Calculus

Problem 1

Find composite functions and their domains for f(x)=xf(x)=\sqrt{x} and g(x)=8x+9g(x)=8x+9. (a) fgf \circ g: 8x+9\sqrt{8x+9}, domain x98x \geq -\frac{9}{8}. (b) gfg \circ f: 8x+98\sqrt{x}+9.

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Problem 2

Find the min/max value and axis of symmetry for f(x)=3x2+12x8f(x)=-3 x^{2}+12 x-8. The yy-value of the extremum is the solution.

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Problem 3

Find the xx-intercepts and zeros of f(x)=(x+2)2(x4)3(x3)f(x)=(x+2)^{2}(x-4)^{3}(x-3). Express the intercepts and zeros as ordered pairs.

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Problem 4

Find the value of tt that satisfies the equation V(t)=4t327t2V(t)=4t^3-27t^2.

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Problem 5

Find the rr value for the equation y=4.95e0.0542ty=4.95 e^{0.0542 t} and explain the type of change it represents.

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Problem 6

Find the value of xx that satisfies the equation ln(x3)+ln(3x+1)=8\ln(x-3) + \ln(3x+1) = 8.

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Problem 7

Integrate (t+7)(t+7) to find F(x)F(x), then demonstrate the Second Fundamental Theorem by differentiating F(x)F(x).

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Problem 8

Find the center of the power series xn/n3\sum x^n / n^3. The options are: a) 2, b) 3, c) 1/21/2, d) -1, e) 1/2-1/2, f) -1, g) 1, h) -3, i) 0.

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Problem 9

Find the values of g(x)=(14)xg(x) = (\frac{1}{4})^x for x=2,1,0,1,2x = -2, -1, 0, 1, 2.

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Problem 10

Compute the area between the graph of ff and the xx-axis on interval II. a) f(x)=3x6,I=[2;6]f(x)=3x-6, I=[-2; 6] b) f(x)=cos(x),I=[0;2π]f(x)=\cos(x), I=[0; 2\pi]

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Problem 11

Evaluate the derivative ddx[(ff)(x)]\frac{d}{d x}[(f \circ f)(x)] at x=1x=1 given the provided table of f(x)f(x) and f(x)f'(x) values.

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Problem 12

Solve ln(2x1)=8\ln (2x-1)=8. Round the solution to the nearest thousandth.

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Problem 13

Find the equation of a function f(x)f(x) whose range decreases in 1<x<1-1<x<1 and increases in x>1x>1 and x<1x<-1. A. f(x)=x22x+1f(x)=x^{2}-2x+1 B. f(x)=x2+2x+1f(x)=x^{2}+2x+1 C. f(x)=x3+3x+5f(x)=x^{3}+3x+5 D. f(x)=x33x+5f(x)=x^{3}-3x+5

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Problem 14

Solve for xx in the equation ln(x2)3=9\frac{\ln (x-2)}{3}=9.

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Problem 15

Differentiate e3x(sinx+2cosx)e^{3x}(\sin x + 2\cos x) with respect to xx.

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Problem 16

Find the derivative of ln(x2)\ln \left(\frac{x}{2}\right).

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Problem 17

The oven's temperature, FF (in ^{\circ}F), increases by 3030 per minute according to the line of best fit F=30t+70F=30t+70. What does the slope of 3030 represent?

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Problem 18

Find the derivative k(x)k'(x) of k(x)=4x2+10x36(4x37x2)k(x) = \sqrt[6]{-4 x^{2} + 10 x^{3}} \cdot (4 x^{3} - 7 x^{2}).

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Problem 19

Find the best decimal approximation for 62\sqrt{62}.

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Problem 20

Find the most general antiderivative of the function f(x)=3ex+8sec2xf(x) = 3e^x + 8\sec^2 x. Answer: F(x)=3ex8tanx+CF(x) = 3e^x - 8\tan x + C.

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Problem 21

Solve the equation 7ln(x)ln(x2)=17 \ln (x) - \ln (x^{2}) = 1 for the unknown variable xx.

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Problem 22

Compute the indefinite integrals with constant of integration CC: sin(t)dt\int -\sin(t) dt, cos(t)dt\int -\cos(t) dt.

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Problem 23

Evaluate f(1)f(-1) and solve f(x)=3f(x)=3 for the function f(x)=x+5f(x)=\sqrt{x+5}.

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Problem 24

Find the area of the region between the lines y=xy=x and y=2xy=2\sqrt{x}. The area is 43\frac{4}{3}.

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Problem 25

Describe the transformations of g(x)=xg(x) = |x| to get f(x)=2x3+5f(x) = -2|x-3| + 5. Find the domain and range of f(x)f(x).

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Problem 26

Find the derivatives of XX, 3xex3xe^x, and xexxe^x.

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Problem 27

Find the average rate of change of f(x)=x(x+2)(x3)f(x)=-x(x+2)(x-3) from x=2x=2 to x=3x=3. This gives the drug concentration decrease rate in ΔfΔx\frac{\Delta f}{\Delta x} ppm/hour. State the integer answer.

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Problem 28

Find the gradient of the function f(x,y)=xexy2+cosy2f(x, y) = x e^{xy^2} + \cos y^2.

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Problem 29

Graph f(x)=x2(45x)3f(x) = x^2(4 - 5x)^3 and estimate any relative extrema.

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Problem 30

Find the value of yy given the equation y=a+blnxy = a + b \ln x, where a=50.0391615882a = 50.0391615882, b=9.67624973418b = 9.67624973418, and x=19x = 19.

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Problem 31

Find the rate of change of sales with respect to advertising spending for the function S(x)=0.002x3+0.7x2+7x+500S(x) = -0.002x^3 + 0.7x^2 + 7x + 500 on the interval 0x2000 \leq x \leq 200. Determine if sales are increasing faster at $110,000\$110,000 or $160,000\$160,000 in advertising spend.

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Problem 32

Find the derivative of y=4t38t+1y = \frac{4t - 3}{8t + 1} using the quotient rule. The solution is dydt=32(8t+1)2\frac{d y}{d t} = \frac{32}{(8t + 1)^2}.

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Problem 33

Find the volume of the solid generated by revolving the region bounded by y=tan1xy=\tan^{-1} x, y=0y=0, and x=1x=1 about the yy-axis.

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Problem 34

Use trapezoidal rule with 4 intervals to approximate the area under f(x)=x+1f(x)=\sqrt{x+1} on [2,4.5][2,4.5]. Round answer to 2 decimal places.

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Problem 35

Continuous function f(x)f(x) has values {7,9,8,2,8}\{7, -9, 8, 2, -8\} at x={0,10,20,30,40}x=\{0, 10, 20, 30, 40\}. Guarantee roots between x=0x=0 and x=10x=10, x=20x=20 and x=30x=30, x=30x=30 and x=40x=40, but not between x=10x=10 and x=20x=20.

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Problem 36

Find critical values and relative extrema of g(x)=x3+12x19g(x) = -x^3 + 12x - 19. Select A if the critical value(s) exist, and fill in the answer box. Select B if the function has no critical values.

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Problem 37

Find the product and ratio of f(x)=x1/2f(x)=x^{1/2} and g(x)=2x3+1g(x)=\sqrt[3]{2x}+1, and rationalize the denominator of the ratio.

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Problem 38

Determine if Rolle's Theorem applies to f(x)=sin7xf(x) = \sin 7x on [π/7,2π/7][\pi/7, 2\pi/7]. If so, find the point(s) guaranteed to exist. Select A and fill in the answer if Rolle's Theorem applies, otherwise select B.

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Problem 39

Find the inverse function f1(x)f^{-1}(x) if f(x)=ex+12f(x) = e^{x+1} - 2.

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Problem 40

Calculate the arc length of y=14x212lnxy=\frac{1}{4} x^{2}-\frac{1}{2} \ln x on [1,3e][1,3 e].

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Problem 41

Find the length of the curve y=log(ex1ex+1)y = \log\left(\frac{e^{x} - 1}{e^{x} + 1}\right) from x=1x = 1 to x=2x = 2.

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Problem 42

Find the vertical asymptotes and holes of the rational function f(x)=x7x29x+14f(x) = \frac{x-7}{x^{2}-9 x+14}. B. Vertical asymptote(s) at x=3,5x=3, 5 and hole(s) at x=7x=7.

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Problem 43

Determine if 06e2xdx\int_{0}^{\infty} 6 e^{-2 x} d x converges or diverges. If convergent, calculate its value.

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Problem 44

Calculate the 2nd and 3rd order Taylor polynomials for f(x)=8tan(x)f(x) = 8 \tan(x) centered at a=0a = 0. Express the polynomials using symbolic notation and fractions.

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Problem 45

Evaluate the function y=(14)xy = (\frac{1}{4})^{x} for x=3x = 3.

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Problem 46

Find the relative extrema of f(x)=4x+x2f(x) = -4 - x + x^2. Identify the intervals where the function is increasing and decreasing, and sketch the graph.

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Problem 47

Finde die Ableitung der Funktion f(x)=x3+2x2x9f(x)=x^{3}+2 x^{2}-x-9.

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Problem 48

Approximate e\sqrt{e} using a Maclaurin polynomial for exe^{x} with a maximum error of 0.01.

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Problem 49

Find the derivative of y=8+t2y = \sqrt{8 + t^2}.

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Problem 50

Find the limit of the function 2log8(3x)-2 \log_{8}(3-x) as xx approaches 3 from the left.

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Problem 51

Compose and differentiate y=e5xy = e^{5\sqrt{x}}. Identify the inner function u=g(x)u=g(x) and the outer function y=f(u)y=f(u).
(f(u),g(x))=(eu,x)(f(u), g(x)) = (e^u, \sqrt{x})
dydx=25e5x12x\frac{dy}{dx} = 25e^{5\sqrt{x}}\frac{1}{2\sqrt{x}}

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Problem 52

Calculate the average rate of change of f(x)=5x33xf(x) = -5x^3 - 3x on the interval [2,4][2, 4].

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Problem 53

Find the derivative of y=6xy=6x. Options: A. xx, B. 0, C. x2x^{2}, D. 66.

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Problem 54

Solve for xx in the equation lnx74lnx=1\ln x^{7} - 4 \ln x = 1.

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Problem 55

Find the average value of x39xx^3 - 9x on the interval 1x3-1 \leq x \leq 3.

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Problem 56

Find the equation of the function f(x)f(x) where f(x)=1x+3f(x)=\frac{1}{x+3}.

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Problem 57

Evaluate f(x)=(x5)2f(x)=(x-5)^{2} at x=3x=3, then graph the function using the point (3,f(3))(3, f(3)).

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Problem 58

Find the interval(s) where s(t)=t33+7t226t+2s(t) = -\frac{t^{3}}{3} + \frac{7t^{2}}{2} - 6t + 2 describes a slowing hummingbird, where t0t \geq 0 is time in seconds and ss is position in yards.

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Problem 59

Calculate the average fuel consumption rate from 6 AM to 3 PM given the formula f(t)=0.9t20.2t0.4+10f(t) = 0.9t^2 - 0.2t^{0.4} + 10, where tt is the time in AM and f(t)f(t) is the fuel consumption in barrels.

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Problem 60

Find the local max/min of f(x)=6+x+x2x3f(x)=6+x+x^{2}-x^{3}, and the intervals where it's increasing/decreasing. State answers to 2 decimal places.

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Problem 61

Find the value of the polynomial P(x)=4x29x+2P(x)=4x^2-9x+2 when x=3x=3.

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Problem 62

Find the points x^0\hat{x}_{0} and x~0\tilde{x}_{0} where the linear function L(x)=18x+13L(x)=18x + 13 approximates f(x)=9x390x+157f(x)=9x^3 - 90x + 157 and g(x)=9ln(5x)27x+22g(x)=9\ln(5x) - 27x + 22, respectively. Compute the derivatives, set them equal to L(x)L'(x), and check the yy-intercept.

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Problem 63

The derivative f(a)f'(a) represents the instantaneous rate of change of f(x)f(x) at x=ax=a. This is the slope of the tangent line to f(x)f(x) at (a,f(a))(a, f(a)). True or False?

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Problem 64

Find the interval where the cubic function f(x)=x39xf(x) = x^3 - 9x has a positive average rate of change.

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Problem 65

Find the inverse of x3\sqrt{x-3} and determine its domain, including any restrictions from the original function.

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Problem 66

Solve the equation (e3)2x8=e4x+5(e^{3})^{2x-8} = e^{-4x+5}

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Problem 67

Finden Sie die Ableitung der Funktion f(x)=x(3x2)f(x)=\sqrt{x} \cdot(3 x-2).

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Problem 68

Find the derivative of q(x)=log8(6x3+3x+1)q(x)=\log_{8}(6x^{3}+3x+1).

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Problem 69

Find the minimum value of the function y=3x212x+10y = 3x^2 - 12x + 10.

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Problem 70

Finden Sie den Wert von xx, der die Gleichung eex=eexe^{e} x=e \cdot e^{x} erfüllt.

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Problem 71

Find dydx\frac{dy}{dx} and d2ydx2\frac{d^2y}{dx^2} for x=t2+at+a2,y=13t32at2x=t^2+at+a^2, y=\frac{1}{3}t^3-2at^2 (where a>0a>0) and determine the values of tt where the curve is concave upward.

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Problem 72

Solve lnc=15\ln c = 15 and write the exact solution set. Also provide approximate solutions to 4 decimal places.

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Problem 73

Find the value of tt in the equation t=ln(1,00014,954)ln0.8t=\frac{\ln \left(\frac{1,000}{14,954}\right)}{\ln 0.8}.

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Problem 74

Find the rate of change of the area of a square as its sides increase at 6 m/sec6 \mathrm{~m} / \mathrm{sec}, when the sides are 20 m20 \mathrm{~m} and 26 m26 \mathrm{~m} long.

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Problem 75

Find the value of (fg)(3)(f \circ g)(3) where f(x)=7x24xf(x) = 7x^2 - 4x and g(x)=5x9g(x) = 5x - 9.

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Problem 76

Simplify the expression: log10005\log \sqrt[5]{1000}.

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Problem 77

Find the range of the function f(x)=x+53f(x) = \sqrt{x+5} - 3 using graphing technology.

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Problem 78

Simplify lnxe2x\ln \frac{x}{e^{2 x}} by expressing it as a sum or difference of logarithms, and represent powers as factors.

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Problem 79

Round log742\log_{7} 42 to the nearest thousand.

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Problem 80

Find the value of xx that satisfies the infinite series equation n=12x5n=20\sum_{n=1}^{\infty} 2 x^{5 n} = 20.

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Problem 81

Find the point on the curve y=tanxy=\tan x closest to (1,1), correct to two decimal places.

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Problem 82

Find cc such that f(c)=0f'(c)=0 for f(x)=e1x2f(x)=e^{1-x^2} and determine if f(x)f(x) has a local extremum at x=cx=c.

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Problem 83

Solve the logarithmic equation ln(7x3)5=3\ln (7 x-3)-5=-3 for the value of xx.

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Problem 84

Identify the function represented by the power series k=1xkk\sum_{k=1}^{\infty} \frac{x^{k}}{k}. Find the corresponding Taylor series f(x)f(x).

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Problem 85

Find the difference between exe^{-x} and e2xe^{-2x}.

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Problem 86

Evaluate the function f(x)=3(13)x3f(x)=3\left(\frac{1}{3}\right)^{x}-3 at x=2x=-2. What is the numerical value of f(2)f(-2)?

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Problem 87

Find the first derivative of 5x5 \sqrt{x}.

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Problem 88

Simplify lne3+2lne+e3ln42eln2\ln e^{3} + 2 \ln \sqrt{e} + e^{3 \ln 4} - 2 e^{-\ln 2}.

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Problem 89

Find the numerical value of tanh(0)\tanh(0) and tanh(1)\tanh(1), rounded to five decimal places.

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Problem 90

Solve the equation e4t+1et23=10\frac{e^{4 t+1}}{e^{t-2}}-3=10 and select the correct answer.

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Problem 91

A rubber band exerts a restoring force F=20x+0.3x2F=-20x+0.3x^2 (in N) when stretched xx (in m). The work done stretching it from x=0x=0 to x=5x=5 is: a) 190 J, b) 356 J, c) 238 J, d) 119 J, e) 309 J.

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Problem 92

Find the derivative of y=log8(9xx5)y=\log_{8}(9-x-x^{5}).

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Problem 93

A company's total cost, C(t)=6040etC(t)=60-40 e^{-t}, where tt is time in years. Find: a) C(t)C'(t), b) C(0)C'(0), c) C(6)C'(6), d) limtC(t)\lim_{t\to\infty} C(t) and limtC(t)\lim_{t\to\infty} C'(t).

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Problem 94

Find the natural logarithm of the reciprocal of the mathematical constant ee.

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Problem 95

Use f(x)=4x2f(x)=4x^2 to complete the table. Enter the answer as a whole number, fraction, or decimal rounded to 2 decimals. For fractions, use 12\frac{1}{2} or 12-\frac{1}{2} format.

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Problem 96

Determine the dependent and independent variables in the population function P(y)=0.015y22.105y+278.090P(y)=0.015 y^{2}-2.105 y+278.090.

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Problem 97

Solve for xx without a calculator: lnx3=4\frac{\ln x}{3} = 4

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Problem 98

Find the derivative of y=4x2+2xy=4x^2+2\sqrt{x} evaluated at x=1x=1.

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Problem 99

Find the mean and standard deviation of Y=4X5Y=4X-5 when XX has mean =29=29 and standard deviation =6=6.

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Problem 100

Find the antiderivative of dsdt=11t(5t27)3\frac{d s}{d t}=11 t\left(5 t^{2}-7\right)^{3}.

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