Calculus

Problem 2001

Find ff such that f(x)=8x5,f(9)=0f^{\prime}(x)=8 x-5, f(9)=0. f(x)=f(x)=

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Problem 2002

Use geometry to evaluate the definite integral. 022dx\int_{0}^{2} 2 d x

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Problem 2003

Evaluate. 15(2x2+5)dx\int_{1}^{5}\left(2 x^{2}+5\right) d x

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Problem 2004

(x7+x5x4)6(7x6+5x44x3)dx(x7+x5x4)6(7x6+5x44x3)dx=\begin{array}{c} \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x \\ \int\left(x^{7}+x^{5}-x^{4}\right)^{6}\left(7 x^{6}+5 x^{4}-4 x^{3}\right) d x= \end{array} \square (Type an exact answer. Use parentheses to clearly denote the argument

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Problem 2005

Note: Figure not drawn to scale.
A long insulating tube of length LL has inner radius rar_{a} and outer radius rbr_{b}, where LrbL \gg r_{b}, as shown in the figure. The tube has a nonuniform charge distributed throughout its volume. The charge density ρ\rho as a function of the distance rr from the tube's central axis is ρ(r)=βr3\rho(r)=\beta r^{3} for ra<r<rbr_{a}<r<r_{b}, where β\beta is a positive constant. lgnoring edge effects, what is the electric field EE in the range ra<r<rbr_{a}<r<r_{b} inside the tube? (A) 0 (B) β4εar(r4ra4)\frac{\beta}{4 \varepsilon_{a} r}\left(r^{4}-r_{a}^{4}\right) (C) β5ε0r(r5ra5)\frac{\beta}{5 \varepsilon_{0} r}\left(r^{5}-r_{a}^{5}\right) (D) β5ε0r(rb5r5)\frac{\beta}{5 \varepsilon_{0}{ }^{r}}\left(r_{b}^{5}-r^{5}\right) Question 6 of 24 Back Next

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Problem 2006

Exercice 1: Soit ff une fonction numérique de var iablex définie par : f(x)=xf(x)=\sqrt{x} et Soit (Cf)\left(C_{f}\right) sa courbe sur un repére (O;i;j)(O ; \vec{i} ; \vec{j}) 1) Donner le domaine de définition dela fonction ff. 2) Etudier la parité dela fonction ff. 2) a-Calculer limx+f(x)\lim _{x \rightarrow+\infty} f(x). bb-Etudier labranche inf inie de (Cf)\left(C_{f}\right) au voi sin\sin agede ++\infty. 4) a-Etudier la dérivabilité de fà droite de 0 . bb-Etudier les var iations dela fonction ff. 5) Tracer (Cf)\left(C_{f}\right).

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Problem 2007

1e2lnppdp\int_{1}^{e^{2}} \frac{\ln p}{p} d p

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Problem 2008

Is there a number a such that the following limit exists? (If an answer does not exist, enter DNE.) limx23x2+ax+a+9x2+x2\lim _{x \rightarrow-2} \frac{3 x^{2}+a x+a+9}{x^{2}+x-2}
Find the value aa. a=7a=-7
Evaluate the limit. 55

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Problem 2009

23. [0/1 Points] DETAILS MY NOTES SCALCET9M 2.6.016. PREVIOUS ANSWERS
Find the limit. (If the limit is infinite, enter ' \infty ' or '- \infty ', as appropriate. If the limit does not otherwise exist, enter DNE.) limx(52x+7)\lim _{x \rightarrow \infty}\left(\frac{-5}{2 x+7}\right)
Need Help? Read It Submit Answer

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Problem 2010

Find yy^{\prime}. y=(2x3)3(x2+5)4y=\sqrt{(2 x-3)^{3}\left(x^{2}+5\right)^{4}} ++
Choose the correct answer below. A. y=(x2+5)(11x212x+15)2x3y^{\prime}=\frac{\left(x^{2}+5\right)\left(11 x^{2}-12 x+15\right)}{\sqrt{2 x-3}} B. y=2x3(x2+5)(11x212x+15)y^{\prime}=\sqrt{2 x-3}\left(x^{2}+5\right)\left(11 x^{2}-12 x+15\right) C. y=x2+5(2x3)(10x2+6x30)y^{\prime}=\sqrt{x^{2}+5}(2 x-3)\left(10 x^{2}+6 x-30\right) D. y=(2x3)(10x2+6x30)x2+5y^{\prime}=\frac{(2 x-3)\left(10 x^{2}+6 x-30\right)}{\sqrt{x^{2}+5}}

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Problem 2011

Differentiate. y=lnxx5dydx=\begin{array}{c} y=\frac{\ln x}{x^{5}} \\ \frac{d y}{d x}=\square \end{array} \square

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Problem 2012

[3] limx2x+2x3+8\lim _{x \rightarrow 2^{-}} \frac{x+2}{x^{3}+8}

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Problem 2013

P57. (3 Use Stokes' Theorem to evaluate the line integral C(6x2+z22xz3)dx+(2+3z2)dy+(x23x2z2)dz\oint_{\mathcal{C}}\left(-6 x^{2}+z^{2}-2 x z^{3}\right) d x+\left(-2+3 z^{2}\right) d y+\left(x^{2}-3 x^{2} z^{2}\right) d z as a surface integral where C\mathcal{C} is the boundary of the surface with parametrization r(s,t)=\vec{r}(s, t)= s,t,2s3t+6\langle s, t,-2 s-3 t+6\rangle where 0s10 \leq s \leq 1 and 0t20 \leq t \leq 2. Answer. -30

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Problem 2014

Evaluate the given definite integral. 24(3x2x35)dx\int_{2}^{4}\left(3 x^{2}-\frac{x^{3}}{5}\right) d x 24(3x2x35)dx=\int_{2}^{4}\left(3 x^{2}-\frac{x^{3}}{5}\right) d x= \square (Simplify your answer.)

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Problem 2015

Make the given substitution to evaluate the indefinite integral. 7(7x+14)4dx,u=7x+147(7x+14)4dx=\begin{array}{l} \int 7(7 x+14)^{4} d x, u=7 x+14 \\ \int 7(7 x+14)^{4} d x=\square \end{array}

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Problem 2016

32. What is the minimum value of f(x)=xln(x)f(x)=x \ln (x) ?

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Problem 2017

[3 marks] (b) Consider a function f(x)=12xf(x)=\frac{1}{2-\sqrt{x}}. Find limxf(x)\lim _{x \rightarrow \infty} f(x) and state the equation of horizontal asymptotes for f(x)f(x). [3 marks]

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Problem 2018

Use the substitution formula abf(g(x))g(x)dx=g(a)g(b)f(u)du\int_{a}^{b} f(g(x)) g^{\prime}(x) d x=\int_{g(a)}^{g(b)} f(u) d u where g(x)=ug(x)=u, to evaluate the following integrals. (a) 02πcosz6+sinzdz\int_{0}^{2 \pi} \frac{\cos z}{\sqrt{6+\sin z}} d z (b) ππcosz6+sinzdz\int_{-\pi}^{\pi} \frac{\cos z}{\sqrt{6+\sin z}} d z (a) 02πcosz6+sinzdz=0\int_{0}^{2 \pi} \frac{\cos z}{\sqrt{6+\sin z}} d z=0 (Simplify your answer. Type an integer or a fraction.) (b) ππcosz6+sinzdz=\int_{-\pi}^{\pi} \frac{\cos z}{\sqrt{6+\sin z}} d z= \square (Simplify your answer. Type an integer or a fraction.)

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Problem 2019

Find the total area of the shaded regions.
The area is \square . (Simplify your answer.)

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Problem 2020

(b) Given y=32x5y=3^{2 x-5}. Find d2ydx2\frac{d^{2} y}{d x^{2}} when x=3x=3. Give the answer in the form of logarithm.
3. The function f(x)=x36x2+9x3f(x)=x^{3}-6 x^{2}+9 x-3 is defined on the interval [0,5][0,5]. Find the [4 marks] critical points of f(x)f(x) on this interval and determine whether the critical points are local minimum or maximum. [7 marks]

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Problem 2021

2. (a) Consider the parametric equations, x=t2x=t^{2}, and y=t33ty=t^{3}-3 t. [3 marks] Evaluate dydx\frac{d y}{d x} when t=3t=\sqrt{3}. [5 marks] (b) Given y=32x5y=3^{2 x-5}. Find d2ydx2\frac{d^{2} y}{d x^{2}} when x=3x=3. Give the answer in the form of logarithm. [4 marks]

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Problem 2022

Given the function ff and point a below, complete parts (a)-(c). f(x)=76x,a=16f(x)=7-6 x, a=\frac{1}{6}
1 (x) - 6 b. Graph f(x)f(x) and f1(x)f^{-1}(x) together. Choose the correct graph below. A. B. C. D. c. Evaluate dfdx\frac{d f}{d x} at x=ax=a and df1dx\frac{d f^{-1}}{d x} at x=f(a)x=f(a) to show that df1dxx=f(a)=1(df/dx)x=a\left.\frac{d f^{-1}}{d x}\right|_{x=f(a)}=\frac{1}{\left.(d f / d x)\right|_{x=a}} dfdxx=16=\left.\frac{d f}{d x}\right|_{x=\frac{1}{6}}= \square df1dxx=f(16)=\left.\frac{d f^{-1}}{d x}\right|_{x=f\left(\frac{1}{6}\right)}= \square (Simplify your answers. Use integers or fractions for any numbers in the expressions.)

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Problem 2023

45. Let ff be the function with derivative given by f(x)=x22xf^{\prime}(x)=x^{2}-\frac{2}{x}. On which of the following intervals is ff decreasing?

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Problem 2024

Exercice 1: Soit ff une fonction numérique de var iablex définie par : f(x)=x2f(x)=\sqrt{x^{2}} et Soit (Cf)\left(C_{f}\right) sa courbe sur un repére (O;i;j)(O ; \vec{i} ; \vec{j}) 1) Donner ledomaine de définition dela fonction ff. 2) Etudier la parité dela fonction ff. 2) a-Calculer limx+f(x)\lim _{x \rightarrow+\infty} f(x). bb-Etudier labranche inf inie de (Cf)\left(C_{f}\right) au voi sin\sin agede ++\infty. 4) a-Etudier la dérivabilité de f à droite de 0 . bb-Etudier les var iations dela fonction ff. 5) Tracer ( Cf)\left.C_{f}\right).

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Problem 2025

Consider the function f(x)=xe5x,0x2f(x)=x e^{-5 x}, \quad 0 \leq x \leq 2. This function has an absolute minimum value equal to: \square which is attained at x=x= \square and an absolute maximum value equal to: which is attained at x=x= \square

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Problem 2026

Given that u=v24u=v^{2}-4, find ddv(3u52sinv)\frac{d}{d v}\left(3 u^{5}-2 \sin v\right) in terms of only vv.

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Problem 2027

54. The function ff is defined by f(x)=ex(x2+2x)f(x)=e^{-x}\left(x^{2}+2 x\right). At what values of xx does ff have a relative maximum? (A) x=2+2x=-2+\sqrt{2} and x=22x=-2-\sqrt{2}

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Problem 2028

Using three rectangles and the midpoint rule, estimate the area above the xx-axis and under the function f(x)=1(cosπx3)6f(x)=1-\left(\cos \frac{\pi x}{3}\right)^{6} over the interval [0,3][0,3]. Leave your answer as an exact value. Note that f(x)0f(x) \geq 0 for all xx.

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Problem 2029

Find the area of the region enclosed by the curves y=x24xy=x^{2}-4 x and y=x2+6xy=-x^{2}+6 x.
The area of the region enclosed by the curves is \square (Type an integer or a simplified fraction.)

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Problem 2030

21 Mark for Review
The function mm is given bs m(x)=log10e+log10(x1)m(x)=\log _{10} e+\log _{10}\left(x^{-1}\right). Which of the following statements about mm is true? (A) mm is increasing, the graph of mm is concave up, and limxm(x)=log10e\lim _{x \rightarrow-\infty} m(x)=\log _{10} e. (B) mm is increasing, the graph of mm is concave down, and limx0+m(x)=\lim _{x \rightarrow 0^{+}} m(x)=-\infty. (C) mm is decreasing, the graph of mm is concave up, and limx0+m(x)=\lim _{x \rightarrow 0^{+}} m(x)=\infty. (D) mm is decreasing, the graph of mm is concave down, and limxm(x)=log10e\lim _{x \rightarrow-\infty} m(x)=-\log _{10} e

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Problem 2031

Find the total area of the shaded regions.
The area is \square \square. (Simplify your answer.)

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Problem 2032

Evaluate the indefinite integral by using the substitution u=x2+19\mathrm{u}=\mathrm{x}^{2}+19. 2x(x2+19)10dx2x(x2+19)10dx=\begin{array}{l} \int 2 x\left(x^{2}+19\right)^{-10} d x \\ \int 2 x\left(x^{2}+19\right)^{-10} d x= \end{array}

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Problem 2033

The acceleration function (in m/s2\mathrm{m} / \mathrm{s}^{2} ) and the initial velocity v(0)v(0) are given for a particle moving along a line. a(t)=2t+4,v(0)=5,0t5a(t)=2 t+4, \quad v(0)=-5, \quad 0 \leq t \leq 5 (a) Find the velocity at time tt. v(t)=v(t)= \square m/s\mathrm{m} / \mathrm{s} (b) Find the distance traveled during the given time interval.

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Problem 2034

Find the limit: limx0(e3x1x)\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right).

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Problem 2035

Bestimme die erste und zweite Ableitung der Funktionen: a) f(x)=ex+1f(x)=e^{x}+1, b) f(x)=ex+xf(x)=e^{x}+x, c) f(x)=ex+2x2f(x)=e^{x}+2 x^{2}, d) f(x)=ex+1f(x)=-e^{x}+1, e) f(x)=2ex+3x2f(x)=2 e^{x}+3 x^{2}, f) f(x)=5ex0,5x3f(x)=-5 e^{x}-0,5 x^{3}, g) f(x)=12(exx3)f(x)=-\frac{1}{2}(e^{x}-x^{3}), h) f(x)=14ex+sin(x)f(x)=\frac{1}{4}e^{x}+\sin(x).

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Problem 2036

Bestimmen Sie die erste Ableitung für die folgenden Funktionen und überprüfen Sie mit dem GTR: a) 2x(4x1)2 x \cdot(4 x-1), b) (5x+3)(x+2)(5 x+3) \cdot(x+2), c) (25x)(x+2)(2-5 x) \cdot(x+2), d) 2xex2 x \cdot e^{x}, e) (4x+2)ex(4 x+2) \cdot e^{x}, f) (6x+1)ex(6 x+1) \cdot e^{x}.

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Problem 2037

Find the marginal cost MCM C and average cost ACA C from the total cost function TC=300ln(q+30)+150T C=300 \ln (q+30)+150.

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Problem 2038

Find the tangent line equation for f(x)=xf(x) = \sqrt{x} at the point (1,1).

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Problem 2039

Find the limit: limΔx0[89(4+Δx)](28)Δx\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}.

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Problem 2040

Find the function f(x)f(x) and the number cc given the limit: limΔx0[89(4+Δx)](28)Δx\lim _{\Delta x \rightarrow 0} \frac{[8-9(4+\Delta x)]-(-28)}{\Delta x}

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Problem 2041

An object is launched with an initial velocity of 48 ft/s from 80 ft. Find average velocity from t=0t=0 to t=2t=2 and t=2t=2 to t=4t=4. Use h(t)=16t2+48t+80h(t)=-16 t^{2}+48 t+80.

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Problem 2042

연속 함수 f(x)f(x)가 주어지고, g(x)=0xf(t)dtx4f(t)dtg(x)=\int_{0}^{x} f(t) d t-\int_{x}^{4} f(t) d tx=2x=2에서 0일 때, 124f(x)dx\int_{\frac{1}{2}}^{4} f(x) d x의 값은?

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Problem 2043

Find the limit as xx approaches 1110\frac{11}{10} from the right: limx1110+(15x1110x)\lim _{x \rightarrow \frac{11}{10}^{+}}\left(\frac{15 x}{11-10 x}\right).

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Problem 2044

Find F(x)F^{\prime}(x) using the first principle if F(x)=x23xF(x)=x^{2}-3x.

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Problem 2045

Find the average rate of change of h(t)=cotth(t)=\cot t over the intervals: a. [3π4,5π4]\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right] b. [5π6,3π2]\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right]

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Problem 2046

Calculate the average rate of change of h(t)=cotth(t)=\cot t over the intervals: a. [3π4,5π4]\left[\frac{3 \pi}{4}, \frac{5 \pi}{4}\right], b. [5π6,3π2]\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right].

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Problem 2047

Find the limits: 1) limx3x29x23x \lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}-3 x} 2) limx33x46x+12x5+4x3 \lim _{x \rightarrow 3} \frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}

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Problem 2048

Find the average rate of change of h(t)=cotth(t)=\cot t over [5π6,3π2]\left[\frac{5 \pi}{6}, \frac{3 \pi}{2}\right].

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Problem 2049

Find the limit as xx approaches 3 for the expression x29x23x\frac{x^{2}-9}{x^{2}-3x}.

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Problem 2050

Simplify the expression: 3x46x+12x5+4x3\frac{3 x^{4}-6 x+12}{x^{5}+4 x^{3}}

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Problem 2051

Differentiate 3xy=16x3x33xy = \sqrt{16x} - \frac{3}{x^3} with respect to xx.

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Problem 2052

Differentiate y=6x4x+53x2y=\frac{6 x^{4}-x+5}{3 x^{2}} and express the answer with positive exponents.

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Problem 2053

Find the derivative of yy with respect to xx. y=(1+5x)e5xdydx=\begin{array}{l} y=(1+5 x) e^{-5 x} \\ \frac{d y}{d x}=\square \end{array} \square

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Problem 2054

Find the 2019th 2019^{\text {th }} derivative of y=sin(2x+1)+5x6+20x100y=\sin (2 x+1)+5 x^{6}+20 x^{100}

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Problem 2055

Suppose f(x)=x3+4,x[0,1]f(x)=x^{3}+4, x \in[0,1]. (a) Find the slope of the secant line connecting the points (x,y)=(0,4)(x, y)=(0,4) and (1,5)(1,5). (b) Find a number c(0,1)c \in(0,1) such that f(c)f^{\prime}(c) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in ( 0,1 ).

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Problem 2056

Find the derivative of yy with respect to xx. y=ln(x12)dydx=\begin{array}{l} y=\ln \left(x^{12}\right) \\ \frac{d y}{d x}=\square \end{array}

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Problem 2057

Use the graph below to find the largest value of δ>0\delta>0 such that for all x,f(x)L<εx,|f(x)-L|<\varepsilon whenever 0<xc<δ0<|x-c|<\delta. f(x)=3xc=1L=3ε=0.5\begin{aligned} f(x) & =\frac{3}{\sqrt{-x}} \\ c & =-1 \\ L & =3 \\ \varepsilon & =0.5 \end{aligned}
The largest value of δ\delta is \square (Simplify your answer.)

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Problem 2058

Find yy^{\prime \prime} for y=(2+1x)3y=\left(2+\frac{1}{x}\right)^{3} y=y^{\prime \prime}=

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Problem 2059

(1.72) Show that (AB×r)=A×B\nabla(A \cdot B \times r)=A \times B where rr is the position vector, AA and BB are constant vectors.

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Problem 2060

3. [15] a) Find the 5th 5^{\text {th }} degree Taylor polynomial of 8x56x4+3x3+27x2+32x128 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12 about x=0x=0 and [6] about x=20x=20, then simplify each result. What do you observe? b) Find the 100th 100^{\text {th }} degree Taylor polynomial of 8x56x4+3x3+27x2+32x128 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12 about x=0x=0 [6] and about x=20x=20, then simplify each result. What do you observe? c) What can you conclude about the nth n^{\text {th }} degree Taylor polynomial about x=ax=a of a polynomial of [3] degree mm, where nmn \geq m ?

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Problem 2061

The figure shows the electric field inside a cylinder of radius R=3.3 mmR=3.3 \mathrm{~mm}. The field strength is increasing with time as E=1.0×108t2 V/mE=1.0 \times 10^{8} t^{2} \mathrm{~V} / \mathrm{m}, where tt is in s . The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for t<0t<0. (Figure 1)
Part A
Part B
Find an expression for the magnetic field strength as a function of time at a distance r<Rr<R from the center. Express your answer in teslas as a multiple of product of distance rr and time tt.

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Problem 2062

6. A force of 50.N50 . \mathrm{N} is required to stretch a spring 5.0 cm horizontally from its equitiblum position. How much work is needed to stretch the spring another 2.0 cm from this point?

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Problem 2063

Evaluate the indefinite integral. x(4x+5)8dx\int x(4 x+5)^{8} d x

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Problem 2064

Quiz \#8 (covers past 2 weeks) Remaining Time: 00:08:37 - Question 1
1 point Number Help
Use the Fundamental Theorem to evaluate 0πsin(2x)dx\int_{0}^{\pi} \sin (2 x) d x
Make sure your value looks reasonable! Number

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Problem 2065

Find the integral of the function f(x)=e3x2e3x f(x) = e^{3x} \sqrt{2-e^{3x}} .

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Problem 2066

The function f(x)=(4x1)e3xf(x)=(4 x-1) e^{-3 x} has one critical number. Find it.

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Problem 2067

Consider 6x7(4x8+9)6dx\int 6 x^{7}\left(4 x^{8}+9\right)^{6} d x What is the most appropriate u-substitution? Select only ONE answer. u=(4x8+9)6u=\left(4 x^{8}+9\right)^{6} u=6x7u=6 x^{7} u=4x8+9u=4 x^{8}+9 u=xu=x

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Problem 2068

stion 5
Consider x2+1x3+3xdx\int \frac{x^{2}+1}{x^{3}+3 x} d x What is the most efficient u-substitution? Select only ONE answer. u=x3+3xu=x^{3}+3 x u=x+1\mathrm{u}=\mathrm{x}+1 u=x\mathrm{u}=\mathrm{x} u=x3u=x^{3} y=x2+1y=x^{2}+1 y=x2y=x^{2}

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Problem 2069

Let f(x)=x312x2+45x+6f(x)=x^{3}-12 x^{2}+45 x+6. Find the open intervals on which ff is increasing (decreasing). Then determine the xx-coordinates of all relative maxima (minima).
1. ff is increasing on the intervals \square

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Problem 2070

limx(2)2xx+2\lim _{x \rightarrow(-2)} \frac{2-|x|}{x+2}

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Problem 2071

Find the area under y=4cos(x)y=4 \cos (x) and above y=4sin(x)y=4 \sin (x) for π2x3π2\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}. (Note that this area may not be defined over the entire interval.) area == \square

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Problem 2072

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to xx or yy. Then find the area of the region. y=7cosx,y=(4sec(x))2,x=π/4,x=π/4y=7 \cos x, y=(4 \sec (x))^{2}, x=-\pi / 4, x=\pi / 4

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Problem 2073

The velocity function is v(t)=t2+6t8v(t)=-t^{2}+6 t-8 for a particle moving along a line. Find the displacement and the distance traveled by the particle during the time interval [-2,6]. displacement = 120.67-120.67 \square distance traveled = 92.67 \square

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Problem 2074

If the tangent line equation to the curve sin(xy)=yx12\sin (x y)=\sqrt{y}-x-\frac{1}{\sqrt{2}} at (0,12)\left(0, \frac{1}{2}\right) is y=ax+12y=a x+\frac{1}{2} then a=a=

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Problem 2075

Completed: 9 of 11 My score: 7.63/117.63 / 11 pts (69.32\%)
Given the graph of the positive velocity of an object moving along a line, what is the geometric representation of its displacement over a time interval [a, b]?
Let the horizontal axis measure time in terms of t . Choose the correct answer below. A. The object's displacement can be represented geometrically as the area between the graph and the t -axis from t=a\mathrm{t}=\mathrm{a} to t=b\mathrm{t}=\mathrm{b}. B. The object's displacement can be represented geometrically as the difference between the value of the graph at t=at=a and the value of the graph at t=bt=b. C. The object's displacement can be represented geometrically as the average value of the graph between t=at=a and t=bt=b. D. The object's displacement can be represented geometrically as the slope of the graph between t=at=a and t=bt=b.

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Problem 2076

If x2+y2=13x^{2}+y^{2}=13 and dy dt=8\frac{\mathrm{d} y}{\mathrm{~d} t}=8, find dx dt\frac{\mathrm{d} x}{\mathrm{~d} t} when y=3y=3 and x=2x=2. dx dt(x,y)=(2,3)=\left.\frac{\mathrm{d} x}{\mathrm{~d} t}\right|_{(x, y)=(2,3)}=

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Problem 2077

H.w Solve: y+2y+y=4exlnx\quad y^{\prime \prime}+2 y^{\prime}+y=4 e^{-x} \ln x

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Problem 2078

Given the function f(x)=2.5xe0.1x+2.64 f(x) = -2.5 x \cdot e^{-0.1 x} + 2.64 , find the antiderivative (indefinite integral) of the function.

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Problem 2079

Soit ff la fonction définie par: f(x)=1+x1xf(x)=\frac{\sqrt{1+x}-1}{x} 1) Déterminer DjD_{j} 2). Verifier que xDff(x)=11+x+1\forall x \in D_{f} f(x)=\frac{1}{1+\sqrt{x+1}} (3). Montrer que ff est décropssante sur DfD_{f} 4). Montrer que xDf0<f(x)1\forall x \in D_{f} 0<f(x) \leq 1 5) Calculer f(1)f(-1) est dédule que 1 est une valeur maximale de la fonction ff sur DfD_{f}

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Problem 2080

A weight on a vertical spring is given an initial upward velocity of 5 cm/sec5 \mathrm{~cm} / \mathrm{sec} from a point 7 cm below equilibrium. Assume that the contstant ω\omega has a value of 0.1 . Write the formula for the location of the weight at time tt. x=x=
Find the location of the weight 5 seconds after it is set in motion. centimeters

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Problem 2081

In Problems 1-32, use a table or a graph to investigate each limit.
1. limx2(x24x+1)\lim _{x \rightarrow 2}\left(x^{2}-4 x+1\right)
2. limx2x2+3x+2\lim _{x \rightarrow 2} \frac{x^{2}+3}{x+2}
3. limx12x1+x2\lim _{x \rightarrow-1} \frac{2 x}{1+x^{2}}
4. lims2s(s24)\lim _{s \rightarrow 2} s\left(s^{2}-4\right)
5. limxπ3cosx4\lim _{x \rightarrow \pi} 3 \cos \frac{x}{4}
6. limtπ/9sin(3t)\lim _{t \rightarrow \pi / 9} \sin (3 t)
7. limxπ/22secx3\lim _{x \rightarrow \pi / 2} 2 \sec \frac{x}{3}
8. limxπ/2tanxπ/22\lim _{x \rightarrow \pi / 2} \tan \frac{x-\pi / 2}{2}
9. limx2ex2/2\lim _{x \rightarrow-2} e^{-x^{2} / 2}
10. limx0ex+12x+3\lim _{x \rightarrow 0} \frac{e^{x}+1}{2 x+3}
11. limx0ln(x+1)\lim _{x \rightarrow 0} \ln (x+1)
12. limtelnt3\lim _{t \rightarrow e} \ln t^{3}
13. limx3x216x4\lim _{x \rightarrow 3} \frac{x^{2}-16}{x-4}
14. limx2x24x+2\lim _{x \rightarrow 2} \frac{x^{2}-4}{x+2}
15. limxπ/2sin(2x)\lim _{x \rightarrow \pi / 2} \sin (2 x)
16. limxπ/2cos(xπ)\lim _{x \rightarrow \pi / 2} \cos (x-\pi)
17. limx011+x2\lim _{x \rightarrow 0} \frac{1}{1+x^{2}}
18. limx01x21\lim _{x \rightarrow 0} \frac{1}{x^{2}-1}
19. limx0+(1ex)\lim _{x \rightarrow 0^{+}}\left(1-e^{-x}\right)
20. limx0(1+ex)\lim _{x \rightarrow 0^{-}}\left(1+e^{x}\right)
21. limx42x4\lim _{x \rightarrow 4^{-}} \frac{2}{x-4}
22. limx3+1x3\lim _{x \rightarrow 3^{+}} \frac{1}{x-3}
23. limx121x\lim _{x \rightarrow 1^{-}} \frac{2}{1-x}
24. limx2+42x\lim _{x \rightarrow 2^{+}} \frac{4}{2-x}
25. limx111x2\lim _{x \rightarrow 1^{-}} \frac{1}{1-x^{2}}
26. limx2+2x24\lim _{x \rightarrow 2^{+}} \frac{2}{x^{2}-4}
27. limx31(x3)2\lim _{x \rightarrow 3} \frac{1}{(x-3)^{2}}
28. limx01x2x2\lim _{x \rightarrow 0} \frac{1-x^{2}}{x^{2}}
29. limx0x2+93x2\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}}
30. limx0x2+42x\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x}
31. limx011x2x2\lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}}
32. limx02x22x\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x}

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Problem 2082

29. limx0x2+93x2\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}}
30. limx0x2+42x\lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x}
31. limx011x2x2\lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}}
32. limx02x22x\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x}

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Problem 2083

limx0sinx1+cosx\lim_{x \rightarrow 0} \frac{\sin x}{1+\cos x}

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Problem 2084

Evaluate the infinite series by identifying it as the value of a derivative of a geometric series. n=1n4n=\sum_{n=1}^{\infty} \frac{n}{4^{n}}= \square Hint: Write it as f(14)f^{\prime}\left(\frac{1}{4}\right) where f(x)=n=0xnf(x)=\sum_{n=0}^{\infty} x^{n}. Question Help: D Post to forum

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Problem 2085

Taylor Series Formula n=0f(n)(a)(xa)nn!\sum_{n=0}^{\infty} \frac{f^{(n)}(a)(x-a)^{n}}{n!}
Find the Taylor Series for the following functions at the given value by using the definition and finding the derivatives at the given value.
1. f(x)=exf(x)=e^{x} at a=ea=e
4. f(x)=cos(2x)f(x)=\cos (2 x) at a=π4a=\frac{\pi}{4}
2. f(x)=e2xf(x)=e^{2 x} at a=3a=3
5. f(x)=1xf(x)=\frac{1}{x} \quad at a=7\quad a=7
3. f(x)=cos(x)f(x)=\cos (x) at a=πa=-\pi
6. f(x)=ln(x)f(x)=\ln (x) at a=ea=e

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Problem 2086

Lamonte is going to invest in an account paying an interest rate of 4\% compounded continuously. How much would Lamonte need to invest, to the nearest cent, for the value of the account to reach \$12,300 in 8 years?

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Problem 2087

32. limx02x22x\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x}

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Problem 2088

Find the derivative of the function y=1x2arcCos(x) y = \sqrt{1-x^{2}} \, \operatorname{arcCos}(x) .

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Problem 2089

Find the derivative of the function f(x)=sin2(2x2)+3tan(ex) f(x) = \sin^2(2-x^2) + 3 \tan(e^x) .

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Problem 2090

dydt=0.17y(1y400)y(0)=5\begin{array}{l}\frac{d y}{d t}=0.17 y\left(1-\frac{y}{400}\right) \\ y(0)=5\end{array}

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Problem 2091

Let f:R2R,(x,y)x2y2f: \mathbb{R}^{2} \rightarrow \mathbb{R},(x, y) \mapsto x^{2}-y^{2}, and let SS be the circle of radius 1 around the orig Find the extrema of fSf \mid S.

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Problem 2092

Exercises - find the derivative of y=f(x)y=f(x) at x=0x=0 where yy is determined by y5+2yx3x7=0y^{5}+2 y-x-3 x^{7}=0 - find the tangent line of x216+y29=1\frac{x^{2}}{16}+\frac{y^{2}}{9}=1 at (2,332)\left(2, \frac{3 \sqrt{3}}{2}\right)

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Problem 2093

Determine whether the following statements are true and give an explanation or counterexample. Assume that f,ff, f^{\prime}, and ff^{\prime \prime} and are continuous functions for all real numbers. Complete parts (a) through (e) below. a. Decide whether the statement f(x)f(x)dx=12(f(x))2+C\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}+C is correct. Choose the correct answer below. A. True; f(x)f(x)dx=12(f(x))2+C\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}+C B. False; f(x)f(x)dx=2(f(x))2+C\int f(x) f^{\prime}(x) d x=2(f(x))^{2}+C C. False; f(x)f(x)dx=12(f(x))2(f(x))2+C\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))^{2}\left(f^{\prime \prime}(x)\right)^{2}+C D. False; f(x)f(x)dx=12(f(x))+C\int f(x) f^{\prime}(x) d x=\frac{1}{2}(f(x))+C

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Problem 2094

Exemple (1) Calculez le revenu total gagné sur un contrat de 4 ans où la fonction du revenu continue est c(t)=90000+11000tc(t)=90000+11000 t (2) Calculez la valeur actualisée de ce revenu si le taux d'inflation est 1%1 \%.

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Problem 2095

Find the derivative of the function g(x)=lnx24 g(x) = \ln \left| x^2 - 4 \right| .

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Problem 2096

2. Find the integral of f(x,y,z)=zf(x, y, z)=z over the region RR, where RR is the pyramid with a square base having vertices (1,1,0),(1,1,0),(1,1,0)(1,1,0),(1,-1,0),(-1,-1,0), and (1,1,0)(-1,1,0) and with its top at (0,0,2)(0,0,2).

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Problem 2097

Evaluate the indefinite integral. x6cos3(x7)sin2(x7)dx=x\int x^{6} \cos ^{3}\left(x^{7}\right) \sin ^{2}\left(x^{7}\right) d x=x

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Problem 2098

The answer above is NOT correct.
Evaluate 45(4t22t5)dt\int_{4}^{5}\left(\frac{4}{t^{2}}-\frac{2}{t^{5}}\right) d t using the Fundamental Theorem of Calculus, Part 2. Use exact values. 45(4t22t5)dt=4922500\int_{4}^{5}\left(\frac{4}{t^{2}}-\frac{2}{t^{5}}\right) d t=\frac{492}{2500} help (numbers).

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Problem 2099

Suppose that a particle moves along a straight line with a velocity v(t)=42tv(t)=4-2 t, where tt is in the interval [0,8][0,8]. Find the displacement of the particle up to t=8t=8 and the total distance traveled up to t=8t=8.
Total displacement == \square Total distance == \square

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Problem 2100

Evaluate: (7yesin(x))dx+[15xsin(y3+8y)]dy\oint\left(7 y-e^{\sin (x)}\right) d x+\left[15 x-\sin \left(y^{3}+8 y\right)\right] d y using one method. CC is the boundary of the graph of a circle of radius 3 .

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