin (t)$ , $f(0)=-9$ , and $f(\pi)=9$ .

Calculate the integral $\int_{1}^{2} \frac{\ln x^{3}}{x} d x$ and select the correct answer from the options.

Find the integral of $\frac{1}{1+e^{x}}$ . Choose the correct answer from the options given.

Find the slope of the secant line for $f(x) = 17 \cos x$ between $x = \frac{\pi}{2}$ and $x = \pi$ .

Find the derivative of $y=2^{x}$ . Options: $y' = 2^{x} \ln 2$ , $y' = 2^{x} \ln x$ , $y' = \frac{2^{x}}{\ln 2}$ , $y' = 2^{-} x \ln 2$ , None.

Find the derivative of $y=\frac{t}{\ln t}$ . Choose the correct option from the list.

Find the derivative of $y=t \cdot \ln t$ . Choose the correct answer from the options provided.

Find the derivative of $x e^{x} + e^{x}$ . What is the result?

Find the derivative of $y=\frac{\ln t}{t}$ . Choose the correct option from the list.

Find the derivative of $x e^{x} - e^{x}$ . What is it?

Find the derivative of $-x e^{x}+e^{x}$ . What is it?

Evaluate the integral from 5 to 25: $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

Find the derivative of $y=\frac{\ln t}{t}$ .

Find the integral of $\frac{2}{1+e^{x}}$ . Choose the correct answer from the options provided.

Evaluate the integral $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ and choose the correct answer from the options.

Find the derivative of $-x e^{x}+e^{x}$ . What is it?

Find the integral of $\frac{2 e^{x}}{1+e^{x}}$ . Choose the correct answer from the options given.

Find the derivative of $y=3^{x}$ . Options: $y'=\frac{3^{x}}{\ln 3}$ , $y'=3^{x} \ln$ , $y'=3^{x} \ln x$ , None, $y'=3^{-x} \ln 3$ .

Find the integral of $\frac{1}{1+e^{x}}$ . Choose the correct answer from the options provided.

Evaluate the integral from 3 to 9 of $\frac{1}{x(\ln x)^{2}} \, dx$ . What is the result?

Find the derivative of $y=3^{-x}$ . Choices: $y'=-3^{-x} \ln 3$ , $y'=\frac{3^{x}}{\ln 3}$ , etc.

Evaluate the integral $\int \frac{1}{1+e^{x}}$ and choose the correct answer from the options given.

Find the integral of $\frac{2 e^{x}}{1+e^{x}}$ . What is the result?

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{5 x}$ .

Find the limit as $x$ approaches -4 for $3x^3 - 2x^2 + 3x + 8$ or state if it doesn't exist.

Find the derivative of $y=2^{-x}$ . What is $y^{\prime}$ ? Options: $-2^{-x} \ln 2$ , $\frac{2^{x}}{\ln 2}$ , $2^{x} \ln x$ , None, $2^{-x} \ln 2$ .

Calculate the integral from 5 to 25 of $\frac{1}{x(\ln x)^{2}} \, dx$ . Options include $\frac{1}{\ln 2}$ , $\ln 25$ , etc.

Find the derivative of $y=3^{-x}$ . Choose the correct option.

Find the limit: $\lim _{x \rightarrow 4} \frac{-4 x}{\sqrt{3 x-8}}$ . Choose A for the limit value or B if it does not exist.

Evaluate the integral from 5 to 25: $\int_{5}^{25} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

Find the integral of $\frac{e^{x}}{1+e^{x}}$ . Choose the correct answer from the options provided.

Evaluate the integral from 2 to 4: $\int_{2}^{4} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

Find the derivative of $y = t \cdot \ln(t)$ . Choose the correct option from the list.

Find the derivative of $x e^{x} - e^{x}$ . What is the result?

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{7+h}-\frac{1}{7}}{h}$ and simplify your answer.

Find the limit: $\lim _{x \rightarrow 2} \frac{4 x^{2}+7 x+6}{8 x-4}$ or state if it doesn't exist.

Find the limit: $\lim _{y \rightarrow 0.25^{-}} \frac{4 y-1}{|4 y^{3}-y^{2}|}$ .

Find the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$ . Simplify it to $\lim _{x \rightarrow-5} \square$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(12+h)^{2}-144}{h}$ and simplify your answer.

Find the limit: $\lim _{h \rightarrow 0} \frac{4}{\sqrt{36+5 h}+9}$ . A: Simplify your answer; B: Limit does not exist.

Find the limit: $\lim _{x \rightarrow 169} \frac{\sqrt{x}-13}{x-169}$ or state if it doesn't exist.

Find the limit: $\lim _{t \rightarrow 5}\left(t^{2}+8 t+4\right)$ . A. Simplify your answer. B. It doesn't exist.

Find the limit as $x$ approaches 5 for $2x - 8$ . Is it A. $2x - 8 = \square$ or B. limit does not exist?

Evaluate the integral $\int_{3}^{9} \frac{d x}{x(\ln x)^{2}}$ . What is the result?

Find the limit as $x$ approaches 6 for the constant function 8. What is the result? A. $\lim _{x \rightarrow 6} 8=\square$ B. Limit does not exist.

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-3 x-4}{x-4}$ . Simplify it to $\lim _{x \rightarrow 4}(\square)$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\sqrt{256+h}-16}{h}$ or state if it doesn't exist.

Given the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+9, & x<-9 \\ \sqrt{x+9}, & x \geq-9\end{array}\right.$ , find the limits as $x$ approaches -9 from the left, right, and overall.

Find the average velocity of the object given by $s(t)=-7 t^{2}+28 t$ over these intervals: (a) $[1,6]$ , (b) $[1,5]$ , (c) $[1,4]$ , (d) $[1,1+h]$ where $h>0$ .

Given $f(x)=\left\{\begin{array}{ll}x^{2}+9, & x<-9 \\ \sqrt{x+9}, & x \geq-9\end{array}\right.$ , find the limits as $x$ approaches -9.

Find average velocities for $s(t)=-16 t^{2}+106 t$ over intervals $[1,2]$ , $[1,1.5]$ , $[1,1.1]$ , $[1,1.01]$ , $[1,1.001]$ . What is the instantaneous velocity at $t=1$ ?

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

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

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

Evaluate the integral from 2 to 16: $\int_{2}^{16} \frac{d x}{2 x \sqrt{\ln x}}$ . Choose the correct answer.

Find the gradient $\nabla f$ of $f(x, y, z)=x^{2}+2xy-z^{3}$ at the point $(1,1,1)$ . Choices: a. $4 i+j$ , b. $4 i+2 j-3 k$ , c. $4 i-j$ , d. None.

A hemispherical tank has water depth $h$ . Volume is $V=\pi\left(30-\frac{2}{3} h\right)(h+1)^{2}$ . At $h=10$ , find $\frac{dh}{dt}$ when water flows in at $6\pi \mathrm{~m}^{3}/\mathrm{s}$ .