Calculus

Problem 14101

Determine if the function $f(x) = \{6x^2 - 5 \text{ for } x \leq 3, 15x + 9 \text{ for } x > 3\}$ is continuous at $x=3$ .

See Solution

Problem 14102

Evaluate the integral $\int x \sin \frac{1}{11} x \, dx$ using integration by parts. Which option simplifies it?

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Problem 14103

Find the average rate of change of revenue $\mathrm{R}(x)=600 x-0.3 x^{2}$ from 100 to 200 units. Also, differentiate $6+5 x-x^{2}$ using first principles.

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Problem 14104

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{\cos (2 x)-1}$ .

See Solution

Problem 14105

Evaluate the limit as $x$ approaches -2 for the expression $7x^{2} + 8$ . What is the result?

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Problem 14106

Differentiate the function $6 + 5x - x^{2}$ using the first principle of differentiation.

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Problem 14107

Evaluate the limit as $x$ approaches -2 for the expression $7 x^{2} + 8$ .

See Solution

Problem 14108

Evaluate the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x-5}$ . Options: A. 0 B. -5 C. -1 D. $\infty$

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Problem 14109

A spherical tumor's radius grows by $\frac{1}{5}$ cm/week. Find the volume's growth rate when the radius is 3 cm.

See Solution

Problem 14110

Evaluate the integral $\int x \sin \frac{1}{4} x \, dx$ using integration by parts. Choose the correct new integral form.

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Problem 14111

Find $\frac{d y}{d x}$ for $x^{2}+y^{2}=x y+19$ at $x=5, y=2$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 14112

Evaluate the integral $\int_{1}^{e} \ln 22 x \, dx = \square($ Type an exact answer.)

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Problem 14113

Find $\frac{d y}{d x}$ for $\sqrt[3]{x}+\sqrt[3]{y}=6$ at $x=1$ , $y=8$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 14114

Find $\frac{d p}{d x}$ for the demand equation $x=\sqrt{500-p^{2}}$ at $p=20$ .

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Problem 14115

Evaluate the integral $\int x^{2} e^{10 x} d x$ using integration by parts. Choose the correct answer.

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Problem 14116

Evaluate the integral $\int x \sin \frac{1}{4} x \, dx$ using integration by parts. Which option simplifies it?

See Solution

Problem 14117

Find the limit: $\lim _{x \rightarrow 0} \frac{5}{x}=$ A. 0 B. 1 C. 5 D. $\infty$

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Problem 14118

Find the volume change rate of a sphere with radius 6 in, given $\frac{dr}{dt} = \frac{4}{3}$ in/min. Use $\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}$ .

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Problem 14119

Evaluate the integral $\int 10 x \ln 9 x \, dx$ using integration by parts. Which option simplifies it? A. $5 x \ln (5 x^{2}) - \int(9 x) \, dx$ B. $5 x \ln (5 x) - \int(9 x) \, dx$ C. $5 x^{2} \ln (9 x) - \int(5 x) \, dx$ D. $9 x \ln (9 x) - \int(5 x^{2}) \, dx$

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Problem 14120

Derive the reduction formula using integration by parts: $\int x^{n} \cos a x \, dx=\frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1} \sin a x \, dx$ for $a \neq 0$ . What is the formula for integration by parts? A. $\int u \, dv = uv - \int v \, du$ B. $\int u \, dv = \int u \, du - \int v \, dv$ C. $\int u \, dv = uv - \int u \, dv$ D. $\int u \, dv = \int u \, du + \int v \, dv$

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Problem 14121

Evaluate the integral $\int 8 x e^{7 x} d x$ using integration by parts. Choose the correct answer.

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Problem 14122

Evaluate the integral: $\int x^{5} e^{5 x} d x$ using the reduction formula for $e^{a x}$ .

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Problem 14123

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x(y-5)^{2}=8$ .

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Problem 14124

Use implicit differentiation on $y^{3}-y^{2}+y-1=x$ to find $\frac{d y}{d x}$ .

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Problem 14125

Test if the series $\sum_{n=1}^{\infty} \frac{\sin (n \pi / 6)}{1+n \sqrt{n}}$ is absolutely convergent, conditionally convergent, or divergent.

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Problem 14126

Evaluate the integral using integration by parts: $\int_{1}^{e} \ln 22 x \, dx = \square($ Type an exact answer.)

See Solution

Problem 14127

Evaluate the integral: $\int x \sin \frac{1}{4} x \, dx = \square$

See Solution

Problem 14128

Find the predicted population in 4 years using the model $P(t)=219,000 e^{-0.021 t}$ . Round to the nearest person.

See Solution

Problem 14129

Calculate the expectation $\mathbb{R}$ and variance $\mathbb{V}$ for the PDF $\rho(x)=\frac{2}{\pi} \frac{1}{x^{2}+1}$ on $[0,+\infty)$ . Check for convergence first.

See Solution

Problem 14130

Find the storage time $t$ that maximizes the wine value $V(t)=2000+60 \sqrt{t}-10 t$ , for $0 \leq t \leq 25$ . $t=\square \mathrm{yr}$

See Solution

Problem 14131

Find the predicted population in 9 years using the model $P(t)=458,000 e^{-0.015 t}$ . Round to the nearest person.

See Solution

Problem 14132

Find the tax rate $t$ that maximizes revenue, given $S(t)=7-6 \sqrt[3]{t}$ . Round your answer to three decimal places.

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Problem 14133

For the function $f(x)=\frac{2}{x+5}$ , which limit behavior is correct as $x$ approaches certain values?

See Solution

Problem 14134

Find the learning rate at 4 hours and 8 hours, given $y=25 \sqrt[3]{x^{2}}$ .

See Solution

Problem 14135

Determine where the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ is increasing or decreasing.

See Solution

Problem 14136

Find the local max and min of the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ .

See Solution

Problem 14137

Determine the horizontal asymptote of the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ .

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Problem 14138

Find the sensitivity $S$ , defined as the rate of change of reaction $R$ with respect to brightness $x$ , given $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ . $R^{\prime}(x)=$

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Problem 14139

Analyze the function $f(x)=\frac{2x^2}{x^2 -1}$ :
1. Find and simplify $f'(x)$ ; determine where $f'(x)>0$ and $f'(x)<0$ .
2. Identify critical numbers and check for local max/min.
3. Find and simplify $f''(x)$ ; determine where $f''(x)>0$ and $f''(x)<0$ .
4. Identify concavity and points of inflection.

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Problem 14140

Find the sensitivity $S$ of the reaction $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ . Compute $R'(x)$ .

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Problem 14141

Evaluate if the following limits are indeterminate forms given:
1. $\lim _{x \rightarrow a} f(x)=0$
2. $\lim _{x \rightarrow a} g(x)=0$
3. $\lim _{x \rightarrow a} h(x)=1$
4. $\lim _{x \rightarrow a} p(x)=\infty$
5. $\lim _{x \rightarrow a} q(x)=\infty$
(a) $\lim _{x \rightarrow a}[f(x)-p(x)]$ (b) $\lim _{x \rightarrow a}[p(x)-q(x)]$ (c) $\lim _{x \rightarrow a}[p(x)+q(x)]$
If indeterminate, enter INDETERMINATE.

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Problem 14142

Find the derivative of the function $y=x^{2}(4x^{2}-9x)$ . What is $y'$ ?

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Problem 14143

Find the derivative of the function $f(x)=\sqrt{x}-8 \sqrt[3]{x}$ , so $f^{\prime}(x)=$

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Problem 14144

Find the derivative of $f(x)=\frac{10}{\sqrt[3]{x}}+2 \cos x$ .

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Problem 14145

Differentiate $h(t)=\frac{t^{2}-4 t+3}{t+1}$ with respect to $t$ . Find $h^{\prime}(t)=\square$ .

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Problem 14146

Differentiate the function $h(t)=\frac{t^{2}-3 t+5}{t+4}$ with respect to $t$ : $h^{\prime}(t)=\square$ .

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Problem 14147

Find the absolute maximum of the function $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ on the interval $0 \leq x \leq 8$ .

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Problem 14148

Find the absolute maximum of the function $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ for $0 \leqslant x \leqslant 8$ . What is $x$ ?

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Problem 14149

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 7 x}{1-\cos 2 x}.$

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Problem 14150

Find $\frac{d y}{d x}$ for the equation $y^{9} \ln(x^{4}) + 5x^{7} - y^{10} = -7$ .

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Problem 14151

Find $\frac{d y}{d x}$ for the equation $y^{2} \ln (x^{10}) - 10 x^{6} - y^{3} = -4$ .

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Problem 14152

Find the rate of change of the area $A=s^{2}$ with respect to $s$ when $s=4$ meters. What is $A^{\prime}(4)$ ?

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Problem 14153

Find the average rate of change of $f(t)=6 t^{2}-1$ over $[5,5.1]$ and compare it with the rates at the endpoints.

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Problem 14154

Bestimme die Ableitung von $f(x)=x^{2}+2$ an den gegebenen Stellen und trage die Werte in ein Koordinatensystem ein.

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Problem 14155

Find the derivative of $g(x)=(3 x-5)(1-7 x)$ using the Product Rule: $g^{\prime}(x)=$

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Problem 14156

Calculate the average rate of change of $f(t)=8t+1$ over $[2,6]$ and compare it with the rates at the endpoints.

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Problem 14157

Use a graphing tool to estimate and find the limit of $f(x)=x^{2}-x \sqrt{x(x-1)}$ as $x \to \infty$ .

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Problem 14158

Find the derivative of $g(x)=\frac{7 \sin x}{e^{x}}$ using the Quotient Rule: $g^{\prime}(x)=\square$ .

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Problem 14159

Find the average rate of change of $f(t)=5t^2-2$ from $t=4$ to $t=4.1$ , and the instantaneous rates at both points.

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Problem 14160

Find the derivative of the function $y=e^{x}-\cot x$ . What is $y^{\prime}$ ?

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Problem 14161

Find the average rate of change of $f(t)=5 t^{2}-2$ over $[4,4.1]$ and compare with instantaneous rates at endpoints.

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Problem 14162

Find the derivative of $f(x)=4 e^{x} \cos x$ using the Product Rule. What is $f^{\prime}(x)$ ?

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Problem 14163

Find the derivative $f^{\prime}(x)$ and evaluate it at $c$ for the function $f(x)=\frac{x^{2}-4}{x-3}$ .

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Problem 14164

Find $f^{\prime}(x)$ and $f^{\prime}(1)$ for the function $f(x)=\frac{x^{2}-4}{x-3}$ .

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Problem 14165

Find $f^{\prime}(x)$ and $f^{\prime}(0)$ for $f(x)=(x^{5}+3x)(2x^{4}+2x-4)$ .

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Problem 14166

Evaluate the integral $\int \sqrt{\cot 3 x} \csc ^{4} 3 x \, dx$ .

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Problem 14167

Find the derivative of the function $y=7x \sin(x) + x^2 e^x$ . What is $y'$ ?

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Problem 14168

A culture starts with 420 bacteria, growing as $P(t)=420\left(5+\frac{3 t}{25+t^{2}}\right)$ . Find $P^{\prime}(t)$ and $P^{\prime}(2)$ .

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Problem 14169

Integrate $(\csc 4 x + \cot 4 x)^{2}$ with respect to $x$ .

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Problem 14170

Find $f^{\prime}(x)$ and $f^{\prime}(c)$ for the function $f(x)=x \cos (x)$ where $c=\frac{\pi}{4}$ .

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Problem 14171

Find the growth rate $P^{\prime}(t)$ when $t=2$ for $P^{\prime}(t)=420\left(\frac{3}{25+t}-\frac{6t^{2}}{(25+t^{2})^{2}}\right)$ . Round to two decimal places.

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Problem 14172

Show that $g(1) \leq \mu_n \leq g\left(\frac{n+1}{n}\right) - g\left(\frac{1}{n}\right)$ given the equations for $g'(x)$ , $g(x)$ , and $\mu_n$ .

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Problem 14173

Berechnen Sie das Integral von $f(x)=3 \sqrt{x}$ im Intervall $[0, 3]$ .

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Problem 14174

Berechne das Integral von $f(x) = \frac{1}{\sqrt{x}}$ im Intervall [1; 4] oder den Flächeninhalt unter dieser Kurve.

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Problem 14175

Calculate the integral $\int x^{2} \sqrt{9-4 x^{2}} \, dx$ .

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Problem 14176

Bestimme die Monotonieintervalle für die Funktionen $f(x)=6,5 x^{2}+1$ und $f(x)=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}$ .

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Problem 14177

Find the integral: $\int \frac{d x}{x \sqrt{x^{2}-9}}$ .

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Problem 14178

Evaluate the integral: $\int \frac{d x}{x \sqrt{x^{2}-9}}$ .

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Problem 14179

Berechnen Sie die Fläche unter der Funktion $f(x)=\frac{4}{x^{2}}$ im Intervall $[-3, -1]$ .

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Problem 14180

Evaluate the integral $\int \frac{d x}{49-25 x^{2}}$ .

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Problem 14181

Find the derivative of the function $f(x)=\frac{1-5x-x^{2}}{x^{2}-3}$ . What is $f'(x)$ ?

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Problem 14182

Find the derivative of $f(x)=3 x^{2} \ln(3 x)$ .

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Problem 14183

Evaluate the integral: $\int \frac{d x}{(x+3)^{2}-25}$ .

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Problem 14184

Find the derivative of the function $f(x)=\frac{1-5 x-x^{2}}{x^{2}-3}$ . What is $f^{\prime}(x)$ ?

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Problem 14185

Find the derivative of $f(x)=6 x^{2} \ln(6 x)$ .

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Problem 14186

Evaluate the integral $\int \frac{x \arcsin x}{\sqrt{1-x^{2}}} d x$ .

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Problem 14187

Solve the differential equation: $(y^{2}+y) \, dx + x \, dy = 0$ .

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Problem 14188

Find the derivative of the function $y=x(8^{-2x})$ .

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Problem 14189

Find the derivative of $h(x)=\sin(7x)\cos(7x)$ .

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Problem 14190

Find the derivative $f^{\prime}(x)=\frac{-64 x^{2}+4096}{(x^{2}+64)^{2}}$ and evaluate it at $x=-4$ . Then, find the tangent line at $(-4,-\frac{16}{5})$ .

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Problem 14191

Analyze the function $y=\frac{x^{2}}{x^{2}+75}$ : find intercepts, extrema, inflection points, and asymptotes.

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Problem 14192

Analyze and sketch the graph of $y=\frac{x^{2}}{x^{2}+75}$ . Find intercepts, extrema, inflection points, and asymptotes.

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Problem 14193

Determine if Rolle's Theorem applies to $f(x)=(x-2)(x-3)(x-8)$ on $[2,8]$ . If yes, find $c$ where $f'(c)=0$ .

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Problem 14194

Find the integral of the function $(2^{3x} + 1)^{2}$ with respect to $x$ .

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Problem 14195

Bestimmen Sie $F$ zu $f$ mit $F(1)=100$ : (a) $f(x)=2x$ , (b) $f(x)=x^{2}$ , (c) $f(x)=5$ , (d) $f(x)=-x$ , (e) $f(x)=-10$ .

See Solution

Problem 14196

Determine if Rolle's Theorem applies to $f(x)=(x-2)(x-3)(x-8)$ on $[2,8]$ . If yes, find $c$ where $f'(c)=0$ .

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Problem 14197

Find the derivative of $h(x)=(3x-2)^{2}$ and simplify it to $h^{\prime}(x)=18x-12$ .

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Problem 14198

Finde die Stammfunktionen für die folgenden Funktionen: (a) $f(x)=2 x+x^{3}$ , (b) $f(x)=-3 x^{2}+\frac{1}{2} x^{3}$ , (c) $f(x)=\frac{2}{3} x^{-4}+x^{-2}$ , (d) $f(x)=-\frac{1}{4} x^{-5}-2 x^{-4}+2 x^{3}$ , (e) $f(x)=\sqrt{x}+x^{3}$ , (f) $f(x)=3 x^{-4}-2 x^{-2}+7 x$ .

See Solution

Problem 14199

Find the derivative of $g(x)=\left(\frac{9 x^{2}-4}{4 x+9}\right)^{-2}$ .

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Problem 14200

Find the temperature below which the reaction $2 \mathrm{HNO}_3(\mathrm{aq})+\mathrm{NO}(\mathrm{g}) \rightarrow 3 \mathrm{NO}_2(\mathrm{g})+\mathrm{H}_2\mathrm{O}(\mathrm{l})$ is nonspontaneous, given $\Delta H=+193 \mathrm{~kJ}$ and $\Delta S=+285 \mathrm{~J/K}$ . Report without decimals.

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Calculus

Problem 14101

Determine if the function f(x)={6x2−5 for x≤3,15x+9 for x>3}f(x) = \{6x^2 - 5 \text{ for } x \leq 3, 15x + 9 \text{ for } x > 3\}f(x)={6x2−5 for x≤3,15x+9 for x>3} is continuous at x=3x=3x=3.

Problem 14102

Evaluate the integral ∫xsin⁡111x dx\int x \sin \frac{1}{11} x \, dx∫xsin111​xdx using integration by parts. Which option simplifies it?

Problem 14103

Find the average rate of change of revenue R(x)=600x−0.3x2\mathrm{R}(x)=600 x-0.3 x^{2}R(x)=600x−0.3x2 from 100 to 200 units. Also, differentiate 6+5x−x26+5 x-x^{2}6+5x−x2 using first principles.

Problem 14104

Find the limit: lim⁡x→01−cos⁡(5x)cos⁡(2x)−1\lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{\cos (2 x)-1}limx→0​cos(2x)−11−cos(5x)​.

Problem 14105

Evaluate the limit as xxx approaches -2 for the expression 7x2+87x^{2} + 87x2+8. What is the result?

Problem 14106

Differentiate the function 6+5x−x26 + 5x - x^{2}6+5x−x2 using the first principle of differentiation.

Problem 14107

Evaluate the limit as xxx approaches -2 for the expression 7x2+87 x^{2} + 87x2+8.

Problem 14108

Evaluate the limit: lim⁡x→−5x2−25x−5\lim _{x \rightarrow-5} \frac{x^{2}-25}{x-5}limx→−5​x−5x2−25​. Options: A. 0 B. -5 C. -1 D. ∞\infty∞

Problem 14109

A spherical tumor's radius grows by 15\frac{1}{5}51​ cm/week. Find the volume's growth rate when the radius is 3 cm.

Problem 14110

Evaluate the integral ∫xsin⁡14x dx\int x \sin \frac{1}{4} x \, dx∫xsin41​xdx using integration by parts. Choose the correct new integral form.

Problem 14111

Find dydx\frac{d y}{d x}dxdy​ for x2+y2=xy+19x^{2}+y^{2}=x y+19x2+y2=xy+19 at x=5,y=2x=5, y=2x=5,y=2. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 14112

Evaluate the integral ∫1eln⁡22x dx=□(\int_{1}^{e} \ln 22 x \, dx = \square(∫1e​ln22xdx=□( Type an exact answer.)

Problem 14113

Find dydx\frac{d y}{d x}dxdy​ for x3+y3=6\sqrt[3]{x}+\sqrt[3]{y}=63x​+3y​=6 at x=1x=1x=1, y=8y=8y=8. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 14114

Find dpdx\frac{d p}{d x}dxdp​ for the demand equation x=500−p2x=\sqrt{500-p^{2}}x=500−p2​ at p=20p=20p=20.

Problem 14115

Evaluate the integral ∫x2e10xdx\int x^{2} e^{10 x} d x∫x2e10xdx using integration by parts. Choose the correct answer.

Problem 14116

Evaluate the integral ∫xsin⁡14x dx\int x \sin \frac{1}{4} x \, dx∫xsin41​xdx using integration by parts. Which option simplifies it?

Problem 14117

Find the limit: lim⁡x→05x=\lim _{x \rightarrow 0} \frac{5}{x}=limx→0​x5​= A. 0 B. 1 C. 5 D. ∞\infty∞

Problem 14118

Find the volume change rate of a sphere with radius 6 in, given drdt=43\frac{dr}{dt} = \frac{4}{3}dtdr​=34​ in/min. Use dVdt=4πr2drdt\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}dtdV​=4πr2dtdr​.

Problem 14119

Problem 14120

Problem 14121

Evaluate the integral ∫8xe7xdx\int 8 x e^{7 x} d x∫8xe7xdx using integration by parts. Choose the correct answer.

Problem 14122

Evaluate the integral: ∫x5e5xdx\int x^{5} e^{5 x} d x∫x5e5xdx using the reduction formula for eaxe^{a x}eax.

Problem 14123

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x(y−5)2=8x(y-5)^{2}=8x(y−5)2=8.

Problem 14124

Use implicit differentiation on y3−y2+y−1=xy^{3}-y^{2}+y-1=xy3−y2+y−1=x to find dydx\frac{d y}{d x}dxdy​.

Problem 14125

Test if the series ∑n=1∞sin⁡(nπ/6)1+nn\sum_{n=1}^{\infty} \frac{\sin (n \pi / 6)}{1+n \sqrt{n}}∑n=1∞​1+nn​sin(nπ/6)​ is absolutely convergent, conditionally convergent, or divergent.

Problem 14126

Evaluate the integral using integration by parts: ∫1eln⁡22x dx=□(\int_{1}^{e} \ln 22 x \, dx = \square(∫1e​ln22xdx=□( Type an exact answer.)

Problem 14127

Evaluate the integral: ∫xsin⁡14x dx=□\int x \sin \frac{1}{4} x \, dx = \square∫xsin41​xdx=□

Problem 14128

Find the predicted population in 4 years using the model P(t)=219,000e−0.021tP(t)=219,000 e^{-0.021 t}P(t)=219,000e−0.021t. Round to the nearest person.

Problem 14129

Calculate the expectation R\mathbb{R}R and variance V\mathbb{V}V for the PDF ρ(x)=2π1x2+1\rho(x)=\frac{2}{\pi} \frac{1}{x^{2}+1}ρ(x)=π2​x2+11​ on [0,+∞)[0,+\infty)[0,+∞). Check for convergence first.

Problem 14130

Find the storage time ttt that maximizes the wine value V(t)=2000+60t−10tV(t)=2000+60 \sqrt{t}-10 tV(t)=2000+60t​−10t, for 0≤t≤250 \leq t \leq 250≤t≤25. t=□yr t=\square \mathrm{yr} t=□yr

Problem 14131

Find the predicted population in 9 years using the model P(t)=458,000e−0.015tP(t)=458,000 e^{-0.015 t}P(t)=458,000e−0.015t. Round to the nearest person.

Problem 14132

Find the tax rate ttt that maximizes revenue, given S(t)=7−6t3S(t)=7-6 \sqrt[3]{t}S(t)=7−63t​. Round your answer to three decimal places.

Problem 14133

For the function f(x)=2x+5f(x)=\frac{2}{x+5}f(x)=x+52​, which limit behavior is correct as xxx approaches certain values?

Problem 14134

Find the learning rate at 4 hours and 8 hours, given y=25x23y=25 \sqrt[3]{x^{2}}y=253x2​.

Problem 14135

Determine where the function f(x)=1+7x−6x2f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}f(x)=1+x7​−x26​ is increasing or decreasing.

Problem 14136

Find the local max and min of the function f(x)=1+7x−6x2f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}f(x)=1+x7​−x26​.

Problem 14137

Determine the horizontal asymptote of the function f(x)=1+7x−6x2f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}f(x)=1+x7​−x26​.

Problem 14138

Find the sensitivity SSS, defined as the rate of change of reaction RRR with respect to brightness xxx, given R(x)=40+24x0.41+4x0.4R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}R(x)=1+4x0.440+24x0.4​. R′(x)=R^{\prime}(x)= R′(x)=

Problem 14139

Problem 14140

Find the sensitivity SSS of the reaction R(x)=40+24x0.41+4x0.4R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}R(x)=1+4x0.440+24x0.4​. Compute R′(x)R'(x)R′(x).

Problem 14141

Problem 14142

Determine if the function $f(x) = \{6x^2 - 5 \text{ for } x \leq 3, 15x + 9 \text{ for } x > 3\}$ is continuous at $x=3$ .

Evaluate the integral $\int x \sin \frac{1}{11} x \, dx$ using integration by parts. Which option simplifies it?

Find the average rate of change of revenue $\mathrm{R}(x)=600 x-0.3 x^{2}$ from 100 to 200 units. Also, differentiate $6+5 x-x^{2}$ using first principles.

Find the limit: $\lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{\cos (2 x)-1}$ .

Evaluate the limit as $x$ approaches -2 for the expression $7x^{2} + 8$ . What is the result?

Differentiate the function $6 + 5x - x^{2}$ using the first principle of differentiation.

Evaluate the limit as $x$ approaches -2 for the expression $7 x^{2} + 8$ .

Evaluate the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x-5}$ . Options: A. 0 B. -5 C. -1 D. $\infty$

A spherical tumor's radius grows by $\frac{1}{5}$ cm/week. Find the volume's growth rate when the radius is 3 cm.

Evaluate the integral $\int x \sin \frac{1}{4} x \, dx$ using integration by parts. Choose the correct new integral form.

Find $\frac{d y}{d x}$ for $x^{2}+y^{2}=x y+19$ at $x=5, y=2$ . What is $\frac{d y}{d x}=\square$ ?

Evaluate the integral $\int_{1}^{e} \ln 22 x \, dx = \square($ Type an exact answer.)

Find $\frac{d y}{d x}$ for $\sqrt[3]{x}+\sqrt[3]{y}=6$ at $x=1$ , $y=8$ . What is $\frac{d y}{d x}=\square$ ?

Find $\frac{d p}{d x}$ for the demand equation $x=\sqrt{500-p^{2}}$ at $p=20$ .

Evaluate the integral $\int x^{2} e^{10 x} d x$ using integration by parts. Choose the correct answer.

Evaluate the integral $\int x \sin \frac{1}{4} x \, dx$ using integration by parts. Which option simplifies it?

Find the limit: $\lim _{x \rightarrow 0} \frac{5}{x}=$ A. 0 B. 1 C. 5 D. $\infty$

Find the volume change rate of a sphere with radius 6 in, given $\frac{dr}{dt} = \frac{4}{3}$ in/min. Use $\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}$ .

Evaluate the integral $\int 8 x e^{7 x} d x$ using integration by parts. Choose the correct answer.

Evaluate the integral: $\int x^{5} e^{5 x} d x$ using the reduction formula for $e^{a x}$ .

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x(y-5)^{2}=8$ .

Use implicit differentiation on $y^{3}-y^{2}+y-1=x$ to find $\frac{d y}{d x}$ .

Test if the series $\sum_{n=1}^{\infty} \frac{\sin (n \pi / 6)}{1+n \sqrt{n}}$ is absolutely convergent, conditionally convergent, or divergent.

Evaluate the integral using integration by parts: $\int_{1}^{e} \ln 22 x \, dx = \square($ Type an exact answer.)

Evaluate the integral: $\int x \sin \frac{1}{4} x \, dx = \square$

Find the predicted population in 4 years using the model $P(t)=219,000 e^{-0.021 t}$ . Round to the nearest person.

Calculate the expectation $\mathbb{R}$ and variance $\mathbb{V}$ for the PDF $\rho(x)=\frac{2}{\pi} \frac{1}{x^{2}+1}$ on $[0,+\infty)$ . Check for convergence first.

Find the storage time $t$ that maximizes the wine value $V(t)=2000+60 \sqrt{t}-10 t$ , for $0 \leq t \leq 25$ . $t=\square \mathrm{yr}$

Find the predicted population in 9 years using the model $P(t)=458,000 e^{-0.015 t}$ . Round to the nearest person.

Find the tax rate $t$ that maximizes revenue, given $S(t)=7-6 \sqrt[3]{t}$ . Round your answer to three decimal places.

For the function $f(x)=\frac{2}{x+5}$ , which limit behavior is correct as $x$ approaches certain values?

Find the learning rate at 4 hours and 8 hours, given $y=25 \sqrt[3]{x^{2}}$ .

Determine where the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ is increasing or decreasing.

Find the local max and min of the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ .

Determine the horizontal asymptote of the function $f(x)=1+\frac{7}{x}-\frac{6}{x^{2}}$ .

Find the sensitivity $S$ , defined as the rate of change of reaction $R$ with respect to brightness $x$ , given $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ . $R^{\prime}(x)=$

Find the sensitivity $S$ of the reaction $R(x)=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$ . Compute $R'(x)$ .

Find the derivative of the function $y=x^{2}(4x^{2}-9x)$ . What is $y'$ ?

Find the derivative of the function $f(x)=\sqrt{x}-8 \sqrt[3]{x}$ , so $f^{\prime}(x)=$

Find the derivative of $f(x)=\frac{10}{\sqrt[3]{x}}+2 \cos x$ .

Differentiate $h(t)=\frac{t^{2}-4 t+3}{t+1}$ with respect to $t$ . Find $h^{\prime}(t)=\square$ .

Differentiate the function $h(t)=\frac{t^{2}-3 t+5}{t+4}$ with respect to $t$ : $h^{\prime}(t)=\square$ .

Find the absolute maximum of the function $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ on the interval $0 \leq x \leq 8$ .

Find the absolute maximum of the function $f(x)=\frac{\sqrt{x}}{108+x^{2}}$ for $0 \leqslant x \leqslant 8$ . What is $x$ ?

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 7 x}{1-\cos 2 x}.$

Find $\frac{d y}{d x}$ for the equation $y^{9} \ln(x^{4}) + 5x^{7} - y^{10} = -7$ .

Find $\frac{d y}{d x}$ for the equation $y^{2} \ln (x^{10}) - 10 x^{6} - y^{3} = -4$ .

Find the rate of change of the area $A=s^{2}$ with respect to $s$ when $s=4$ meters. What is $A^{\prime}(4)$ ?

Find the average rate of change of $f(t)=6 t^{2}-1$ over $[5,5.1]$ and compare it with the rates at the endpoints.

Bestimme die Ableitung von $f(x)=x^{2}+2$ an den gegebenen Stellen und trage die Werte in ein Koordinatensystem ein.

Find the derivative of $g(x)=(3 x-5)(1-7 x)$ using the Product Rule: $g^{\prime}(x)=$

Calculate the average rate of change of $f(t)=8t+1$ over $[2,6]$ and compare it with the rates at the endpoints.

Use a graphing tool to estimate and find the limit of $f(x)=x^{2}-x \sqrt{x(x-1)}$ as $x \to \infty$ .

Find the derivative of $g(x)=\frac{7 \sin x}{e^{x}}$ using the Quotient Rule: $g^{\prime}(x)=\square$ .

Find the average rate of change of $f(t)=5t^2-2$ from $t=4$ to $t=4.1$ , and the instantaneous rates at both points.

Find the derivative of the function $y=e^{x}-\cot x$ . What is $y^{\prime}$ ?

Find the average rate of change of $f(t)=5 t^{2}-2$ over $[4,4.1]$ and compare with instantaneous rates at endpoints.

Find the derivative of $f(x)=4 e^{x} \cos x$ using the Product Rule. What is $f^{\prime}(x)$ ?

Find the derivative $f^{\prime}(x)$ and evaluate it at $c$ for the function $f(x)=\frac{x^{2}-4}{x-3}$ .

Find $f^{\prime}(x)$ and $f^{\prime}(1)$ for the function $f(x)=\frac{x^{2}-4}{x-3}$ .

Find $f^{\prime}(x)$ and $f^{\prime}(0)$ for $f(x)=(x^{5}+3x)(2x^{4}+2x-4)$ .

Evaluate the integral $\int \sqrt{\cot 3 x} \csc ^{4} 3 x \, dx$ .

Find the derivative of the function $y=7x \sin(x) + x^2 e^x$ . What is $y'$ ?

A culture starts with 420 bacteria, growing as $P(t)=420\left(5+\frac{3 t}{25+t^{2}}\right)$ . Find $P^{\prime}(t)$ and $P^{\prime}(2)$ .

Integrate $(\csc 4 x + \cot 4 x)^{2}$ with respect to $x$ .

Find $f^{\prime}(x)$ and $f^{\prime}(c)$ for the function $f(x)=x \cos (x)$ where $c=\frac{\pi}{4}$ .

Find the growth rate $P^{\prime}(t)$ when $t=2$ for $P^{\prime}(t)=420\left(\frac{3}{25+t}-\frac{6t^{2}}{(25+t^{2})^{2}}\right)$ . Round to two decimal places.

Show that $g(1) \leq \mu_n \leq g\left(\frac{n+1}{n}\right) - g\left(\frac{1}{n}\right)$ given the equations for $g'(x)$ , $g(x)$ , and $\mu_n$ .

Berechnen Sie das Integral von $f(x)=3 \sqrt{x}$ im Intervall $[0, 3]$ .

Berechne das Integral von $f(x) = \frac{1}{\sqrt{x}}$ im Intervall [1; 4] oder den Flächeninhalt unter dieser Kurve.