Calculus

Problem 1

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

See Solution

Problem 2

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

See Solution

Problem 3

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

See Solution

Problem 4

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

See Solution

Problem 5

Find the $r$ value for the equation $y=4.95 e^{0.0542 t}$ and explain the type of change it represents.

See Solution

Problem 6

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

See Solution

Problem 7

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

See Solution

Problem 8

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

See Solution

Problem 9

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

See Solution

Problem 10

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

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Problem 11

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

See Solution

Problem 12

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

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Problem 13

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

See Solution

Problem 14

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

See Solution

Problem 15

Differentiate $e^{3x}(\sin x + 2\cos x)$ with respect to $x$ .

See Solution

Problem 16

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

See Solution

Problem 17

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

See Solution

Problem 18

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

See Solution

Problem 19

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

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Problem 20

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

See Solution

Problem 21

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

See Solution

Problem 22

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

See Solution

Problem 23

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

See Solution

Problem 24

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

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Problem 25

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

See Solution

Problem 26

Find the derivatives of $X$ , $3xe^x$ , and $xe^x$ .

See Solution

Problem 27

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

See Solution

Problem 28

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

See Solution

Problem 29

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

See Solution

Problem 30

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

See Solution

Problem 31

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

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Problem 32

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

See Solution

Problem 33

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

See Solution

Problem 34

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

See Solution

Problem 35

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

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Problem 36

Find critical values and relative extrema of $g(x) = -x^3 + 12x - 19$ . 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 $f(x)=x^{1/2}$ and $g(x)=\sqrt[3]{2x}+1$ , and rationalize the denominator of the ratio.

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Problem 38

Determine if Rolle's Theorem applies to $f(x) = \sin 7x$ on $[\pi/7, 2\pi/7]$ . 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 $f^{-1}(x)$ if $f(x) = e^{x+1} - 2$ .

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Problem 40

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

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Problem 41

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

See Solution

Problem 42

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

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Problem 43

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

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Problem 44

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

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Problem 45

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

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Problem 46

Find the relative extrema of $f(x) = -4 - x + x^2$ . 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 $f(x)=x^{3}+2 x^{2}-x-9$ .

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Problem 48

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

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Problem 49

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

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Problem 50

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

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Problem 51

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

See Solution

Problem 52

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

See Solution

Problem 53

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

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Problem 54

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

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Problem 55

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

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Problem 56

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

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Problem 57

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

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Problem 58

Find the interval(s) where $s(t) = -\frac{t^{3}}{3} + \frac{7t^{2}}{2} - 6t + 2$ describes a slowing hummingbird, where $t \geq 0$ is time in seconds and $s$ 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 $f(t) = 0.9t^2 - 0.2t^{0.4} + 10$ , where $t$ is the time in AM and $f(t)$ is the fuel consumption in barrels.

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Problem 60

Find the local max/min of $f(x)=6+x+x^{2}-x^{3}$ , 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 $P(x)=4x^2-9x+2$ when $x=3$ .

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Problem 62

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

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Problem 63

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

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Problem 64

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

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Problem 65

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

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Problem 66

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

See Solution

Problem 67

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

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Problem 68

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

See Solution

Problem 69

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

See Solution

Problem 70

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

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Problem 71

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

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Problem 72

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

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Problem 73

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

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Problem 74

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

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Problem 75

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

See Solution

Problem 76

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

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Problem 77

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

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Problem 78

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

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Problem 79

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

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Problem 80

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

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Problem 81

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

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Problem 82

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

See Solution

Problem 83

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

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Problem 84

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

See Solution

Problem 85

Find the difference between $e^{-x}$ and $e^{-2x}$ .

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Problem 86

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

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Problem 87

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

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Problem 88

Simplify $\ln e^{3} + 2 \ln \sqrt{e} + e^{3 \ln 4} - 2 e^{-\ln 2}$ .

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Problem 89

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

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Problem 90

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

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Problem 91

A rubber band exerts a restoring force $F=-20x+0.3x^2$ (in N) when stretched $x$ (in m). The work done stretching it from $x=0$ to $x=5$ 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 $y=\log_{8}(9-x-x^{5})$ .

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Problem 93

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

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Problem 94

Find the natural logarithm of the reciprocal of the mathematical constant $e$ .

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Problem 95

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

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Problem 96

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

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Problem 97

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

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Problem 98

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

See Solution

Problem 99

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

See Solution

Problem 100

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

See Solution

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Calculus

Problem 1

Find composite functions and their domains for f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=8x+9g(x)=8x+9g(x)=8x+9. (a) f∘gf \circ gf∘g: 8x+9\sqrt{8x+9}8x+9​, domain x≥−98x \geq -\frac{9}{8}x≥−89​. (b) g∘fg \circ fg∘f: 8x+98\sqrt{x}+98x​+9.

Problem 2

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

Problem 3

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

Problem 4

Find the value of ttt that satisfies the equation V(t)=4t3−27t2V(t)=4t^3-27t^2V(t)=4t3−27t2.

Problem 5

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

Problem 6

Find the value of xxx that satisfies the equation ln⁡(x−3)+ln⁡(3x+1)=8\ln(x-3) + \ln(3x+1) = 8ln(x−3)+ln(3x+1)=8.

Problem 7

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

Problem 8

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

Problem 9

Find the values of g(x)=(14)xg(x) = (\frac{1}{4})^xg(x)=(41​)x for x=−2,−1,0,1,2x = -2, -1, 0, 1, 2x=−2,−1,0,1,2.

Problem 10

Compute the area between the graph of fff and the xxx-axis on interval III. a) f(x)=3x−6,I=[−2;6]f(x)=3x-6, 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]f(x)=cos(x),I=[0;2π]

Problem 11

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

Problem 12

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

Problem 13

Problem 14

Solve for xxx in the equation ln⁡(x−2)3=9\frac{\ln (x-2)}{3}=93ln(x−2)​=9.

Problem 15

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

Problem 16

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

Problem 17

The oven's temperature, FFF (in ∘^{\circ}∘F), increases by 303030 per minute according to the line of best fit F=30t+70F=30t+70F=30t+70. What does the slope of 303030 represent?

Problem 18

Find the derivative k′(x)k'(x)k′(x) of k(x)=−4x2+10x36⋅(4x3−7x2)k(x) = \sqrt[6]{-4 x^{2} + 10 x^{3}} \cdot (4 x^{3} - 7 x^{2})k(x)=6−4x2+10x3​⋅(4x3−7x2).

Problem 19

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

Problem 20

Find the most general antiderivative of the function f(x)=3ex+8sec⁡2xf(x) = 3e^x + 8\sec^2 xf(x)=3ex+8sec2x. Answer: F(x)=3ex−8tan⁡x+CF(x) = 3e^x - 8\tan x + CF(x)=3ex−8tanx+C.

Problem 21

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

Problem 22

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

Problem 23

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

Problem 24

Find the area of the region between the lines y=xy=xy=x and y=2xy=2\sqrt{x}y=2x​. The area is 43\frac{4}{3}34​.

Problem 25

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

Problem 26

Find the derivatives of XXX, 3xex3xe^x3xex, and xexxe^xxex.

Problem 27

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

Problem 28

Find the gradient of the function f(x,y)=xexy2+cos⁡y2f(x, y) = x e^{xy^2} + \cos y^2f(x,y)=xexy2+cosy2.

Problem 29

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

Problem 30

Find the value of yyy given the equation y=a+bln⁡xy = a + b \ln xy=a+blnx, where a=50.0391615882a = 50.0391615882a=50.0391615882, b=9.67624973418b = 9.67624973418b=9.67624973418, and x=19x = 19x=19.

Problem 31

Problem 32

Find the derivative of y=4t−38t+1y = \frac{4t - 3}{8t + 1}y=8t+14t−3​ using the quotient rule. The solution is dydt=32(8t+1)2\frac{d y}{d t} = \frac{32}{(8t + 1)^2}dtdy​=(8t+1)232​.

Problem 33

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

Problem 34

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

Problem 35

Problem 36

Find critical values and relative extrema of g(x)=−x3+12x−19g(x) = -x^3 + 12x - 19g(x)=−x3+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.

Problem 37

Find the product and ratio of f(x)=x1/2f(x)=x^{1/2}f(x)=x1/2 and g(x)=2x3+1g(x)=\sqrt[3]{2x}+1g(x)=32x​+1, and rationalize the denominator of the ratio.

Problem 38

Determine if Rolle's Theorem applies to f(x)=sin⁡7xf(x) = \sin 7xf(x)=sin7x on [π/7,2π/7][\pi/7, 2\pi/7][π/7,2π/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.

Problem 39

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

Problem 40

Calculate the arc length of y=14x2−12ln⁡xy=\frac{1}{4} x^{2}-\frac{1}{2} \ln xy=41​x2−21​lnx on [1,3e][1,3 e][1,3e].

Problem 41

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

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

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

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

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

Find the $r$ value for the equation $y=4.95 e^{0.0542 t}$ and explain the type of change it represents.

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

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

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

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

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

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

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

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

Differentiate $e^{3x}(\sin x + 2\cos x)$ with respect to $x$ .

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

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

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

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

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

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

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

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

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

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

Find the derivatives of $X$ , $3xe^x$ , and $xe^x$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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