Calculus

Problem 32101

Find the derivative of $f(x)=4 \sqrt{4 x^{2}+7}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 32102

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(8-9 x)^{8}$ .

See Solution

Problem 32103

Find the missing expression in the equation: $\frac{d}{d x} \ln \left(x^{7}+6\right)=\frac{1}{x^{7}+6} ?$

See Solution

Problem 32104

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{7 x}$ .

See Solution

Problem 32105

Determine if the following series converge or diverge, and explain your reasoning: a. $\sum_{n=1}^{\infty} \frac{\sin n}{n^{2}}$ b. $\sum_{n=1}^{\infty} \frac{(2 n)!}{n! n!}$ c. $\sum_{n=1}^{\infty} \frac{3^{n}}{(2 n)^{n}}$ d. $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3}{n}$

See Solution

Problem 32106

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 \ln(1+2 x^{2})$ .

See Solution

Problem 32107

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

See Solution

Problem 32108

Find the 3rd order Taylor polynomial for $f(x)=2 \sqrt{x}$ at $a=1$ .

See Solution

Problem 32109

Find the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{4^{n}}$ . When does it converge conditionally?

See Solution

Problem 32110

Find the surface area generated by revolving $x=\frac{e^{y}+e^{-y}}{2}$ from $y=0$ to $y=\ln 7$ around the $y$ -axis.
Area $S = \square$ .

See Solution

Problem 32111

Find $f^{\prime}(x)$ , the tangent line at $x=3$ for $f(x)=(6x-2)^{1/2}$ , and where the tangent is horizontal.

See Solution

Problem 32112

Find the surface area of a cone by revolving $y=\frac{r}{h} x$ from $0$ to $h$ around the $x$ -axis.

See Solution

Problem 32113

Find $f^{\prime}(x)$ and the tangent line at $x=3$ for $f(x)=(6x-2)^{1/2}$ . Where is the tangent line horizontal?

See Solution

Problem 32114

Find the derivative of $y(x)=\sqrt{x^{2}+10}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 32115

Find the derivative of $5\left(t^{2}+7 t\right)^{-6}$ . What is $\frac{d}{d t} 5\left(t^{2}+7 t\right)^{-6}=\square$ ?

See Solution

Problem 32116

Find $f^{\prime}(x)$ , the tangent line at $x=2$ , and where the tangent line is horizontal for $f(x)=(8x-7)^{1/2}$ .

See Solution

Problem 32117

Find the Taylor series at $x=0$ for the following functions: a. $f(x)=\frac{4}{4-x}$ , b. $f(x)=1-\cos 2x$ .

See Solution

Problem 32118

Evaluate $F^{\prime}(5)$ for the function $F(x)=\int_{4}^{x^{3}} \ln (t) d t$ .

See Solution

Problem 32119

Find the derivative and simplify: $\frac{d}{d x} \frac{\ln (3+x)}{x^{7}}$ . Choose the correct answer.

See Solution

Problem 32120

Find the derivative $f^{\prime}(x)$ and the tangent line equation at $x=1$ for $f(x)=\frac{x}{(6x-5)^{4}}$ .

See Solution

Problem 32121

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4x-7)^{9}}$ .

See Solution

Problem 32122

Evaluate the derivative: $\frac{d}{d r}\left(\frac{\int_{-6}^{r} f(t) d t}{7 r^{8}}\right)$ .

See Solution

Problem 32123

Find the derivative $g^{\prime}(x)$ for the function $g(x)=2 x e^{7 x}$ .

See Solution

Problem 32124

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4 x-7)^{9}}$ .

See Solution

Problem 32125

Find the derivative of the function: $e^{2x} (e^{4x} - 3)^2$ .

See Solution

Problem 32126

Find the area of the surface formed by revolving $y=\sqrt{2x-x^{2}}$ from $x=1$ to $x=1.5$ about the $x$ -axis. Set up the integral: $s=\int_{1}^{1.5} 2\pi y \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} \, dx$ .

See Solution

Problem 32127

Find $f^{\prime}(x)$ and the tangent line equation at $x=3$ for $f(x)=\frac{x}{(2 x-5)^{5}}$ .

See Solution

Problem 32128

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4 x-7)^{9}}$ .

See Solution

Problem 32129

A 9-N force stretches a rubber band. How far? Also, set up the integral for work done in joules.

See Solution

Problem 32130

Determine if the series $200 + 40 + 8 + 1.6 + \ldots$ is convergent or divergent. If convergent, find the sum.

See Solution

Problem 32131

Find $f^{\prime}(x)$ and the tangent line equation at $x=3$ for $f(x)=\frac{x}{(2 x-5)^{5}}$ .

See Solution

Problem 32132

Is the series $\sum_{d=1}^{\infty}-15(2)^{d-1}$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 32133

Find the derivative of $7(t^{2}+3t)^{-5}$ . What is $\frac{d}{dt} 7(t^{2}+3t)^{-5}=\square$ ?

See Solution

Problem 32134

Is the series $\sum_{k=2}^{\infty} \frac{5}{2}\left(\frac{1}{3}\right)^{k-1}$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 32135

A climber will lift a $30-\mathrm{m}$ rope weighing $0.5 \mathrm{~N/m}$ . Set up the work integral: $w=\int \square d x$ .

See Solution

Problem 32136

Find the derivative of $5(t^{2}+7t)^{-4}$ . What is $\frac{d}{dt} 5(t^{2}+7t)^{-4} = \square$ ?

See Solution

Problem 32137

A climber hauls a $40\text{-m}$ rope weighing $0.75\text{ N/m}$ . Set up the integral for the work done in joules.

See Solution

Problem 32138

Find the limits: (a) $\lim _{x \rightarrow 0^{+}}(1+\sin 3 x)^{1 / x}$ , (b) $\lim _{x \rightarrow \infty} x^{2}(1-e^{1 / x})$ , (c) $\lim _{x \rightarrow+\infty}(\sqrt{x}-\ln(x^{2}+1))$ , (d) $\lim _{x \rightarrow 0^{+}}(\frac{1}{x}-\frac{1}{1-\cos x})$ .

See Solution

Problem 32139

Find the derivative $f^{\prime}(x)$ and the values of $x$ where the tangent line of $f(x)=x^{7}(x-10)^{3}$ is horizontal.

See Solution

Problem 32140

Évaluez les intégrales suivantes : (a) $\int_{0}^{\pi} e^{x} \sin x d x$ , (b) $\int \frac{\sqrt{1+\sqrt{t}}}{\sqrt{t}} d t$ , (c) $\int_{-1}^{1}\left(\frac{3 y+5}{1+y^{2}}\right) d y$ , (d) $\int_{0}^{\pi / 2}\left(\frac{\sin x}{1+\cos x}\right) d x$ , (e) $\int_{0}^{2}\left|x^{2}-1\right| d x$ , (f) $\int 9 x^{2} \ln x d x$ .

See Solution

Problem 32141

Find $f^{\prime}(x)$ and the values of $x$ where the tangent line of $f(x)=\sqrt{x^{2}-12 x+44}$ is horizontal.

See Solution

Problem 32142

Find the derivative of $5x(x^{5}+1)^{8}$ . What is $\frac{d}{d x}\left[5 x\left(x^{5}+1\right)^{8}\right]$ ?

See Solution

Problem 32143

Find $f^{\prime}(x)$ and the $x$ values where the tangent line is horizontal for $f(x)=\sqrt{x^{2}-12 x+44}$ .

See Solution

Problem 32144

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Calculate $C^{\prime}(24)$ .

See Solution

Problem 32145

Find the partial derivatives $f_{x x}, f_{x y}, f_{y y}$ for $f(x, y)=e^{4 x}-\sin(x+y^{2})-\sqrt{x y}$ .

See Solution

Problem 32146

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Calculate $C^{\prime}(x)$ and interpret.

See Solution

Problem 32147

Find $y^{\prime}$ for $2 x + 9 y + 5 = 0$ by implicit differentiation and solving for $y$ . What is the implicit result?

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Problem 32148

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Analyze $C'(x)$ at $x=24$ and $x=42$ . Is cost increasing or decreasing?

See Solution

Problem 32149

Find the pressure at point $A$ in a tank with 0.6 m water above it. Options: A) $3.85 \mathrm{kPa}$ B) $5.88 \mathrm{kPa}$ C) $7.84 \mathrm{kPa}$ D) $9.80 \mathrm{kPa}$

See Solution

Problem 32150

Calculate the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ and find $C'(42)$ .

See Solution

Problem 32151

Find $y^{\prime}$ for $9x + 5y + 3 = 0$ by implicit differentiation and direct differentiation. What is $y$ ?

See Solution

Problem 32152

Find $y^{\prime}$ for $9 x + 5 y + 3 = 0$ by implicit differentiation and direct differentiation. What is the implicit result?

See Solution

Problem 32153

Find the partial derivatives: $f(x, y, z)=\sin^{-1}(xy)-\sin(yz)$ ; compute $f_{xx}, f_{yz}, f_{xyz}$ .

See Solution

Problem 32154

Find $y^{\prime}$ for $9 x + 5 y + 3 = 0$ using implicit differentiation and direct differentiation.

See Solution

Problem 32155

Differentiate $2 x^{3}-9 y-11=0$ implicitly to find $y^{\prime}$ , then solve for $y$ and differentiate directly.

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Problem 32156

Find $y^{\prime}$ using implicit differentiation for $y - 2x^{2} + 7 = 0$ and evaluate at $(2,1)$ .

See Solution

Problem 32157

Find $y^{\prime}$ using implicit differentiation for $5 x^{2}-y^{2}-9=0$ and evaluate at $(3,6)$ .

See Solution

Problem 32158

Find $y^{\prime}$ using implicit differentiation for $y^{2}+2 y+3 x=0$ and evaluate at the point $(-1,1)$ .

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Problem 32159

A circle is in a square. With a radius of 5 m decreasing at 2 m/min and square sides of 19 m increasing at 2 m/min, find the rate of change of the area outside the circle but inside the square in square meters per minute.

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Problem 32160

Differentiate and solve: $y=\frac{(2 e^{2 x}+e^{3})^{2}}{e^{3 x}}$

See Solution

Problem 32161

Calculate the integral $\int_{0}^{2}|8 x-10| d x$ .

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Problem 32162

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for $G(x)$ , the antiderivative of $g(x)=e^{-5x}$ passing through $(0,6)$ .

See Solution

Problem 32163

Find the nonzero value of $x$ where the second derivative of $f(x) = 4x^6 - 8x^5$ is zero. Provide a decimal answer with at least three decimal places.

See Solution

Problem 32164

Find $z^{\prime}$ using implicit differentiation for $xz - 4 = 0$ and evaluate it at $(4,1)$ .

See Solution

Problem 32165

Find the value of $x$ where the first derivative of $g(x)=7x-10\ln(x)$ is zero. Provide your answer as a decimal with three places.

See Solution

Problem 32166

Find the derivative $f^{\prime}(x)$ for the function $f(x)=6 x-5 \ln x^{4}$ .

See Solution

Problem 32167

An inverted pyramid is filled with water at 70 cm³/s. Find the water level rise rate when it's at 4 cm high.

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Problem 32168

Solve the equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=6$ , $s^{\prime}(0)=7$ and find $s(\pi)$ .

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Problem 32169

Find the value of $x$ where the first derivative of $g(x)=3x-3\ln(x)$ is zero. Provide your answer as a decimal with three decimal places.

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Problem 32170

Find the antiderivative of $f(x)=x^{3}-1$ with $F(3)=7$ . What is $F(0)$ ? Provide the answer as a decimal.

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Problem 32171

Find the nonzero $x$ where the second derivative of $f(x)=7x^{6}-10x^{5}$ equals zero. Provide your answer as a decimal with three decimal places.

See Solution

Problem 32172

Find the antiderivative of $f(x)=x^{3}-9$ with $F(3)=6$ . What is $F(0)$ as a decimal?

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Problem 32173

Find the limit of $R_{n}=\frac{2}{n} \sum_{i=1}^{n}\left[2\left(6+i \frac{2}{n}\right)^{2}-3\left(6+i \frac{2}{n}\right)^{5}\right]$ as $n \rightarrow \infty$ as $\int_{a}^{b} f(x) dx$ . Determine $a, b, f(x)$ .

See Solution

Problem 32174

Analyze the inequality $1-\frac{x^{2}}{6}<\frac{x \sin x}{2-2 \cos x}<1$ near $x=0$ and find $\lim _{x \rightarrow 0} \frac{x \sin x}{2-2 \cos x}$ . Graph $y=1-\frac{x^{2}}{6}$ , $y=\frac{x \sin x}{2-2 \cos x}$ , and $y=1$ for $-2 \leq x \leq 2$ and discuss the behavior as $x \rightarrow 0$ .

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Problem 32175

Find the average slope of $f(x)=2 x^{4}+5 x^{3}-x^{2}-3 x+1$ between $x=1$ and $x=3$ , and the instantaneous slopes at those points.

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Problem 32176

Find the antiderivative of $f(x)=x^{3}-9$ with $F(3)=6$ . What is $F(0)$ ? Provide the answer as a decimal.

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Problem 32177

1. Find $\lim_{x \to 0} f(x)$ given $\sqrt{5-2x^{2}} \leq f(x) \leq \sqrt{5-x^{2}}$ for $-1 \leq x \leq 1$ .
2. Find $\lim_{x \to 0} g(x)$ given $2-x^{2} \leq g(x) \leq 2 \cos x$ for all $x$ .
3. a. What does the inequality $1-\frac{x^{2}}{6}<\frac{x \sin x}{2-2 \cos x}<1$ imply about $\lim_{x \to 0} \frac{x \sin x}{2-2 \cos x}$ ?
b. Graph $y=1-\frac{x^{2}}{6}$ , $y=\frac{x \sin x}{2-2 \cos x}$ , and $y=1$ for $-2 \leq x \leq 2$ and comment on the behavior as $x \to 0$ .
4. a. What does the inequality $\frac{1}{2}-\frac{x^{2}}{24}<\frac{1-\cos x}{x^{2}}<\frac{1}{2}$ imply about $\lim_{x \to 0} \frac{1-\cos x}{x^{2}}$ ?
b. Graph $y=\frac{1}{2}-\frac{x^{2}}{24}$ , $y=\frac{1-\cos x}{x^{2}}$ , and $y=\frac{1}{2}$ for $-2 \leq x \leq 2$ and comment on the behavior as $x \to 0$ .

See Solution

Problem 32178

Find the limit as $x$ approaches infinity of the antiderivative $G(x)$ of $g(x)=e^{-4x}$ , passing through $(0,11)$ .

See Solution

Problem 32179

Find the derivative of $3 \ln (4-3x)^{2}$ with respect to $x$ .

See Solution

Problem 32180

Find the value of $t$ where the local maximum of $f(t)=\int_{0}^{t} \frac{x^{2}+11 x+30}{1+\cos ^{2}(x)} d x$ occurs.

See Solution

Problem 32181

Find the general antiderivative of $f(x)=x^{3}-1$ . Given $F(3)=2$ , find $F(0)$ as a decimal.

See Solution

Problem 32182

Find the value of $x$ where the first derivative of $g(x)=7x-6\ln(x)$ is zero. Provide the answer as a decimal with at least three decimal places.

See Solution

Problem 32183

Solve the equation $s''(t)=\cos(t)-\sin(t)$ with $s(0)=9$ , $s'(0)=4$ , and find $s(\pi)$ as a decimal.

See Solution

Problem 32184

Find the linear approximation of $f(x, y)=\sqrt{x^{2}+y^{2}}$ at the point $(0,-3)$ .

See Solution

Problem 32185

Find the derivative of $\ln(9x + 4)$ with respect to $x$ .

See Solution

Problem 32186

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{3}{n(n+3)}$ .

See Solution

Problem 32187

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

See Solution

Problem 32188

Check if the series converges or diverges: $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2 n}$

See Solution

Problem 32189

Evaluate the series $\sum_{n=3}^{\infty} \frac{e^{-n}}{n^{2}+2n}$ .

See Solution

Problem 32190

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

See Solution

Problem 32191

Evaluate the series $\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}$ .

See Solution

Problem 32192

Does the series $\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}$ converge or diverge? Choose a comparison test option.

See Solution

Problem 32193

Find $f'(x)$ for $y=\ln(5x)$ .

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Problem 32194

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\cos ^{2}(n)}{n^{3 / 2}}$ . Options: a) none b) diverges c) converges with $\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}$ d) converges with $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ .

See Solution

Problem 32195

Leite die Funktion $b(x)=(1+x) \cdot(9-12 x+4 x^{2})$ mit der Produktregel ab: $b'(x) = \frac{d}{dx}[(1+x) \cdot(9-12 x+4 x^{2})]$ .

See Solution

Problem 32196

Check if the series converges or diverges: $\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}+1\right)^{2}$ .

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Problem 32197

Evaluate the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}$ .

See Solution

Problem 32198

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

See Solution

Problem 32199

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

See Solution

Problem 32200

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

See Solution

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Calculus

Problem 32101

Find the derivative of f(x)=44x2+7f(x)=4 \sqrt{4 x^{2}+7}f(x)=44x2+7​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 32102

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(8−9x)8f(x)=(8-9 x)^{8}f(x)=(8−9x)8.

Problem 32103

Find the missing expression in the equation: ddxln⁡(x7+6)=1x7+6?\frac{d}{d x} \ln \left(x^{7}+6\right)=\frac{1}{x^{7}+6} ?dxd​ln(x7+6)=x7+61​?

Problem 32104

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=e7xf(x)=e^{7 x}f(x)=e7x.

Problem 32105

Problem 32106

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=7ln⁡(1+2x2)f(x)=7 \ln(1+2 x^{2})f(x)=7ln(1+2x2).

Problem 32107

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=(x6+1)−4f(x)=(x^{6}+1)^{-4}f(x)=(x6+1)−4.

Problem 32108

Find the 3rd order Taylor polynomial for f(x)=2xf(x)=2 \sqrt{x}f(x)=2x​ at a=1a=1a=1.

Problem 32109

Find the radius and interval of convergence for the series ∑n=0∞(x−1)n4n\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{4^{n}}∑n=0∞​4n(x−1)n​. When does it converge conditionally?

Problem 32110

Find the surface area generated by revolving x=ey+e−y2x=\frac{e^{y}+e^{-y}}{2}x=2ey+e−y​ from y=0y=0y=0 to y=ln⁡7y=\ln 7y=ln7 around the yyy-axis. Area S=□S = \squareS=□.

Problem 32111

Find f′(x)f^{\prime}(x)f′(x), the tangent line at x=3x=3x=3 for f(x)=(6x−2)1/2f(x)=(6x-2)^{1/2}f(x)=(6x−2)1/2, and where the tangent is horizontal.

Problem 32112

Find the surface area of a cone by revolving y=rhxy=\frac{r}{h} xy=hr​x from 000 to hhh around the xxx-axis.

Problem 32113

Find f′(x)f^{\prime}(x)f′(x) and the tangent line at x=3x=3x=3 for f(x)=(6x−2)1/2f(x)=(6x-2)^{1/2}f(x)=(6x−2)1/2. Where is the tangent line horizontal?

Problem 32114

Find the derivative of y(x)=x2+10y(x)=\sqrt{x^{2}+10}y(x)=x2+10​. What is dydx\frac{d y}{d x}dxdy​?

Problem 32115

Find the derivative of 5(t2+7t)−65\left(t^{2}+7 t\right)^{-6}5(t2+7t)−6. What is ddt5(t2+7t)−6=□\frac{d}{d t} 5\left(t^{2}+7 t\right)^{-6}=\squaredtd​5(t2+7t)−6=□?

Problem 32116

Find f′(x)f^{\prime}(x)f′(x), the tangent line at x=2x=2x=2, and where the tangent line is horizontal for f(x)=(8x−7)1/2f(x)=(8x-7)^{1/2}f(x)=(8x−7)1/2.

Problem 32117

Find the Taylor series at x=0x=0x=0 for the following functions: a. f(x)=44−xf(x)=\frac{4}{4-x}f(x)=4−x4​, b. f(x)=1−cos⁡2xf(x)=1-\cos 2xf(x)=1−cos2x.

Problem 32118

Evaluate F′(5)F^{\prime}(5)F′(5) for the function F(x)=∫4x3ln⁡(t)dtF(x)=\int_{4}^{x^{3}} \ln (t) d tF(x)=∫4x3​ln(t)dt.

Problem 32119

Find the derivative and simplify: ddxln⁡(3+x)x7\frac{d}{d x} \frac{\ln (3+x)}{x^{7}}dxd​x7ln(3+x)​. Choose the correct answer.

Problem 32120

Find the derivative f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=1x=1x=1 for f(x)=x(6x−5)4f(x)=\frac{x}{(6x-5)^{4}}f(x)=(6x−5)4x​.

Problem 32121

Find f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=2x=2x=2 for f(x)=x(4x−7)9f(x)=\frac{x}{(4x-7)^{9}}f(x)=(4x−7)9x​.

Problem 32122

Evaluate the derivative: ddr(∫−6rf(t)dt7r8)\frac{d}{d r}\left(\frac{\int_{-6}^{r} f(t) d t}{7 r^{8}}\right)drd​(7r8∫−6r​f(t)dt​).

Problem 32123

Find the derivative g′(x)g^{\prime}(x)g′(x) for the function g(x)=2xe7xg(x)=2 x e^{7 x}g(x)=2xe7x.

Problem 32124

Find f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=2x=2x=2 for f(x)=x(4x−7)9f(x)=\frac{x}{(4 x-7)^{9}}f(x)=(4x−7)9x​.

Problem 32125

Find the derivative of the function: e2x(e4x−3)2e^{2x} (e^{4x} - 3)^2e2x(e4x−3)2.

Problem 32126

Find the area of the surface formed by revolving y=2x−x2y=\sqrt{2x-x^{2}}y=2x−x2​ from x=1x=1x=1 to x=1.5x=1.5x=1.5 about the xxx-axis. Set up the integral: s=∫11.52πy1+(dydx)2 dxs=\int_{1}^{1.5} 2\pi y \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} \, dxs=∫11.5​2πy1+(dxdy​)2​dx.

Problem 32127

Find f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=3x=3x=3 for f(x)=x(2x−5)5f(x)=\frac{x}{(2 x-5)^{5}}f(x)=(2x−5)5x​.

Problem 32128

Find f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=2x=2x=2 for f(x)=x(4x−7)9f(x)=\frac{x}{(4 x-7)^{9}}f(x)=(4x−7)9x​.

Problem 32129

A 9-N force stretches a rubber band. How far? Also, set up the integral for work done in joules.

Problem 32130

Determine if the series 200+40+8+1.6+…200 + 40 + 8 + 1.6 + \ldots200+40+8+1.6+… is convergent or divergent. If convergent, find the sum.

Problem 32131

Find f′(x)f^{\prime}(x)f′(x) and the tangent line equation at x=3x=3x=3 for f(x)=x(2x−5)5f(x)=\frac{x}{(2 x-5)^{5}}f(x)=(2x−5)5x​.

Problem 32132

Is the series ∑d=1∞−15(2)d−1\sum_{d=1}^{\infty}-15(2)^{d-1}d=1∑∞​−15(2)d−1 convergent or divergent? If convergent, find the sum.

Problem 32133

Find the derivative of 7(t2+3t)−57(t^{2}+3t)^{-5}7(t2+3t)−5. What is ddt7(t2+3t)−5=□\frac{d}{dt} 7(t^{2}+3t)^{-5}=\squaredtd​7(t2+3t)−5=□?

Problem 32134

Is the series ∑k=2∞52(13)k−1\sum_{k=2}^{\infty} \frac{5}{2}\left(\frac{1}{3}\right)^{k-1}∑k=2∞​25​(31​)k−1 convergent or divergent? If convergent, find the sum.

Problem 32135

A climber will lift a 30−m30-\mathrm{m}30−m rope weighing 0.5 N/m0.5 \mathrm{~N/m}0.5 N/m. Set up the work integral: w=∫□dxw=\int \square d xw=∫□dx.

Problem 32136

Find the derivative of 5(t2+7t)−45(t^{2}+7t)^{-4}5(t2+7t)−4. What is ddt5(t2+7t)−4=□\frac{d}{dt} 5(t^{2}+7t)^{-4} = \squaredtd​5(t2+7t)−4=□?

Problem 32137

A climber hauls a 40-m40\text{-m}40-m rope weighing 0.75 N/m0.75\text{ N/m}0.75 N/m. Set up the integral for the work done in joules.

Problem 32138

Problem 32139

Find the derivative f′(x)f^{\prime}(x)f′(x) and the values of xxx where the tangent line of f(x)=x7(x−10)3f(x)=x^{7}(x-10)^{3}f(x)=x7(x−10)3 is horizontal.

Problem 32140

Problem 32141

Find f′(x)f^{\prime}(x)f′(x) and the values of xxx where the tangent line of f(x)=x2−12x+44f(x)=\sqrt{x^{2}-12 x+44}f(x)=x2−12x+44​ is horizontal.

Find the derivative of $f(x)=4 \sqrt{4 x^{2}+7}$ . What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(8-9 x)^{8}$ .

Find the missing expression in the equation: $\frac{d}{d x} \ln \left(x^{7}+6\right)=\frac{1}{x^{7}+6} ?$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{7 x}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 \ln(1+2 x^{2})$ .

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

Find the 3rd order Taylor polynomial for $f(x)=2 \sqrt{x}$ at $a=1$ .

Find the radius and interval of convergence for the series $\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{4^{n}}$ . When does it converge conditionally?

Find the surface area generated by revolving $x=\frac{e^{y}+e^{-y}}{2}$ from $y=0$ to $y=\ln 7$ around the $y$ -axis.
Area $S = \square$ .

Find $f^{\prime}(x)$ , the tangent line at $x=3$ for $f(x)=(6x-2)^{1/2}$ , and where the tangent is horizontal.

Find the surface area of a cone by revolving $y=\frac{r}{h} x$ from $0$ to $h$ around the $x$ -axis.

Find $f^{\prime}(x)$ and the tangent line at $x=3$ for $f(x)=(6x-2)^{1/2}$ . Where is the tangent line horizontal?

Find the derivative of $y(x)=\sqrt{x^{2}+10}$ . What is $\frac{d y}{d x}$ ?

Find the derivative of $5\left(t^{2}+7 t\right)^{-6}$ . What is $\frac{d}{d t} 5\left(t^{2}+7 t\right)^{-6}=\square$ ?

Find $f^{\prime}(x)$ , the tangent line at $x=2$ , and where the tangent line is horizontal for $f(x)=(8x-7)^{1/2}$ .

Find the Taylor series at $x=0$ for the following functions: a. $f(x)=\frac{4}{4-x}$ , b. $f(x)=1-\cos 2x$ .

Evaluate $F^{\prime}(5)$ for the function $F(x)=\int_{4}^{x^{3}} \ln (t) d t$ .

Find the derivative and simplify: $\frac{d}{d x} \frac{\ln (3+x)}{x^{7}}$ . Choose the correct answer.

Find the derivative $f^{\prime}(x)$ and the tangent line equation at $x=1$ for $f(x)=\frac{x}{(6x-5)^{4}}$ .

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4x-7)^{9}}$ .

Evaluate the derivative: $\frac{d}{d r}\left(\frac{\int_{-6}^{r} f(t) d t}{7 r^{8}}\right)$ .

Find the derivative $g^{\prime}(x)$ for the function $g(x)=2 x e^{7 x}$ .

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4 x-7)^{9}}$ .

Find the derivative of the function: $e^{2x} (e^{4x} - 3)^2$ .

Find the area of the surface formed by revolving $y=\sqrt{2x-x^{2}}$ from $x=1$ to $x=1.5$ about the $x$ -axis. Set up the integral: $s=\int_{1}^{1.5} 2\pi y \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} \, dx$ .

Find $f^{\prime}(x)$ and the tangent line equation at $x=3$ for $f(x)=\frac{x}{(2 x-5)^{5}}$ .

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x}{(4 x-7)^{9}}$ .

Determine if the series $200 + 40 + 8 + 1.6 + \ldots$ is convergent or divergent. If convergent, find the sum.

Find $f^{\prime}(x)$ and the tangent line equation at $x=3$ for $f(x)=\frac{x}{(2 x-5)^{5}}$ .

Is the series $\sum_{d=1}^{\infty}-15(2)^{d-1}$ convergent or divergent? If convergent, find the sum.

Find the derivative of $7(t^{2}+3t)^{-5}$ . What is $\frac{d}{dt} 7(t^{2}+3t)^{-5}=\square$ ?

Is the series $\sum_{k=2}^{\infty} \frac{5}{2}\left(\frac{1}{3}\right)^{k-1}$ convergent or divergent? If convergent, find the sum.

A climber will lift a $30-\mathrm{m}$ rope weighing $0.5 \mathrm{~N/m}$ . Set up the work integral: $w=\int \square d x$ .

Find the derivative of $5(t^{2}+7t)^{-4}$ . What is $\frac{d}{dt} 5(t^{2}+7t)^{-4} = \square$ ?

A climber hauls a $40\text{-m}$ rope weighing $0.75\text{ N/m}$ . Set up the integral for the work done in joules.

Find the derivative $f^{\prime}(x)$ and the values of $x$ where the tangent line of $f(x)=x^{7}(x-10)^{3}$ is horizontal.

Find $f^{\prime}(x)$ and the values of $x$ where the tangent line of $f(x)=\sqrt{x^{2}-12 x+44}$ is horizontal.

Find the derivative of $5x(x^{5}+1)^{8}$ . What is $\frac{d}{d x}\left[5 x\left(x^{5}+1\right)^{8}\right]$ ?

Find $f^{\prime}(x)$ and the $x$ values where the tangent line is horizontal for $f(x)=\sqrt{x^{2}-12 x+44}$ .

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Calculate $C^{\prime}(24)$ .

Find the partial derivatives $f_{x x}, f_{x y}, f_{y y}$ for $f(x, y)=e^{4 x}-\sin(x+y^{2})-\sqrt{x y}$ .

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Calculate $C^{\prime}(x)$ and interpret.

Find $y^{\prime}$ for $2 x + 9 y + 5 = 0$ by implicit differentiation and solving for $y$ . What is the implicit result?

Find the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ . Analyze $C'(x)$ at $x=24$ and $x=42$ . Is cost increasing or decreasing?

Find the pressure at point $A$ in a tank with 0.6 m water above it. Options: A) $3.85 \mathrm{kPa}$ B) $5.88 \mathrm{kPa}$ C) $7.84 \mathrm{kPa}$ D) $9.80 \mathrm{kPa}$

Calculate the cost function $C(x)=10+\sqrt{2x+16}$ for $0 \leq x \leq 50$ and find $C'(42)$ .

Find $y^{\prime}$ for $9x + 5y + 3 = 0$ by implicit differentiation and direct differentiation. What is $y$ ?

Find $y^{\prime}$ for $9 x + 5 y + 3 = 0$ by implicit differentiation and direct differentiation. What is the implicit result?

Find the partial derivatives: $f(x, y, z)=\sin^{-1}(xy)-\sin(yz)$ ; compute $f_{xx}, f_{yz}, f_{xyz}$ .

Find $y^{\prime}$ for $9 x + 5 y + 3 = 0$ using implicit differentiation and direct differentiation.

Differentiate $2 x^{3}-9 y-11=0$ implicitly to find $y^{\prime}$ , then solve for $y$ and differentiate directly.

Find $y^{\prime}$ using implicit differentiation for $y - 2x^{2} + 7 = 0$ and evaluate at $(2,1)$ .

Find $y^{\prime}$ using implicit differentiation for $5 x^{2}-y^{2}-9=0$ and evaluate at $(3,6)$ .

Find $y^{\prime}$ using implicit differentiation for $y^{2}+2 y+3 x=0$ and evaluate at the point $(-1,1)$ .

Differentiate and solve: $y=\frac{(2 e^{2 x}+e^{3})^{2}}{e^{3 x}}$

Calculate the integral $\int_{0}^{2}|8 x-10| d x$ .

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for $G(x)$ , the antiderivative of $g(x)=e^{-5x}$ passing through $(0,6)$ .

Find the nonzero value of $x$ where the second derivative of $f(x) = 4x^6 - 8x^5$ is zero. Provide a decimal answer with at least three decimal places.

Find $z^{\prime}$ using implicit differentiation for $xz - 4 = 0$ and evaluate it at $(4,1)$ .

Find the value of $x$ where the first derivative of $g(x)=7x-10\ln(x)$ is zero. Provide your answer as a decimal with three places.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=6 x-5 \ln x^{4}$ .

Solve the equation $s^{\prime \prime}(t)=\cos (t)-\sin (t)$ with $s(0)=6$ , $s^{\prime}(0)=7$ and find $s(\pi)$ .

Find the value of $x$ where the first derivative of $g(x)=3x-3\ln(x)$ is zero. Provide your answer as a decimal with three decimal places.

Find the antiderivative of $f(x)=x^{3}-1$ with $F(3)=7$ . What is $F(0)$ ? Provide the answer as a decimal.

Find the nonzero $x$ where the second derivative of $f(x)=7x^{6}-10x^{5}$ equals zero. Provide your answer as a decimal with three decimal places.

Find the antiderivative of $f(x)=x^{3}-9$ with $F(3)=6$ . What is $F(0)$ as a decimal?