Calculus

Problem 23101

Find the value of the composition $(f \circ g)^{\prime}(2)$ using the given functions $f$ and $g$ .

See Solution

Problem 23102

Find $h^{\prime}(3)$ where $h(x)=\frac{f(x)}{g(x)}$ . Use values from $f$ and $g$ to compute. Round to three decimals or enter DNE.

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Problem 23103

Compute $(f \circ g)^{\prime}(2)$ using the given values of $f$ , $g$ , $f^{\prime}$ , and $g^{\prime}$ .

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Problem 23104

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ given values of $f$ and $f^{\prime}$ .

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Problem 23105

Evaluate the derivative of $(f \circ f)(x)$ at $x=1$ given $f(1)=0$ and $f'(1)=4$ . What is the exact value?

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Problem 23106

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ using the values of $f$ and $f^{\prime}$ given.

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Problem 23107

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=3$ , given $f(3)=2$ and $f^{\prime}(3)=4$ . Round to three decimal places.

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Problem 23108

Determine if $x=3$ is a local max or min and if $x=0$ is a local max or min for $g'(-\frac{x(x-3)}{(x-5)e^{2x}})$ .

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Problem 23109

Find the slope of the tangent line to $y=\tan^{-1}(x)$ at $x=8$ . Round your answer to three decimal places.

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Problem 23110

Find the derivative of $x^{2}+e^{f(x)}+\ln (f(x))$ at $x=3$ , given $f(3)=4$ and $f'(3)=3$ . Round to three decimal places.

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Problem 23111

Evaluate the derivative of $(f \circ f)(x)$ at $x=1$ given values of $f$ and $f'(x)$ . What is the exact value?

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Problem 23112

Find the positive and negative critical numbers of $h(x)=\frac{x}{x^{2}+9}$ and classify them as max, min, or neither.

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Problem 23113

True or false: If $f$ is differentiable, is $\frac{d}{d x}[3+\sqrt{f(\sin (x))}]=\frac{\cos (x) f^{\prime}(\sin (x))}{2 \sqrt{f(\sin (x))}}$ ?

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Problem 23114

Estimate $f(6.4)$ using linear approximation given $f(6)=8$ and $f^{\prime}(6)=5.6$ . Round to three decimal places.

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Problem 23115

Find the slope of the tangent line to $x^{3}+y^{3}=3xy+3$ at $(1,2)$ . Round your answer to three decimal places.

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Problem 23116

Find the slope of the tangent line to $y=\tan^{-1}(x)$ at $x=4$ . Round your answer to three decimal places.

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Problem 23117

Determine the type of critical numbers $x=0$ and $x=3$ for $h$ using the Second Derivative Test with $h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}$ .

See Solution

Problem 23118

Estimate $f(8.5)$ using linear approximation given $f(8)=3.9$ and $f^{\prime}(8)=1.7$ . Round to three decimals.

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Problem 23119

Identify the type of critical points $x=0$ and $x=3$ for $h$ using the Second Derivative Test given $h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}$ .

See Solution

Problem 23120

Find the slope of the tangent line to $x^{3}+y^{3}=3xy+3$ at the point $(1,2)$ . Round your answer to three decimal places.

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Problem 23121

Find $f'(0)$ for the piecewise function $f(x) = \{-6x^2 + 6x \text{ for } x<0, 3x^2 - 4 \text{ for } x \geq 0\}$ .

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Problem 23122

True or false: For a differentiable function $f$ , is $\frac{d}{d x}\left[f\left(x^{3}\right)\right]=f^{\prime}\left(3 x^{2}\right)$ ?

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Problem 23123

Compute $(f \circ g)^{\prime}(2)$ using the given values of $f$ , $g$ , $f^{\prime}$ , and $g^{\prime}$ .

See Solution

Problem 23124

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=4$ , given $f(4)=2$ and $f'(4)=3$ . Round to three decimal places.

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Problem 23125

Estimate $f(8.4)$ using the linear approximation given $f(8)=2.5$ and $f'(8)=7.9$ . Round to three decimal places.

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Problem 23126

Find the positive critical number of $h(x)=\frac{x}{x^{2}+9}$ . Is it a max, min, or neither? Identify the negative critical number $-3$ and its classification.

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Problem 23127

Find the area under the curve $y=9 \sqrt[3]{x}$ from $x=0$ to $x=27$ .

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Problem 23128

Calculate the area between $y=x^{2}+x+4$ and $y=8$ for $2 \leq x \leq 6. Area =$

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Problem 23129

Given the acceleration $a(t)=2 t+2$ and initial velocity $v(0)=-3$ , find the velocity $v(t)$ and distance traveled.

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Problem 23130

Calculate the area between the curves $y=x^{2}-4x+8$ and $y=3$ from $x=2$ to $x=6$ . Area $=$

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Problem 23131

Sketch the area between $y=e^{x}$ , $y=e^{2x}$ , $x=0$ , and $x=\ln 3$ . Find the area.

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Problem 23132

Water flows from a tank at $r(t) = 400 - 8t$ liters/min. Find the total liters flowing out in the first 30 minutes.

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Problem 23133

Find the derivative of the function $F(x)=\int_{\sqrt{x}}^{1} \frac{s^{2}}{4+4 s^{4}} d s$ using the Fundamental Theorem of Calculus. $F'(x)=$

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Problem 23134

Find the average rate of change of $f(x)$ from $x_1=0$ to $x_2=4$ using points (0,9) and (4,17). Answer as an integer or fraction.

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Problem 23135

Find $f'(0)$ for the piecewise function: $f(x)=\begin{cases} 7x-4x^2 & x<0 \\ 5x^2+7x & x \geq 0 \end{cases}$ .

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Problem 23136

Find the derivative $f^{\prime}(0)$ by calculating the left-hand limit $\lim _{x \rightarrow 0^{-}} \square$ and right-hand limit $\lim _{x \rightarrow 0^{+}} \square$ .

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Problem 23137

Find the limit as $x$ approaches 2 from the right: $\lim _{x \rightarrow 2^{+}} \frac{15}{x-2}+3$ .

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Problem 23138

Approximate the area under $f(x)=1-|x|$ on $[-1,1]$ using 10 rectangles at right endpoints. Sketch the graph and Riemann Sum.

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Problem 23139

Evaluate the integral from 0 to 1 of the function $x^{3}-3 x^{2}$ .

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Problem 23140

Find the position function $p(t)$ of a particle with acceleration $a(t)=10 \sin (t)+3 \cos (t)$ , given $p(0)=0$ and $p(2\pi)=12$ .

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Problem 23141

A pot of boiling water cools in a room. Use $T(x)=20+80 e^{-x}$ to find: (a) Temperature after 2 hours. (b) Time to reach $30^{\circ} \mathrm{C}$ .

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Problem 23142

Find the derivative of $x^{6}+\ln(x^{3}+2xy+3y^{2})$ with respect to $x$ .

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Problem 23143

Find the derivative with respect to $y$ of the function $x^{3} \ln(6xy + 3x + y^{5})$ .

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Problem 23144

Find the derivative of $e^{2 x^{2}+11 x y+y^{2}}$ with respect to $x$ .

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Problem 23145

Find the derivative of $200 x^{\frac{1}{2}} y^{\frac{3}{4}}$ with respect to $y$ .

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Problem 23146

Find the derivative of $(x+4) y^{3} e^{x y}$ with respect to $y$ .

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Problem 23147

Find the derivative $\frac{\partial}{\partial x} \left( x^{6} \ln \left(x y+7 x+y^{2}\right) \right)$ .

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Problem 23148

Find the derivative $\frac{\partial}{\partial y}\left(x^{6}+\ln \left(x^{3}+5 x y+3 y^{2}\right)\right)$ .

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Problem 23149

If \ $23,000 is invested at 4.9% interest compounded continuously, how long to quadruple? Use$ A=P e^{rt}$.

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Problem 23150

Find the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}(x^{2}+y^{4}+e^{x^{2}+3 y^{2}})$ .

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Problem 23151

Find the second partial derivative of $x^{7} y^{8}$ with respect to $y$ : $\frac{\partial^{2}}{\partial y^{2}} x^{7} y^{8}$ .

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Problem 23152

Find the second mixed partial derivative of the function $x^{7} y^{6}$ with respect to $x$ and $y$ .

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Problem 23153

Differentiate the function $f(t)=\sqrt{t}(a+b t)$ .

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Problem 23154

Find the second partial derivative of $x^{3} y^{4}$ with respect to $x$ .

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Problem 23155

Evaluate the integral $\int_{-1}^{1}(2 x^{2}+3) \, dx$ using the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule.

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Problem 23156

Find the second mixed partial derivative $\frac{\partial^{2}}{\partial x \partial y} e^{3 x^{2}+11 x y+5 y^{2}}$ .

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Problem 23157

Estimate the integral $\int_{-1}^{1}\left(2 x^{2}+3\right) d x$ using Midpoint, Trapezoidal, and Simpson's Rules with $n=4$ .

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Problem 23158

Find points $(x, y)$ where $f(x, y)=x^{2}-y^{3}+6 x+48 y-3$ has a relative max, min, or saddle point.

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Problem 23159

Classify the critical point of the function $f(x, y)=x^{2}+x y-3 x$ as a relative maximum, minimum, or saddle point.

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Problem 23160

Classify the critical point $(1,1)$ of the function $f(x, y)=4+x^{3}+y^{3}-3 x y$ as max, min, or saddle.

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Problem 23161

Find the other critical point of the function $f(x, y)=-4 x^{2}+8 x+3 y^{3}-9 y$ given one is at $(1,-1)$ .

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Problem 23162

Approximate $\int_{1}^{9} (x^{2}+3x) \, dx$ using the midpoint rule with $n=4$ . (Ans: 360)

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Problem 23163

Compare the account balance after 5 years with continuous vs monthly compounding for an initial amount of \$10000 at 3\% interest.

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Problem 23164

Find $y^{\prime}$ given $\tan (x+y)=\sin (x)$ where $y$ depends on $x$ .

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Problem 23165

Find the production level $x$ that minimizes the average cost given $c(x)=4 x^{3}-56 x^{2}+24,000 x$ .

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Problem 23166

Find the derivative of the function $f(x)=\frac{1}{(x^{3}-\sec(3x^{2}-7))^{3}}$ . What is $f^{\prime}(x)$ ?

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Problem 23167

Solve the initial value problem: $\frac{d y}{d x}=2 x^{-\frac{4}{5}}$ , with $y(-1)=-4$ . Find $y(x)$ .

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Problem 23168

Find the function $y(x)$ where $\frac{d y}{d x}=4 x-7$ and $y(9)=0$ . What is $y(x)$ ?

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Problem 23169

Find the angle $\theta$ that maximizes the volume of the trough shown in the figure.

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Problem 23170

Find the production level $x$ that minimizes the average cost given $c(x)=4x^{3}-56x^{2}+24000x$ . What is the average cost?

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Problem 23171

Find the antiderivative $s$ of $\frac{ds}{dt}=3 \sin 3t - 4 \cos 4t$ with $s(\frac{\pi}{2})=4$ .

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Problem 23172

Find the antiderivative $s(t)$ for $\frac{d s}{d t}=7 \sin 7 t+2 \cos 2 t$ with $s\left(\frac{\pi}{2}\right)=10$ .

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Problem 23173

Find the function $s(t)$ where $\frac{d s}{d t}=4-7 \cos t$ and $s(0)=2$ . What is $s(t)$ ?

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Problem 23174

Find the Riemann sum for $f(x)=x+11x^{2}$ over $[0,1]$ using right endpoints, then take the limit as $n \rightarrow \infty$ .

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Problem 23175

Find the area between $f(x)=x^{2}+2x+10$ and $g(x)=2x^{2}+x-2$ by integrating $|f(x) - g(x)|$ over their intersection points.

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Problem 23176

Evaluate the integral: $\int \frac{5 e^{5 t}}{7+e^{5 t}} d t = \square$

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Problem 23177

Evaluate the integral: $\int_{\frac{1}{5}}^{\frac{1}{3}} \frac{12^{\frac{1}{p}}}{p^{2}} d p=\square$ (Type an exact answer.)

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Problem 23178

Find the derivative of the function $h(x)=x e^{2 x}$ .

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Problem 23179

Find the rate of change of the mosquito population $A(t)=1000 e^{0.3 t}$ after 4 days.

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Problem 23180

How many years ago was a wooden artifact made if it has 25% of carbon-14? (Half-life: 5730 years) Round to the nearest year.

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Problem 23181

A population of 60 declines at $2.57\%$ per year. Find the half-life in years, rounded to four decimal places.

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Problem 23182

Evaluate the integral: $\int_{\sqrt{x}}^{1} \frac{s^{2}}{2+4 s^{4}} d s$ .

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Problem 23183

Find the derivative of the function $f(x)=\ln \left(\frac{x^{2} \sin x}{2 x+1}\right)$ .

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Problem 23184

A population starts at 30 and declines at $2.03\%$ per year. Find the half-life in years, rounded to four decimal places.

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Problem 23185

Find the limit as $x$ approaches -3 for the expression $\frac{x+9}{(x+3)^{2}}$ .

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Problem 23186

Find the second partial derivatives of the function $f(x, y)=6 x^{2}-4 x y^{4}+y^{6}$ : $f_{x x}(x, y)$ and $f_{x y}(x, y)$ .

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Problem 23187

Find the marginal productivity of labor and capital for the function $P(L, K)=26 L^{0.9} K^{0.1}$ with $L=19$ and $K=20$ .

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Problem 23188

A fiber facility's output is given by $z=f(x, y)=11500 x+4500 y+5 x^{2} y-6 x^{3}$ .
A) Find output for 14 skilled and 42 unskilled workers. B) Determine $f_{x}$ . C) Determine $f_{y}$ . D) Find output change rate for skilled workers at 14 and 42. E) Find output change rate for unskilled workers at 14 and 42.

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Problem 23189

A bacteria doubles every 48.8 min. Given $N(t)=300,000 e^{0.014 t}$ , find (a) $N(120)$ and (b) time for $N=10,000,000$ .

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Problem 23190

Find the partial derivatives $f_{x}(x, y)$ , $f_{y}(x, y)$ , $f_{xx}(x, y)$ , and $f_{xy}(x, y)$ for $f(x, y)=-6 x^{6}-3 x^{2} y^{2}+4 y^{5}$ .

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Problem 23191

Differentiate: $f(x)=\ln (3x+2)^{5}$ .

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Problem 23192

Find the limit as $x$ approaches -3 from the left for the expression $\frac{6x+2}{x+3}$ .

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Problem 23193

Calculate $R_{6}$ for $f(x) = \frac{1}{x(x-1)}$ on $[2,5]$ using the right-endpoint Riemann sum.

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Problem 23194

Find the derivative $f^{\prime}(x)$ for the function $f(x)=5x+2\sqrt{x}$ .

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Problem 23195

Evaluate the integral: $\int \frac{d x}{(2 x+11)^{4}}$ using the substitution $u=$ .

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Problem 23196

Find the tangent line equation to $f(x)=\frac{5}{1-3x}$ at the point $P(2, f(2))$ .

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Problem 23197

Differentiate a) $y=\frac{2}{\sin^{-1} x}$ b) $y=\csc^{-1} x^{2}$ .

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Problem 23198

Find the limit as $x$ approaches 4 for the expression $\frac{3}{x-4}\left(1+\frac{2}{x-6}\right)$ .

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Problem 23199

Evaluate the integral $\int x^{3}(x^{4}-6)^{29} dx$ using the substitution $u=x^{4}-6$ .

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Problem 23200

Calculate the Riemann sum $R_{6}$ for $f(x)=\frac{1}{x(x-1)}$ on $[2,5]$ with 6 subdivisions. Specify the method used.

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Calculus

Problem 23101

Find the value of the composition (f∘g)′(2)(f \circ g)^{\prime}(2)(f∘g)′(2) using the given functions fff and ggg.

Problem 23102

Find h′(3)h^{\prime}(3)h′(3) where h(x)=f(x)g(x)h(x)=\frac{f(x)}{g(x)}h(x)=g(x)f(x)​. Use values from fff and ggg to compute. Round to three decimals or enter DNE.

Problem 23103

Compute (f∘g)′(2)(f \circ g)^{\prime}(2)(f∘g)′(2) using the given values of fff, ggg, f′f^{\prime}f′, and g′g^{\prime}g′.

Problem 23104

Evaluate the derivative of (f(x))2(f(x))^{2}(f(x))2 at x=5x=5x=5 given values of fff and f′f^{\prime}f′.

Problem 23105

Evaluate the derivative of (f∘f)(x)(f \circ f)(x)(f∘f)(x) at x=1x=1x=1 given f(1)=0f(1)=0f(1)=0 and f′(1)=4f'(1)=4f′(1)=4. What is the exact value?

Problem 23106

Evaluate the derivative of (f(x))2(f(x))^{2}(f(x))2 at x=5x=5x=5 using the values of fff and f′f^{\prime}f′ given.

Problem 23107

Find the derivative of x2+ef(x)+ln⁡(f(x))x^{2}+e^{f(x)}+\ln(f(x))x2+ef(x)+ln(f(x)) at x=3x=3x=3, given f(3)=2f(3)=2f(3)=2 and f′(3)=4f^{\prime}(3)=4f′(3)=4. Round to three decimal places.

Problem 23108

Determine if x=3x=3x=3 is a local max or min and if x=0x=0x=0 is a local max or min for g′(−x(x−3)(x−5)e2x)g'(-\frac{x(x-3)}{(x-5)e^{2x}})g′(−(x−5)e2xx(x−3)​).

Problem 23109

Find the slope of the tangent line to y=tan⁡−1(x)y=\tan^{-1}(x)y=tan−1(x) at x=8x=8x=8. Round your answer to three decimal places.

Problem 23110

Find the derivative of x2+ef(x)+ln⁡(f(x))x^{2}+e^{f(x)}+\ln (f(x))x2+ef(x)+ln(f(x)) at x=3x=3x=3, given f(3)=4f(3)=4f(3)=4 and f′(3)=3f'(3)=3f′(3)=3. Round to three decimal places.

Problem 23111

Evaluate the derivative of (f∘f)(x)(f \circ f)(x)(f∘f)(x) at x=1x=1x=1 given values of fff and f′(x)f'(x)f′(x). What is the exact value?

Problem 23112

Find the positive and negative critical numbers of h(x)=xx2+9h(x)=\frac{x}{x^{2}+9}h(x)=x2+9x​ and classify them as max, min, or neither.

Problem 23113

True or false: If fff is differentiable, is ddx[3+f(sin⁡(x))]=cos⁡(x)f′(sin⁡(x))2f(sin⁡(x))\frac{d}{d x}[3+\sqrt{f(\sin (x))}]=\frac{\cos (x) f^{\prime}(\sin (x))}{2 \sqrt{f(\sin (x))}}dxd​[3+f(sin(x))​]=2f(sin(x))​cos(x)f′(sin(x))​?

Problem 23114

Estimate f(6.4)f(6.4)f(6.4) using linear approximation given f(6)=8f(6)=8f(6)=8 and f′(6)=5.6f^{\prime}(6)=5.6f′(6)=5.6. Round to three decimal places.

Problem 23115

Find the slope of the tangent line to x3+y3=3xy+3x^{3}+y^{3}=3xy+3x3+y3=3xy+3 at (1,2)(1,2)(1,2). Round your answer to three decimal places.

Problem 23116

Find the slope of the tangent line to y=tan⁡−1(x)y=\tan^{-1}(x)y=tan−1(x) at x=4x=4x=4. Round your answer to three decimal places.

Problem 23117

Determine the type of critical numbers x=0x=0x=0 and x=3x=3x=3 for hhh using the Second Derivative Test with h′′(x)=x(x+5)f(x)(x−7)2h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}h′′(x)=(x−7)2x(x+5)f(x)​.

Problem 23118

Estimate f(8.5)f(8.5)f(8.5) using linear approximation given f(8)=3.9f(8)=3.9f(8)=3.9 and f′(8)=1.7f^{\prime}(8)=1.7f′(8)=1.7. Round to three decimals.

Problem 23119

Identify the type of critical points x=0x=0x=0 and x=3x=3x=3 for hhh using the Second Derivative Test given h′′(x)=x(x+5)f(x)(x−7)2h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}h′′(x)=(x−7)2x(x+5)f(x)​.

Problem 23120

Find the slope of the tangent line to x3+y3=3xy+3x^{3}+y^{3}=3xy+3x3+y3=3xy+3 at the point (1,2)(1,2)(1,2). Round your answer to three decimal places.

Problem 23121

Find f′(0)f'(0)f′(0) for the piecewise function f(x)={−6x2+6x for x<0,3x2−4 for x≥0}f(x) = \{-6x^2 + 6x \text{ for } x<0, 3x^2 - 4 \text{ for } x \geq 0\}f(x)={−6x2+6x for x<0,3x2−4 for x≥0}.

Problem 23122

True or false: For a differentiable function fff, is ddx[f(x3)]=f′(3x2)\frac{d}{d x}\left[f\left(x^{3}\right)\right]=f^{\prime}\left(3 x^{2}\right)dxd​[f(x3)]=f′(3x2)?

Problem 23123

Compute (f∘g)′(2)(f \circ g)^{\prime}(2)(f∘g)′(2) using the given values of fff, ggg, f′f^{\prime}f′, and g′g^{\prime}g′.

Problem 23124

Find the derivative of x2+ef(x)+ln⁡(f(x))x^{2}+e^{f(x)}+\ln(f(x))x2+ef(x)+ln(f(x)) at x=4x=4x=4, given f(4)=2f(4)=2f(4)=2 and f′(4)=3f'(4)=3f′(4)=3. Round to three decimal places.

Problem 23125

Estimate f(8.4)f(8.4)f(8.4) using the linear approximation given f(8)=2.5f(8)=2.5f(8)=2.5 and f′(8)=7.9f'(8)=7.9f′(8)=7.9. Round to three decimal places.

Problem 23126

Find the positive critical number of h(x)=xx2+9h(x)=\frac{x}{x^{2}+9}h(x)=x2+9x​. Is it a max, min, or neither? Identify the negative critical number −3-3−3 and its classification.

Problem 23127

Find the area under the curve y=9x3y=9 \sqrt[3]{x}y=93x​ from x=0x=0x=0 to x=27x=27x=27.

Problem 23128

Calculate the area between y=x2+x+4y=x^{2}+x+4y=x2+x+4 and y=8y=8y=8 for 2≤x≤6.Area=2 \leq x \leq 6. Area = 2≤x≤6.Area=

Problem 23129

Given the acceleration a(t)=2t+2a(t)=2 t+2a(t)=2t+2 and initial velocity v(0)=−3v(0)=-3v(0)=−3, find the velocity v(t)v(t)v(t) and distance traveled.

Problem 23130

Calculate the area between the curves y=x2−4x+8y=x^{2}-4x+8y=x2−4x+8 and y=3y=3y=3 from x=2x=2x=2 to x=6x=6x=6. Area ===

Problem 23131

Sketch the area between y=exy=e^{x}y=ex, y=e2xy=e^{2x}y=e2x, x=0x=0x=0, and x=ln⁡3x=\ln 3x=ln3. Find the area.

Problem 23132

Water flows from a tank at r(t)=400−8tr(t) = 400 - 8tr(t)=400−8t liters/min. Find the total liters flowing out in the first 30 minutes.

Problem 23133

Find the derivative of the function F(x)=∫x1s24+4s4dsF(x)=\int_{\sqrt{x}}^{1} \frac{s^{2}}{4+4 s^{4}} d sF(x)=∫x​1​4+4s4s2​ds using the Fundamental Theorem of Calculus. F′(x)=F'(x)=F′(x)=

Problem 23134

Find the average rate of change of f(x)f(x)f(x) from x1=0x_1=0x1​=0 to x2=4x_2=4x2​=4 using points (0,9) and (4,17). Answer as an integer or fraction.

Problem 23135

Find f′(0)f'(0)f′(0) for the piecewise function: f(x)={7x−4x2x<05x2+7xx≥0f(x)=\begin{cases} 7x-4x^2 & x<0 \\ 5x^2+7x & x \geq 0 \end{cases}f(x)={7x−4x25x2+7x​x<0x≥0​.

Problem 23136

Find the derivative f′(0)f^{\prime}(0)f′(0) by calculating the left-hand limit lim⁡x→0−□\lim _{x \rightarrow 0^{-}} \squarelimx→0−​□ and right-hand limit lim⁡x→0+□\lim _{x \rightarrow 0^{+}} \squarelimx→0+​□.

Problem 23137

Find the limit as xxx approaches 2 from the right: lim⁡x→2+15x−2+3\lim _{x \rightarrow 2^{+}} \frac{15}{x-2}+3limx→2+​x−215​+3.

Problem 23138

Approximate the area under f(x)=1−∣x∣f(x)=1-|x|f(x)=1−∣x∣ on [−1,1][-1,1][−1,1] using 10 rectangles at right endpoints. Sketch the graph and Riemann Sum.

Problem 23139

Evaluate the integral from 0 to 1 of the function x3−3x2x^{3}-3 x^{2}x3−3x2.

Problem 23140

Find the value of the composition $(f \circ g)^{\prime}(2)$ using the given functions $f$ and $g$ .

Find $h^{\prime}(3)$ where $h(x)=\frac{f(x)}{g(x)}$ . Use values from $f$ and $g$ to compute. Round to three decimals or enter DNE.

Compute $(f \circ g)^{\prime}(2)$ using the given values of $f$ , $g$ , $f^{\prime}$ , and $g^{\prime}$ .

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ given values of $f$ and $f^{\prime}$ .

Evaluate the derivative of $(f \circ f)(x)$ at $x=1$ given $f(1)=0$ and $f'(1)=4$ . What is the exact value?

Evaluate the derivative of $(f(x))^{2}$ at $x=5$ using the values of $f$ and $f^{\prime}$ given.

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=3$ , given $f(3)=2$ and $f^{\prime}(3)=4$ . Round to three decimal places.

Determine if $x=3$ is a local max or min and if $x=0$ is a local max or min for $g'(-\frac{x(x-3)}{(x-5)e^{2x}})$ .

Find the slope of the tangent line to $y=\tan^{-1}(x)$ at $x=8$ . Round your answer to three decimal places.

Find the derivative of $x^{2}+e^{f(x)}+\ln (f(x))$ at $x=3$ , given $f(3)=4$ and $f'(3)=3$ . Round to three decimal places.

Evaluate the derivative of $(f \circ f)(x)$ at $x=1$ given values of $f$ and $f'(x)$ . What is the exact value?

Find the positive and negative critical numbers of $h(x)=\frac{x}{x^{2}+9}$ and classify them as max, min, or neither.

True or false: If $f$ is differentiable, is $\frac{d}{d x}[3+\sqrt{f(\sin (x))}]=\frac{\cos (x) f^{\prime}(\sin (x))}{2 \sqrt{f(\sin (x))}}$ ?

Estimate $f(6.4)$ using linear approximation given $f(6)=8$ and $f^{\prime}(6)=5.6$ . Round to three decimal places.

Find the slope of the tangent line to $x^{3}+y^{3}=3xy+3$ at $(1,2)$ . Round your answer to three decimal places.

Find the slope of the tangent line to $y=\tan^{-1}(x)$ at $x=4$ . Round your answer to three decimal places.

Determine the type of critical numbers $x=0$ and $x=3$ for $h$ using the Second Derivative Test with $h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}$ .

Estimate $f(8.5)$ using linear approximation given $f(8)=3.9$ and $f^{\prime}(8)=1.7$ . Round to three decimals.

Identify the type of critical points $x=0$ and $x=3$ for $h$ using the Second Derivative Test given $h^{\prime \prime}(x)=\frac{x(x+5) f(x)}{(x-7)^{2}}$ .

Find the slope of the tangent line to $x^{3}+y^{3}=3xy+3$ at the point $(1,2)$ . Round your answer to three decimal places.

Find $f'(0)$ for the piecewise function $f(x) = \{-6x^2 + 6x \text{ for } x<0, 3x^2 - 4 \text{ for } x \geq 0\}$ .

True or false: For a differentiable function $f$ , is $\frac{d}{d x}\left[f\left(x^{3}\right)\right]=f^{\prime}\left(3 x^{2}\right)$ ?

Compute $(f \circ g)^{\prime}(2)$ using the given values of $f$ , $g$ , $f^{\prime}$ , and $g^{\prime}$ .

Find the derivative of $x^{2}+e^{f(x)}+\ln(f(x))$ at $x=4$ , given $f(4)=2$ and $f'(4)=3$ . Round to three decimal places.

Estimate $f(8.4)$ using the linear approximation given $f(8)=2.5$ and $f'(8)=7.9$ . Round to three decimal places.

Find the positive critical number of $h(x)=\frac{x}{x^{2}+9}$ . Is it a max, min, or neither? Identify the negative critical number $-3$ and its classification.

Find the area under the curve $y=9 \sqrt[3]{x}$ from $x=0$ to $x=27$ .

Calculate the area between $y=x^{2}+x+4$ and $y=8$ for $2 \leq x \leq 6. Area =$

Given the acceleration $a(t)=2 t+2$ and initial velocity $v(0)=-3$ , find the velocity $v(t)$ and distance traveled.

Calculate the area between the curves $y=x^{2}-4x+8$ and $y=3$ from $x=2$ to $x=6$ . Area $=$

Sketch the area between $y=e^{x}$ , $y=e^{2x}$ , $x=0$ , and $x=\ln 3$ . Find the area.

Water flows from a tank at $r(t) = 400 - 8t$ liters/min. Find the total liters flowing out in the first 30 minutes.

Find the derivative of the function $F(x)=\int_{\sqrt{x}}^{1} \frac{s^{2}}{4+4 s^{4}} d s$ using the Fundamental Theorem of Calculus. $F'(x)=$

Find the average rate of change of $f(x)$ from $x_1=0$ to $x_2=4$ using points (0,9) and (4,17). Answer as an integer or fraction.

Find $f'(0)$ for the piecewise function: $f(x)=\begin{cases} 7x-4x^2 & x<0 \\ 5x^2+7x & x \geq 0 \end{cases}$ .

Find the derivative $f^{\prime}(0)$ by calculating the left-hand limit $\lim _{x \rightarrow 0^{-}} \square$ and right-hand limit $\lim _{x \rightarrow 0^{+}} \square$ .

Find the limit as $x$ approaches 2 from the right: $\lim _{x \rightarrow 2^{+}} \frac{15}{x-2}+3$ .

Approximate the area under $f(x)=1-|x|$ on $[-1,1]$ using 10 rectangles at right endpoints. Sketch the graph and Riemann Sum.

Evaluate the integral from 0 to 1 of the function $x^{3}-3 x^{2}$ .

Find the position function $p(t)$ of a particle with acceleration $a(t)=10 \sin (t)+3 \cos (t)$ , given $p(0)=0$ and $p(2\pi)=12$ .

A pot of boiling water cools in a room. Use $T(x)=20+80 e^{-x}$ to find: (a) Temperature after 2 hours. (b) Time to reach $30^{\circ} \mathrm{C}$ .

Find the derivative of $x^{6}+\ln(x^{3}+2xy+3y^{2})$ with respect to $x$ .

Find the derivative with respect to $y$ of the function $x^{3} \ln(6xy + 3x + y^{5})$ .

Find the derivative of $e^{2 x^{2}+11 x y+y^{2}}$ with respect to $x$ .

Find the derivative of $200 x^{\frac{1}{2}} y^{\frac{3}{4}}$ with respect to $y$ .

Find the derivative of $(x+4) y^{3} e^{x y}$ with respect to $y$ .

Find the derivative $\frac{\partial}{\partial x} \left( x^{6} \ln \left(x y+7 x+y^{2}\right) \right)$ .

Find the derivative $\frac{\partial}{\partial y}\left(x^{6}+\ln \left(x^{3}+5 x y+3 y^{2}\right)\right)$ .

If \ $23,000 is invested at 4.9% interest compounded continuously, how long to quadruple? Use$ A=P e^{rt}$.

Find the second partial derivative $\frac{\partial^{2}}{\partial x^{2}}(x^{2}+y^{4}+e^{x^{2}+3 y^{2}})$ .

Find the second partial derivative of $x^{7} y^{8}$ with respect to $y$ : $\frac{\partial^{2}}{\partial y^{2}} x^{7} y^{8}$ .

Find the second mixed partial derivative of the function $x^{7} y^{6}$ with respect to $x$ and $y$ .

Differentiate the function $f(t)=\sqrt{t}(a+b t)$ .

Find the second partial derivative of $x^{3} y^{4}$ with respect to $x$ .

Evaluate the integral $\int_{-1}^{1}(2 x^{2}+3) \, dx$ using the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule.

Find the second mixed partial derivative $\frac{\partial^{2}}{\partial x \partial y} e^{3 x^{2}+11 x y+5 y^{2}}$ .

Estimate the integral $\int_{-1}^{1}\left(2 x^{2}+3\right) d x$ using Midpoint, Trapezoidal, and Simpson's Rules with $n=4$ .

Find points $(x, y)$ where $f(x, y)=x^{2}-y^{3}+6 x+48 y-3$ has a relative max, min, or saddle point.

Classify the critical point of the function $f(x, y)=x^{2}+x y-3 x$ as a relative maximum, minimum, or saddle point.

Classify the critical point $(1,1)$ of the function $f(x, y)=4+x^{3}+y^{3}-3 x y$ as max, min, or saddle.

Find the other critical point of the function $f(x, y)=-4 x^{2}+8 x+3 y^{3}-9 y$ given one is at $(1,-1)$ .

Approximate $\int_{1}^{9} (x^{2}+3x) \, dx$ using the midpoint rule with $n=4$ . (Ans: 360)

Find $y^{\prime}$ given $\tan (x+y)=\sin (x)$ where $y$ depends on $x$ .

Find the production level $x$ that minimizes the average cost given $c(x)=4 x^{3}-56 x^{2}+24,000 x$ .

Find the derivative of the function $f(x)=\frac{1}{(x^{3}-\sec(3x^{2}-7))^{3}}$ . What is $f^{\prime}(x)$ ?

Solve the initial value problem: $\frac{d y}{d x}=2 x^{-\frac{4}{5}}$ , with $y(-1)=-4$ . Find $y(x)$ .

Find the function $y(x)$ where $\frac{d y}{d x}=4 x-7$ and $y(9)=0$ . What is $y(x)$ ?

Find the angle $\theta$ that maximizes the volume of the trough shown in the figure.

Find the production level $x$ that minimizes the average cost given $c(x)=4x^{3}-56x^{2}+24000x$ . What is the average cost?

Find the antiderivative $s$ of $\frac{ds}{dt}=3 \sin 3t - 4 \cos 4t$ with $s(\frac{\pi}{2})=4$ .

Find the antiderivative $s(t)$ for $\frac{d s}{d t}=7 \sin 7 t+2 \cos 2 t$ with $s\left(\frac{\pi}{2}\right)=10$ .

Find the function $s(t)$ where $\frac{d s}{d t}=4-7 \cos t$ and $s(0)=2$ . What is $s(t)$ ?

Find the Riemann sum for $f(x)=x+11x^{2}$ over $[0,1]$ using right endpoints, then take the limit as $n \rightarrow \infty$ .