Calculus

Problem 12101

La courbe $\mathscr{C}$ représente l'audience d'une chaîne de TV de 2000 à 2019. Estimez les téléspectateurs le 1er janvier 2014 et calculez $f^{\prime}(0)$ pour la tangente $(AB)$ avec $A(0 ; 460)$ et $B(3 ; 82)$ .

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Problem 12102

Find the value of $c$ such that $f'( \frac{\pi}{3} ) = 6$ for $f(x) = c \ln(\sin(x))$ .

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Problem 12103

Find the Jacobian $\frac{\partial(x, y, z)}{\partial(s, t, u)}$ for $x=5t-4s+3u$ , $y=5s-4t-4u$ , $z=2s-5t-4u$ .

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Problem 12104

Analyze the function $f(x)=\frac{x^{2}-18}{x^{2}+18}$ : find $x$ -intercepts, asymptotes, extrema, concavity, and sketch the graph.

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Problem 12105

Find the derivatives of $18 \sqrt{2x^{2}}$ and $61 \sqrt{2x^{2}}$ .

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Problem 12106

Check if the function $f(x)$ is continuous at $x=5$ for $f(x)=\begin{cases} x+20 & \text{if } x \geq 5 \\ x^2 & \text{if } x<5 \end{cases}$ .

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Problem 12107

Bestimme die Ableitung von $18 \sqrt{2 x^{2}}$ .

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Problem 12108

Find the first derivative of the function $f(x) = x^{5/3} \ln(x)$ .

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Problem 12109

Bestimme die Ableitung von $f(x) = \sqrt{2 x^{2}}$ .

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Problem 12110

Find the limits: $\lim_{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}$ and $\lim_{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}$ .

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Problem 12111

Untersuchen Sie die Umsatzfunktion $U(t)=-\frac{1}{8} t^{3}+2 t^{2}$ : Maximum finden und prüfen, wann $U(t)$ unter 40\% des Maximalumsatzes fällt.

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Problem 12112

Find the derivative of $f(x)=\frac{9 \ln (x)}{x}$ at $x=e$ .

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Problem 12113

Find the derivative of $y$ with respect to $t$ for $y=e^{(11 \cos t+\ln t)}$ . What is $\frac{\mathrm{dy}}{\mathrm{dt}}$ ?

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Problem 12114

A volcano cone is 40m tall and 80m wide. If height increases by 0.5m/day, find volume change rate in m³/day.

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Problem 12115

Find the limits: $\lim _{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}=\square$ and $\lim _{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}=\square$ .

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Problem 12116

Find the derivative of $y$ with respect to $t$ for $y=e^{(9 \sin t+\ln t)}$ .

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Problem 12117

A volcanic cone with width 80m and height 40m grows at 0.5m/day (height) and 0.6m/day (width). Find its volume increase rate. Volume = $2000 \pi \, m^{3} / \text{day}$ .

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Problem 12118

Find the derivative of $y$ with respect to $x$ for $y=\ln(x^{16})$ .

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Problem 12119

Find the derivative of $y = t(\ln 8 t)^{2}$ with respect to $t$ .

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Problem 12120

Untersuchen Sie die Berührungspunkte der Balken mit einer Funktion für Untersumme und Obersumme. Wo liegt die Differenz?

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Problem 12121

Find zeros of $f(x)=\cos(2x)$ using Newton's Method with $x_1=1.4$ and $x_1=1.5$ . Explain convergence with a diagram.

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Problem 12122

Find the rate of volume change for a sphere with radius $1 \mathrm{~m}$ dissolving at $3 \mathrm{~cm}/\text{hour}$ .

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Problem 12123

Find the tangent line equation $y=m x+b$ for $f(x)=\frac{9 \sin x}{2 \sin x+6 \cos x}$ at $a=\frac{\pi}{6}$ .

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Problem 12124

Find the relative rate of change of $y$ at $x=7$ for the function $\frac{x}{x+5}$ .

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Problem 12125

Find the derivative of $y=\left(x^{2}-4\right)^{5}(3x+5)^{4}$ .

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Problem 12126

Find the derivative of the function $s(x)=4x+6$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ . What is $s^{\prime}(x)=\square$ ?

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Problem 12127

Untersuchen Sie, ob die mittlere Steigung von $h(x)=x(x-1)(x-3)$ im Intervall $[-0,5 \mid 0,5]$ mehr als $10\%$ von der Änderungsrate bei $x_{0}=0$ abweicht.

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Problem 12128

Find the relative rate of change of $y$ at $x=7$ for the function $\frac{x}{x+5}$ . Answer: $\square$

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Problem 12129

Find the marginal-cost function for the total-cost function $c=\frac{9 q^{2}}{q+2}+5000$ .

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Problem 12130

Find the marginal-cost function for the total-cost function $c=\frac{8 q^{2}}{\sqrt{q^{2}+2}}+5000$ .

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Problem 12131

Find the critical numbers for the function $f(\theta)=2 \sec \theta+\tan \theta$ in the interval $0<\theta<2 \pi$ .

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Problem 12132

Find the derivative $j^{\prime}(x)$ for $j(x)=\frac{g(x)}{f(x)}$ at $x=1$ , $x=2$ , and $x=3$ . Use a graph for guidance.

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Problem 12133

Find the limit as $x$ approaches -13 for the expression $\frac{12-\sqrt{x^{2}-25}}{x+13}$ .

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Problem 12134

Find the difference quotient $\frac{f(a+h)-f(a)}{h}$ and the limit $\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ for $f(x)=-6x-1$ .

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Problem 12135

Find the expression that makes this equation true: $\frac{d}{d x}(8 x+7)^{5}=5(8 x+7)^{4} ?$ The missing part is $\square$ .

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Problem 12136

Untersuchen Sie das Verhalten von $f(x)=\frac{4 x-1}{x}$ für $x \to \infty$ und $x \to -\infty$ durch Termvereinfachung.

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Problem 12137

Find the limit as $x$ approaches -13 for $\frac{12-\sqrt{x^{2}-25}}{x+13}$ . What is the answer?

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Problem 12138

Find the expression for ? that makes this equation true: $\frac{d}{d x}(3 x+8)^{4}=4(3 x+8)^{3} ?$

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Problem 12139

Use Newton's method to find the zero of $f(x)=\operatorname{Tan}^{-1} x$ . Show convergence for small $x_1$ and divergence for large $x_1$ . Find the transition value where approximations alternate.

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Problem 12140

Find the missing expression in the equation: $\frac{d}{d x}\left(5-6 x^{2}\right)^{7}=7\left(5-6 x^{2}\right)^{6} ?$

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Problem 12141

Find the limit as $x$ approaches -10 for the expression $\frac{9-\sqrt{x^{2}-19}}{x+10}$ .

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Problem 12142

Find the expression for ? to make the equation valid: $\frac{d}{d x} \ln \left(x^{3}+8\right)=\frac{1}{x^{3}+8} ?$ The missing expression is $\square$ .

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Problem 12143

Find the missing expression in the equation: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{7}+4}=e^{\mathrm{x}^{7}+4} ?$ The missing expression is $\square$ .

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Problem 12144

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(5-7 x)^{15}$ .

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Problem 12145

Find the rate of change of the volume $V$ of a ball with respect to its radius $r$ when $r=10 \mathrm{~m}$ . Use $V=\frac{4}{3} \pi r^{3}$ . Provide the answer in terms of $\pi$ .

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Problem 12146

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{14 x}$ . What is $f^{\prime}(x)=\square$ ?

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Problem 12147

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{6}+8)^{-4}$ .

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Problem 12148

Find the marginal-cost function for $c(q)=19 q^{2}-10 q+5$ and calculate it at $q=2$ , $q=7$ , and $q=16$ .

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Problem 12149

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

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Problem 12150

Find the rate of change of volume $V$ with respect to radius $r$ at $r=2.5 \mathrm{~m}$ for $V=\frac{4}{3} \pi r^{3}$ .

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Problem 12151

Find the marginal-cost function for $C(x)=0.05 x^{3}+0.4 x^{2}+40 x+120$ at $x=1000$ . What is $C^{\prime}(1000)$ ?

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Problem 12152

Find the derivative of $f(x)=8 \sqrt{8 x^{2}+7}$ . What is $f^{\prime}(x)=\square$ ?

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Problem 12153

Find the derivative of the function $f(x)=6 e^{-2 x}$ . What is $f^{\prime}(x)=\square$ ?

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Problem 12154

Find the limit of $f(x)=\frac{x+1}{x^{2}+4}$ as $x \rightarrow \infty$ .

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Problem 12155

Find the derivative of the function $f(x)=8 \ln(1+8 x^{2})$ . What is $f^{\prime}(x)$ ?

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Problem 12156

Find the marginal-revenue function for $r=239q+35q^{2}-2q^{3}$ and evaluate it at $q=10$ , $q=15$ , $q=20$ .

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Problem 12157

Find $f^{\prime}(x)$ , the tangent line at $x=2$ for $f(x)=(12 x-8)^{1/2}$ , and where the tangent is horizontal.

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Problem 12158

Find the average cost per appliance for the first 130 appliances using $c(q)=1400+110q-0.3q^{2}$ and the marginal cost at $q=130$ .

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Problem 12159

Find the slope of $y=\sqrt{7 x}$ at (7,7) and the equation of the tangent line at that point.

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Problem 12160

Find $f^{\prime}(x)$ and the tangent line equation at $x=0$ for $f(x)=7 e^{x^{2}-3 x+7}$ . Where is the tangent horizontal?

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Problem 12161

Find the second derivative of $y=4 t^{-3}+3 t^{2}$ at $t=1$ .

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Problem 12162

Find $f^{\prime \prime \prime}(-3)$ for the function $f(x)=4 e^{x}-x^{3}$ .

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Problem 12163

Find the derivative of $6(t^{2}+9t)^{-3}$ . What is $\frac{d}{dt} 6(t^{2}+9t)^{-3} = \square$ ?

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Problem 12164

Find the derivative $y^{\prime}$ if $y=\ln \left(x^{2}+1\right)^{3 / 2}$ . What is $y^{\prime}=?$

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Problem 12165

Find the marginal-revenue function for the demand equation $p=\frac{9 q+4}{4 q+9}$ , where revenue $=p q$ .

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Problem 12166

Find the derivative of $y=\frac{-4}{(2 x^{3}+5)^{6}}$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 12167

Find the derivative $g^{\prime}(x)$ of the function $g(x)=\sqrt{4 x}$ .

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Problem 12168

Differentiate the function $y=\frac{4x-1}{9x+8}$ .

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Problem 12169

Find the marginal-revenue function for the demand equation $p=80-0.03 q$ , where revenue $=p q$ .

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Problem 12170

Find the limit: $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ .

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Problem 12171

Given the demand equation $p=300-\sqrt{q^{2}+30}$ , find: (a) $\frac{dp}{dq}$ , (b) $\frac{dp/dq}{p}$ , (c) $\frac{dr}{dq}$ .

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Problem 12172

Find the rate of change of income $y=7x^{7/2}+5800$ with respect to education years $x$ . Evaluate at $x=11$ .

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Problem 12173

Find the derivative using the quotient rule for $y=\frac{x^{2}-5 x+3}{x^{2}+9}$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 12174

Find the slope of $y=\sqrt{6x}$ at the point $(6,6)$ and the equation of the tangent line there.

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Problem 12175

Find the rate of change of $c=0.9 q^{2}+3.8 q+4$ at $q=14$ and its percentage rate of change.

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Problem 12176

Find the derivative $y^{\prime}$ for the function $y=\ln \left(x^{8}+3\right)^{3 / 2}$ .

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Problem 12177

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=x(4-x)^{3}$ .

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Problem 12178

Differentiate the function: $y=(6 x^{2}-7)(3 x^{2}-8 x+3)$

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Problem 12179

Find the limit: $\lim _{x \rightarrow x_{0}} \frac{x^{2}-x_{0}^{2}}{x-x_{0}}$ .

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Problem 12180

Find the marginal-revenue function for the demand equation $p=\frac{3 q+4}{4 q+1}$ , where revenue $=p q$ .

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Problem 12181

Bestimme die Ableitung von $f(x)=-3 \cdot \sqrt{\ln (x+1)}$ . Was ist $f^{\prime}(x)$ ?

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Problem 12182

Find the derivative using the quotient rule for $y=\frac{x^{2}-2 x+1}{x^{2}+4}$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 12183

Find the limit of the sequence $\left\{\frac{a_{n}}{b_{n}}+2 c_{n}\right\}$ given that $\lim _{n \rightarrow \infty}a_{n}=6$ , $\lim _{n \rightarrow \infty}b_{n}=-1$ , and $\lim _{n \rightarrow \infty}c_{n}=3$ .

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Problem 12184

Find the surface area of the solid formed by rotating $f(x)=\ln(x)$ for $x \in [1,3]$ about the $x$ -axis.

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Problem 12185

1. A 67 kg snowboarder starts at a 22 m hill with an initial speed of 15 m/s. (a) Find mechanical energy at the top. (b) Find speed midway and at the bottom. (c) Explain the energy transformation down the hill.

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Problem 12186

Find the difference quotient and limit for $f(x)=6x-7$ : a. $\frac{f(a+h)-f(a)}{h}=\square$ , b. $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}=\square$ .

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Problem 12187

Find the limits: a) $\lim_{x \to 0} \frac{1}{x^2}$ , b) $\lim_{x \to -\infty} 2^x$ , c) $\lim_{x \to 0} \frac{1}{\sqrt{x}}$ for $x > 0$ .

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Problem 12188

Given the demand equation $p=200-\sqrt{q^{2}+20}$ , find: (a) the rate of change of $p$ with $q$ , (b) the relative rate of change, (c) the marginal-revenue function.

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Problem 12189

Find the tangent line equation for $f(x)=x^{2}-6$ at the point $(-4,10)$ .

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Problem 12190

Find the relative rate of change of $y$ for $y=\frac{x}{x+6}$ at $x=4$ . The answer is $\square$ .

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Problem 12191

Find the tangent line equation for $f(x)=x^{2}-7$ at the point $(-4,9)$ . $y=\square$

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Problem 12192

Find the cost function $C(x)=5+\sqrt{2x+36}$ for $0 \leq x \leq 50$ . Given $C^{\prime}(32)=\frac{1}{10}$ , interpret the result. A or B?

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Problem 12193

Find the tangent line equation for $f(x)=x^{2}-3$ at the point $(4,13)$ . $y=\square$

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Problem 12194

Find the marginal-revenue function for the demand equation $p=\frac{8 q+5}{6 q+1}$ , where revenue $=p q$ .

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Problem 12195

Calculate the integral $\int a(x) \, dx$ for $a(x)=-\frac{1}{a^{2}} x^{3}+x$ .

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Problem 12196

Find the marginal-cost function from the total-cost function $c=\frac{5 q^{2}}{q+2}+7000$ . Choose A or B.

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Problem 12197

A 110 kg skier descends a frictionless trail from 210 m. Find the work done by gravity and the skier's speed at the bottom.

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Problem 12198

Differentiate the function $y=(7 x^{2}-9)(3 x^{2}-8 x+2)$ . Find $\frac{d y}{d x}=\square$ .

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Problem 12199

Find the derivative $g^{\prime}(x)$ of the function $g(x)=\sqrt{13 x}$ .

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Problem 12200

Find the concentration rate after 1 hour and 5 hours for $C(t) = 3.75 e^{-t}$ , $0 \leq t \leq 5$ . Graph $C$ .

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Calculus

Problem 12101

La courbe C\mathscr{C}C représente l'audience d'une chaîne de TV de 2000 à 2019. Estimez les téléspectateurs le 1er janvier 2014 et calculez f′(0)f^{\prime}(0)f′(0) pour la tangente (AB)(AB)(AB) avec A(0;460)A(0 ; 460)A(0;460) et B(3;82)B(3 ; 82)B(3;82).

Problem 12102

Find the value of ccc such that f′(π3)=6f'( \frac{\pi}{3} ) = 6f′(3π​)=6 for f(x)=cln⁡(sin⁡(x))f(x) = c \ln(\sin(x))f(x)=cln(sin(x)).

Problem 12103

Find the Jacobian ∂(x,y,z)∂(s,t,u)\frac{\partial(x, y, z)}{\partial(s, t, u)}∂(s,t,u)∂(x,y,z)​ for x=5t−4s+3ux=5t-4s+3ux=5t−4s+3u, y=5s−4t−4uy=5s-4t-4uy=5s−4t−4u, z=2s−5t−4uz=2s-5t-4uz=2s−5t−4u.

Problem 12104

Analyze the function f(x)=x2−18x2+18f(x)=\frac{x^{2}-18}{x^{2}+18}f(x)=x2+18x2−18​: find xxx-intercepts, asymptotes, extrema, concavity, and sketch the graph.

Problem 12105

Find the derivatives of 182x218 \sqrt{2x^{2}}182x2​ and 612x261 \sqrt{2x^{2}}612x2​.

Problem 12106

Check if the function f(x)f(x)f(x) is continuous at x=5x=5x=5 for f(x)={x+20if x≥5x2if x<5f(x)=\begin{cases} x+20 & \text{if } x \geq 5 \\ x^2 & \text{if } x<5 \end{cases}f(x)={x+20x2​if x≥5if x<5​.

Problem 12107

Bestimme die Ableitung von 182x218 \sqrt{2 x^{2}}182x2​.

Problem 12108

Find the first derivative of the function f(x)=x5/3ln⁡(x)f(x) = x^{5/3} \ln(x)f(x)=x5/3ln(x).

Problem 12109

Bestimme die Ableitung von f(x)=2x2f(x) = \sqrt{2 x^{2}}f(x)=2x2​.

Problem 12110

Find the limits: lim⁡x→∞x2−18x2+18\lim_{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}limx→∞​x2+18x2−18​ and lim⁡x→−∞x2−18x2+18\lim_{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}limx→−∞​x2+18x2−18​.

Problem 12111

Untersuchen Sie die Umsatzfunktion U(t)=−18t3+2t2U(t)=-\frac{1}{8} t^{3}+2 t^{2}U(t)=−81​t3+2t2: Maximum finden und prüfen, wann U(t)U(t)U(t) unter 40\% des Maximalumsatzes fällt.

Problem 12112

Find the derivative of f(x)=9ln⁡(x)xf(x)=\frac{9 \ln (x)}{x}f(x)=x9ln(x)​ at x=ex=ex=e.

Problem 12113

Find the derivative of yyy with respect to ttt for y=e(11cos⁡t+ln⁡t)y=e^{(11 \cos t+\ln t)}y=e(11cost+lnt). What is dydt\frac{\mathrm{dy}}{\mathrm{dt}}dtdy​?

Problem 12114

A volcano cone is 40m tall and 80m wide. If height increases by 0.5m/day, find volume change rate in m³/day.

Problem 12115

Find the limits: lim⁡x→∞x2−18x2+18=□\lim _{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}=\squarelimx→∞​x2+18x2−18​=□ and lim⁡x→−∞x2−18x2+18=□\lim _{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}=\squarelimx→−∞​x2+18x2−18​=□.

Problem 12116

Find the derivative of yyy with respect to ttt for y=e(9sin⁡t+ln⁡t)y=e^{(9 \sin t+\ln t)}y=e(9sint+lnt).

Problem 12117

A volcanic cone with width 80m and height 40m grows at 0.5m/day (height) and 0.6m/day (width). Find its volume increase rate. Volume = 2000π m3/day2000 \pi \, m^{3} / \text{day}2000πm3/day.

Problem 12118

Find the derivative of yyy with respect to xxx for y=ln⁡(x16)y=\ln(x^{16})y=ln(x16).

Problem 12119

Find the derivative of y=t(ln⁡8t)2y = t(\ln 8 t)^{2}y=t(ln8t)2 with respect to ttt.

Problem 12120

Untersuchen Sie die Berührungspunkte der Balken mit einer Funktion für Untersumme und Obersumme. Wo liegt die Differenz?

Problem 12121

Find zeros of f(x)=cos⁡(2x)f(x)=\cos(2x)f(x)=cos(2x) using Newton's Method with x1=1.4x_1=1.4x1​=1.4 and x1=1.5x_1=1.5x1​=1.5. Explain convergence with a diagram.

Problem 12122

Find the rate of volume change for a sphere with radius 1 m1 \mathrm{~m}1 m dissolving at 3 cm/hour3 \mathrm{~cm}/\text{hour}3 cm/hour.

Problem 12123

Find the tangent line equation y=mx+by=m x+by=mx+b for f(x)=9sin⁡x2sin⁡x+6cos⁡xf(x)=\frac{9 \sin x}{2 \sin x+6 \cos x}f(x)=2sinx+6cosx9sinx​ at a=π6a=\frac{\pi}{6}a=6π​.

Problem 12124

Find the relative rate of change of yyy at x=7x=7x=7 for the function xx+5\frac{x}{x+5}x+5x​.

Problem 12125

Find the derivative of y=(x2−4)5(3x+5)4y=\left(x^{2}-4\right)^{5}(3x+5)^{4}y=(x2−4)5(3x+5)4.

Problem 12126

Find the derivative of the function s(x)=4x+6s(x)=4x+6s(x)=4x+6 using f′(x)=lim⁡h→0f(x+h)−f(x)hf^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}f′(x)=limh→0​hf(x+h)−f(x)​. What is s′(x)=□s^{\prime}(x)=\squares′(x)=□?

Problem 12127

Untersuchen Sie, ob die mittlere Steigung von h(x)=x(x−1)(x−3)h(x)=x(x-1)(x-3)h(x)=x(x−1)(x−3) im Intervall [−0,5∣0,5][-0,5 \mid 0,5][−0,5∣0,5] mehr als 10%10\%10% von der Änderungsrate bei x0=0x_{0}=0x0​=0 abweicht.

Problem 12128

Find the relative rate of change of yyy at x=7x=7x=7 for the function xx+5\frac{x}{x+5}x+5x​. Answer: □\square□

Problem 12129

Find the marginal-cost function for the total-cost function c=9q2q+2+5000c=\frac{9 q^{2}}{q+2}+5000c=q+29q2​+5000.

Problem 12130

Find the marginal-cost function for the total-cost function c=8q2q2+2+5000c=\frac{8 q^{2}}{\sqrt{q^{2}+2}}+5000c=q2+2​8q2​+5000.

Problem 12131

Find the critical numbers for the function f(θ)=2sec⁡θ+tan⁡θf(\theta)=2 \sec \theta+\tan \thetaf(θ)=2secθ+tanθ in the interval 0<θ<2π0<\theta<2 \pi0<θ<2π.

Problem 12132

Find the derivative j′(x)j^{\prime}(x)j′(x) for j(x)=g(x)f(x)j(x)=\frac{g(x)}{f(x)}j(x)=f(x)g(x)​ at x=1x=1x=1, x=2x=2x=2, and x=3x=3x=3. Use a graph for guidance.

Problem 12133

Find the limit as xxx approaches -13 for the expression 12−x2−25x+13\frac{12-\sqrt{x^{2}-25}}{x+13}x+1312−x2−25​​.

Problem 12134

Find the difference quotient f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}hf(a+h)−f(a)​ and the limit lim⁡h→0f(a+h)−f(a)h\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}limh→0​hf(a+h)−f(a)​ for f(x)=−6x−1f(x)=-6x-1f(x)=−6x−1.

Problem 12135

Find the expression that makes this equation true: ddx(8x+7)5=5(8x+7)4?\frac{d}{d x}(8 x+7)^{5}=5(8 x+7)^{4} ?dxd​(8x+7)5=5(8x+7)4? The missing part is □\square□.

Problem 12136

Untersuchen Sie das Verhalten von f(x)=4x−1xf(x)=\frac{4 x-1}{x}f(x)=x4x−1​ für x→∞x \to \inftyx→∞ und x→−∞x \to -\inftyx→−∞ durch Termvereinfachung.

Problem 12137

Find the limit as xxx approaches -13 for 12−x2−25x+13\frac{12-\sqrt{x^{2}-25}}{x+13}x+1312−x2−25​​. What is the answer?

Problem 12138

Find the expression for ? that makes this equation true: ddx(3x+8)4=4(3x+8)3?\frac{d}{d x}(3 x+8)^{4}=4(3 x+8)^{3} ?dxd​(3x+8)4=4(3x+8)3?

Problem 12139

Use Newton's method to find the zero of f(x)=Tan⁡−1xf(x)=\operatorname{Tan}^{-1} xf(x)=Tan−1x. Show convergence for small x1x_1x1​ and divergence for large x1x_1x1​. Find the transition value where approximations alternate.

Problem 12140

La courbe $\mathscr{C}$ représente l'audience d'une chaîne de TV de 2000 à 2019. Estimez les téléspectateurs le 1er janvier 2014 et calculez $f^{\prime}(0)$ pour la tangente $(AB)$ avec $A(0 ; 460)$ et $B(3 ; 82)$ .

Find the value of $c$ such that $f'( \frac{\pi}{3} ) = 6$ for $f(x) = c \ln(\sin(x))$ .

Find the Jacobian $\frac{\partial(x, y, z)}{\partial(s, t, u)}$ for $x=5t-4s+3u$ , $y=5s-4t-4u$ , $z=2s-5t-4u$ .

Analyze the function $f(x)=\frac{x^{2}-18}{x^{2}+18}$ : find $x$ -intercepts, asymptotes, extrema, concavity, and sketch the graph.

Find the derivatives of $18 \sqrt{2x^{2}}$ and $61 \sqrt{2x^{2}}$ .

Check if the function $f(x)$ is continuous at $x=5$ for $f(x)=\begin{cases} x+20 & \text{if } x \geq 5 \\ x^2 & \text{if } x<5 \end{cases}$ .

Bestimme die Ableitung von $18 \sqrt{2 x^{2}}$ .

Find the first derivative of the function $f(x) = x^{5/3} \ln(x)$ .

Bestimme die Ableitung von $f(x) = \sqrt{2 x^{2}}$ .

Find the limits: $\lim_{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}$ and $\lim_{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}$ .

Untersuchen Sie die Umsatzfunktion $U(t)=-\frac{1}{8} t^{3}+2 t^{2}$ : Maximum finden und prüfen, wann $U(t)$ unter 40\% des Maximalumsatzes fällt.

Find the derivative of $f(x)=\frac{9 \ln (x)}{x}$ at $x=e$ .

Find the derivative of $y$ with respect to $t$ for $y=e^{(11 \cos t+\ln t)}$ . What is $\frac{\mathrm{dy}}{\mathrm{dt}}$ ?

Find the limits: $\lim _{x \rightarrow \infty} \frac{x^{2}-18}{x^{2}+18}=\square$ and $\lim _{x \rightarrow -\infty} \frac{x^{2}-18}{x^{2}+18}=\square$ .

Find the derivative of $y$ with respect to $t$ for $y=e^{(9 \sin t+\ln t)}$ .

A volcanic cone with width 80m and height 40m grows at 0.5m/day (height) and 0.6m/day (width). Find its volume increase rate. Volume = $2000 \pi \, m^{3} / \text{day}$ .

Find the derivative of $y$ with respect to $x$ for $y=\ln(x^{16})$ .

Find the derivative of $y = t(\ln 8 t)^{2}$ with respect to $t$ .

Find zeros of $f(x)=\cos(2x)$ using Newton's Method with $x_1=1.4$ and $x_1=1.5$ . Explain convergence with a diagram.

Find the rate of volume change for a sphere with radius $1 \mathrm{~m}$ dissolving at $3 \mathrm{~cm}/\text{hour}$ .

Find the tangent line equation $y=m x+b$ for $f(x)=\frac{9 \sin x}{2 \sin x+6 \cos x}$ at $a=\frac{\pi}{6}$ .

Find the relative rate of change of $y$ at $x=7$ for the function $\frac{x}{x+5}$ .

Find the derivative of $y=\left(x^{2}-4\right)^{5}(3x+5)^{4}$ .

Find the derivative of the function $s(x)=4x+6$ using $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ . What is $s^{\prime}(x)=\square$ ?

Untersuchen Sie, ob die mittlere Steigung von $h(x)=x(x-1)(x-3)$ im Intervall $[-0,5 \mid 0,5]$ mehr als $10\%$ von der Änderungsrate bei $x_{0}=0$ abweicht.

Find the relative rate of change of $y$ at $x=7$ for the function $\frac{x}{x+5}$ . Answer: $\square$

Find the marginal-cost function for the total-cost function $c=\frac{9 q^{2}}{q+2}+5000$ .

Find the marginal-cost function for the total-cost function $c=\frac{8 q^{2}}{\sqrt{q^{2}+2}}+5000$ .

Find the critical numbers for the function $f(\theta)=2 \sec \theta+\tan \theta$ in the interval $0<\theta<2 \pi$ .

Find the derivative $j^{\prime}(x)$ for $j(x)=\frac{g(x)}{f(x)}$ at $x=1$ , $x=2$ , and $x=3$ . Use a graph for guidance.

Find the limit as $x$ approaches -13 for the expression $\frac{12-\sqrt{x^{2}-25}}{x+13}$ .

Find the difference quotient $\frac{f(a+h)-f(a)}{h}$ and the limit $\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ for $f(x)=-6x-1$ .

Find the expression that makes this equation true: $\frac{d}{d x}(8 x+7)^{5}=5(8 x+7)^{4} ?$ The missing part is $\square$ .

Untersuchen Sie das Verhalten von $f(x)=\frac{4 x-1}{x}$ für $x \to \infty$ und $x \to -\infty$ durch Termvereinfachung.

Find the limit as $x$ approaches -13 for $\frac{12-\sqrt{x^{2}-25}}{x+13}$ . What is the answer?

Find the expression for ? that makes this equation true: $\frac{d}{d x}(3 x+8)^{4}=4(3 x+8)^{3} ?$

Use Newton's method to find the zero of $f(x)=\operatorname{Tan}^{-1} x$ . Show convergence for small $x_1$ and divergence for large $x_1$ . Find the transition value where approximations alternate.

Find the missing expression in the equation: $\frac{d}{d x}\left(5-6 x^{2}\right)^{7}=7\left(5-6 x^{2}\right)^{6} ?$

Find the limit as $x$ approaches -10 for the expression $\frac{9-\sqrt{x^{2}-19}}{x+10}$ .

Find the expression for ? to make the equation valid: $\frac{d}{d x} \ln \left(x^{3}+8\right)=\frac{1}{x^{3}+8} ?$ The missing expression is $\square$ .

Find the missing expression in the equation: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{7}+4}=e^{\mathrm{x}^{7}+4} ?$ The missing expression is $\square$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(5-7 x)^{15}$ .

Find the rate of change of the volume $V$ of a ball with respect to its radius $r$ when $r=10 \mathrm{~m}$ . Use $V=\frac{4}{3} \pi r^{3}$ . Provide the answer in terms of $\pi$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=e^{14 x}$ . What is $f^{\prime}(x)=\square$ ?

Find the derivative $f^{\prime}(x)$ for the function $f(x)=(x^{6}+8)^{-4}$ .

Find the marginal-cost function for $c(q)=19 q^{2}-10 q+5$ and calculate it at $q=2$ , $q=7$ , and $q=16$ .

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

Find the rate of change of volume $V$ with respect to radius $r$ at $r=2.5 \mathrm{~m}$ for $V=\frac{4}{3} \pi r^{3}$ .

Find the marginal-cost function for $C(x)=0.05 x^{3}+0.4 x^{2}+40 x+120$ at $x=1000$ . What is $C^{\prime}(1000)$ ?

Find the derivative of $f(x)=8 \sqrt{8 x^{2}+7}$ . What is $f^{\prime}(x)=\square$ ?

Find the derivative of the function $f(x)=6 e^{-2 x}$ . What is $f^{\prime}(x)=\square$ ?

Find the limit of $f(x)=\frac{x+1}{x^{2}+4}$ as $x \rightarrow \infty$ .

Find the derivative of the function $f(x)=8 \ln(1+8 x^{2})$ . What is $f^{\prime}(x)$ ?

Find the marginal-revenue function for $r=239q+35q^{2}-2q^{3}$ and evaluate it at $q=10$ , $q=15$ , $q=20$ .

Find $f^{\prime}(x)$ , the tangent line at $x=2$ for $f(x)=(12 x-8)^{1/2}$ , and where the tangent is horizontal.

Find the average cost per appliance for the first 130 appliances using $c(q)=1400+110q-0.3q^{2}$ and the marginal cost at $q=130$ .

Find the slope of $y=\sqrt{7 x}$ at (7,7) and the equation of the tangent line at that point.

Find $f^{\prime}(x)$ and the tangent line equation at $x=0$ for $f(x)=7 e^{x^{2}-3 x+7}$ . Where is the tangent horizontal?

Find the second derivative of $y=4 t^{-3}+3 t^{2}$ at $t=1$ .

Find $f^{\prime \prime \prime}(-3)$ for the function $f(x)=4 e^{x}-x^{3}$ .

Find the derivative of $6(t^{2}+9t)^{-3}$ . What is $\frac{d}{dt} 6(t^{2}+9t)^{-3} = \square$ ?

Find the derivative $y^{\prime}$ if $y=\ln \left(x^{2}+1\right)^{3 / 2}$ . What is $y^{\prime}=?$

Find the marginal-revenue function for the demand equation $p=\frac{9 q+4}{4 q+9}$ , where revenue $=p q$ .

Find the derivative of $y=\frac{-4}{(2 x^{3}+5)^{6}}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative $g^{\prime}(x)$ of the function $g(x)=\sqrt{4 x}$ .

Differentiate the function $y=\frac{4x-1}{9x+8}$ .

Find the marginal-revenue function for the demand equation $p=80-0.03 q$ , where revenue $=p q$ .

Find the limit: $\lim _{x \rightarrow 3} \frac{3 x^{2}-27}{x-3}$ .

Given the demand equation $p=300-\sqrt{q^{2}+30}$ , find: (a) $\frac{dp}{dq}$ , (b) $\frac{dp/dq}{p}$ , (c) $\frac{dr}{dq}$ .

Find the rate of change of income $y=7x^{7/2}+5800$ with respect to education years $x$ . Evaluate at $x=11$ .

Find the derivative using the quotient rule for $y=\frac{x^{2}-5 x+3}{x^{2}+9}$ . What is $\frac{d y}{d x}=\square$ ?

Find the slope of $y=\sqrt{6x}$ at the point $(6,6)$ and the equation of the tangent line there.