Calculus

Problem 6101

Find the derivative of $f(x)=\frac{1}{2} e^{\frac{1}{3} x+2}$ and simplify it.

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Problem 6102

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme die Steigung in $P(2 \mid f(2))$ , waagerechte Tangenten und Punkte mit $45^{\circ}$ Steigungswinkel.

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Problem 6103

Bestimme die erste Ableitung von $f_{a}$ für die Funktionen: a) $f_{a}(x)=\frac{1}{a} \cdot x^{2}+a x$ , b) $f_{a}(x)=a \cdot \cos (x)+a$ , c) $f_{a}(x)=-a \cdot \sin (x)+a x^{2}$ .

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Problem 6104

Berechne die Ableitungen $f_{a}^{\prime}(x)$ und $f_{a}^{(21)}(x)$ für die Funktionen: a) $f_{a}(x)=\cos (a x)$ , b) $f_{a}(x)=(a x+5)^{21}$ , c) $f_{a}(x)=\sin \left(-a^{2} x\right)$ .

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Problem 6105

Bestimmen Sie die Ableitungen der folgenden Funktionen: a) $f(x)=\frac{1}{(x-1)^{3}}$ , b) $f(x)=\frac{1}{(1-x)^{3}}$ , c) $f(x)=\frac{1}{(3x+2)^{2}}$ , d) $f(x)=\frac{1}{3}(x+2)^{-2}$ , e) $f(x)=\sqrt{x+3}$ , f) $f(t)=\sqrt{3t+1}$ , g) $f(x)=\sqrt{5x^{2}+1}$ , h) $f(t)=\sqrt{\frac{2}{t}}$ , i) $f(x)=2\cos(3x)$ , j) $f(t)=\sin(5t^{3}+1)$ , k) $f(x)=\sqrt{\cos(x)}$ , l) $f(t)=\frac{1}{\sin(t)}$ .

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Problem 6106

Find the total displacement of an object from a mountain-shaped velocity vs. time graph.

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Problem 6107

Zeichnen Sie im Intervall $[-2 ; 2]$ den Graphen einer Funktion $f$ für folgende Integrale: a) $0$ , b) $2$ , c) $-4$ , d) $\pi$ .

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Problem 6108

Find the tangent lines at $x=2$ for: a. $y=f(x)+g(x)$ , b. $y=f(x)-3g(x)$ , c. $y=2f(x)$ given tangents.

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Problem 6109

Find the tangent line equations at $x=4$ for: a. $y=f(x)+g(x)$ b. $y=f(x)-2g(x)$ c. $y=3f(x)$ Given: $f'(4)=3$ and $g'(4)=3$ .

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Problem 6110

Leiten Sie die folgenden Funktionen ab, nutzen Sie die Produktregel oder formen Sie sie um: a) $f(x)=\sqrt{x} \cdot(x+1)^{2}$ b) $f(x)=(x+2)^{3} \cdot x$ c) $f(x)=\frac{1}{x} \cdot\left(x^{4}+3 x^{2}\right)$

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Problem 6111

Berechne die Ableitung mit Produkt- und Kettenregel, dann umformen und mit Potenz- und Summenregel ableiten für: a) $f(x)=(5-x) \cdot(3+x)$ b) $g(x)=3 x \cdot(0,5 x+1)^{2}$ c) $h(x)=\frac{1}{x} \cdot\left(x^{2}+5 x\right)$

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Problem 6112

Zeigen Sie, dass für $b > 0$ gilt: $\int_{0}^{b} x^{3} dx = \frac{1}{4} b^{4}$ . Hinweis: $0^{3}+1^{3}+\ldots+(n-1)^{3}=\frac{1}{4}(n-1)^{2}n^{2}$ .

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Problem 6113

The student claims the derivative of $x^{2}$ is $2 x^{\prime}$ . Identify the mistake and provide the correct derivative.

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Problem 6114

Check if Rolle's theorem applies to $g(x)=8 x^{4}-3 x^{2}$ on $[-3,3]$ and find values $c$ that satisfy it.

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Problem 6115

Check if Rolle's Theorem applies to $f(x)=2 \sin (2 x)$ on $[0,2 \pi]$ and find all $c$ satisfying it.

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Problem 6116

Check if $f(x)=2 \sqrt{x}-2 x$ meets the Mean Value Theorem on $[4,25]$ and find $c$ values.

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Problem 6117

Check if Rolle's theorem applies to $g(x)=8 x^{4}-3 x^{2}$ on $[-3,3]$ and find values $c$ that satisfy it.

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Problem 6118

Leite die Funktion $f(x)=\frac{2}{3} x^{3}+3 x^{2}+4 x$ ab.

See Solution

Problem 6119

Find local extrema of $Q(x)=(x-2)(x-4)(x-5)+7$ by deriving, setting to zero, and analyzing intervals. Graph $Q(x)$ .

See Solution

Problem 6120

Bestimmen Sie den Gesamtinhalt $A$ zwischen $f(x)=\frac{1}{6} x^{4}-\frac{3}{2} x^{2}$ und der x-Achse im Intervall $I=[-3 ; 3,5]$ .

See Solution

Problem 6121

Rabbit population on an island is given by $P(t)=140t-0.3t^{4}+1000$ . Find max population and when it disappears.

See Solution

Problem 6122

Bestimme den Gesamtinhalt $A$ zwischen $f(x) = x^3 - 3x$ und der x-Achse im Intervall $I = [-2, 2.5]$ .

See Solution

Problem 6123

Berechnen Sie die Steigung der Funktion an den angegebenen Stellen: a) $f(x)=x^{2}, x_{0}=2$ b) $f(x)=-3 x^{2}, x_{0}=1$ d) $f(x)=\sqrt{x}, x_{0}=2$ e) $f(x)=2 x^{2}+3 x, x_{0}=3$ .

See Solution

Problem 6124

Berechne die Ableitung von $f(x)=\sqrt{x}$ bei $x_{0}=2$ .

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Problem 6125

Gegeben ist die Funktion $f(x)=x^3-2x^2-x+2$ . Finde die Nullstellen, berechne die Fläche unter $f$ und das Integral von $-1$ bis $2$ .

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Problem 6126

Calculate the integral: $\int (a x^{2} + 6 x) \, dx$ .

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Problem 6127

Untersuchen Sie die Wendepunkte der Funktionen und skizzieren Sie die Graphen: a) $f(x)=\frac{1}{2} x^{3}-\frac{3}{2} x^{2}$ für $-1,5 \leq x \leq 3,5$ b) $f(x)=\frac{1}{8} x^{5}-4$ für $-2 \leq x \leq 2$

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Problem 6128

Find $\frac{d y}{d x}$ given the equation $x y = x + x^{2} - 1$ .

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Problem 6129

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt[3]{x^{3}-3 x^{2}}-x\right)$ .

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Problem 6130

Question 1: Given $Y=Z K+B N$ , find output per worker $y$ in terms of capital per worker $k$ , and determine the marginal and average product of $k$ . Also, derive the dynamic equation of $k$ considering growth rate $n$ and depreciation rate $d$ .

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Problem 6131

Find $f^{\prime}(a)$ for $f(x)=\sqrt[3]{x}$ using the limit definition. Show $f^{\prime}(0)$ doesn't exist.

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Problem 6132

Find the slope of the graph of $f(x)=-6-5 x^{2}$ at the point $(4,-86)$ . If it doesn't exist, enter 'DNE'.

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Problem 6133

Derive the dynamic equation of capital $k$ from the production function $Y=Z K+B N$ with growth rates $n$ and $d$ .

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Problem 6134

Find $f^{\prime}(a)$ for $f(x)=\sqrt[3]{x}$ using $f^{\prime}(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ , $a \neq 0$ .

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Problem 6135

Find the slope of $f(x)=-6-5 x^{2}$ at the point $(4,-86)$ . If no slope, enter 'DNE'.

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Problem 6136

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

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Problem 6137

Find the tangent line equation for $f(x) = \sqrt{x}$ at $(1,1)$ and graph both using a utility.

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Problem 6138

Find the equations of the tangent lines to $f(x)=x^{2}$ that pass through $(1,-3)$ . Negative slope: $y=$ , Positive slope: $y=$ .

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Problem 6139

Find the derivative $\frac{d r}{d \theta}$ for the function $r=\sin \theta \tan \theta$ .

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Problem 6140

Given the production function $Y=Z K+B N$ , find output per worker $y$ as a function of capital per worker $k$ , and determine the marginal and average product of $k$ . Also, derive the dynamic equation of $k$ and state conditions for steady state and endogenous growth.

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Problem 6141

Given the production function $Y=Z K+B N$ , find output per worker $y$ in terms of $k$ , and determine the marginal and average products. Then derive the dynamic equation for $k$ and state conditions for steady state and endogenous growth.

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Problem 6142

Given the function $Y=Z K+B N$ , find output per worker $y$ as a function of capital per worker $k$ . Determine marginal and average products of $k$ . Then, derive the motion equation of $k$ and discuss conditions for steady state vs. endogenous growth.

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Problem 6143

A ball falls from $O$ to $P$ (20 m down) and bounces on a $50^{\circ}$ incline. Find its speed at impact and time to $Q$ .

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Problem 6144

Find the derivative $y'$ , given the function $y=x^{2}$ at the point $(2,4)$ .

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Problem 6145

Find the derivative of the function $y=\frac{6}{7 x^{4}}$ and simplify your answer.

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Problem 6146

Estimate the slope of the tangent line to $y=x^{2/3}$ at (1,1) and verify it analytically. Find $y'$ .

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Problem 6147

Given the production function $Y=Z K+B N$ , find output per worker $y$ as a function of capital per worker $k$ , and determine marginal and average products. Also, derive the dynamic equation of $k$ , conditions for steady state vs. endogenous growth, and analyze growth rates over time. Finally, calculate long-run growth rate of output per worker with $s=0.4, Z=1, B=2, d=0.08$ , and $n=0.02$ , and compare with $B=5$ .

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Problem 6148

Find the tangent line equation for the function $f(x)=-8 x^{4}+11 x^{2}-3$ at the point $(1,0)$ . $y=$

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Problem 6149

Find the derivative of the function $f(x)=\sqrt{x}-10 \sqrt[3]{x}$ . What is $f^{\prime}(x)$ ?

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Problem 6150

Find the derivative of $f(x) = \frac{2x^3 + 6x^2}{x}$ .

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Problem 6151

Find the slope of $f(x)=3(2-x)^{2}$ at the point (7,75) using a graphing utility. Calculate $f^{\prime}(7)$ .

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Problem 6152

Find the integral of $x^{3} e^{x}$ with respect to $x$ .

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Problem 6153

Find the integral of $e^{2 x} \sin (3 x)$ with respect to $x$ .

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Problem 6154

Find the integral of $\sin^{4} x$ with respect to $x$ : $\int \sin^{4} x \, dx$ .

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Problem 6155

Calculate the integral $\int \sin (4 x) \cos (5 x) d x$ .

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Problem 6156

Evaluate the integral: $\int \frac{e^{-x}}{e^{x}-5+4 e^{-x}} d x$

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Problem 6157

Calculate the integral $\int \frac{4 x^{2}-3 x+2}{4 x^{2}-4 x+3} d x$ .

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Problem 6158

Evaluate the integral: $\int \frac{x}{\sqrt{3-2 x-x^{2}}} d x$

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Problem 6159

Determine if the following integrals are convergent or divergent: (i) $\int_{0}^{+\infty} e^{-x^{2}} d x$ , (ii) $\int_{0}^{+\infty} \frac{4}{\sqrt{x}(9+x)} d x$ .

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Problem 6160

Prove that $f(x)=\int_{0}^{1} \frac{1}{1+(t x)^{3}} dt$ is continuous at zero: show $\lim_{x \to 0^{+}} f(x)=f(0)$ .

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Problem 6161

Calculate $I_{0}$ , show $(2n+1)I_{n}=\sqrt{2}-2nI_{n-1}$ , and find $I_{1}$ and $I_{2}$ .

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Problem 6162

Find the limit: $\lim _{x \rightarrow 0^{+}}(1+\sin (4 x))^{\cot x}$ .

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Problem 6163

Find the derivative of $f(t) = \ln(t^4 + 7)$ . What is $f'(t)$ ?

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Problem 6164

Find the derivative $d y / d x$ for $y=e^{x} \sqrt{x^{2}+1}$ . What is $\frac{d y}{d x}=$ ?

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Problem 6165

Find the derivative of the function $f(x)=6 \tan^{-1}(6 e^{x})$ . What is $f^{\prime}(x)$ ?

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Problem 6166

Find the derivative of $q(r) = \frac{4 r}{2 r + 3}$ . Simplify first if helpful. What is $q^{\prime}(r)$ ?

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Problem 6167

Find the derivative of $f(x)=\pi^{\cos x}$ . What is $f^{\prime}(x)$ ?

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Problem 6168

Determine if I'Hôspital's rule applies to evaluate the limit: $\lim _{x \rightarrow-1} \frac{x^{2}+6 x+5}{x^{2}+x}$ .

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Problem 6169

Determine if l'Hôpital's rule applies to evaluate the limit: $\lim _{x \rightarrow-\infty} \frac{9 x^{2}+10 x-1}{8 x^{2}-5 x}$

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Problem 6170

Find the derivative of $h(t)=400-16 t^{2}$ at $(3,256)$ and explain the falling speed of the bundle.

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Problem 6171

Evaluate $\lim _{x \rightarrow 1} \frac{x^{2}-2 x+1}{x^{2}-x}$ using l'Hospital's rule if applicable.

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Problem 6172

Find the derivative of the function $f(t)=\frac{1}{t^{3}}(\cot t+5)$ . What is $f^{\prime}(t)$ ?

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Problem 6173

Find the rate of growth of the area $A(t)=\pi(70+6t)^{2}$ one hour after spotting the oil slick. Express as a derivative.

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Problem 6174

Find the slope of the tangent to $g(x) = 3x^4$ at the point $(-2, 48)$ using $g^{\prime}(-2)$ .

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Problem 6175

Solve the differential equation: $y' - \frac{2y}{x} = x^2 \cos x$ .

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Problem 6176

Find the derivative of the function $s(x)=\left(\frac{3 x-6}{2 x+2}\right)^{3}$ .

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Problem 6177

Differentiate the function $y=\frac{9}{\pi}+\frac{2}{x^{2}+1}$ with respect to $x$ .

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Problem 6178

Find $x$ where the derivative $f^{\prime}(x)=0$ for $f(x)=\frac{x}{x^{2}+9}$ .

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Problem 6179

Find the tangent line equation to $f$ at $x=2$ , where $f(x)=\frac{x-6}{7x-5}$ .

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Problem 6180

Find the rate of fish production change at time $t=7$ months, given $Q(K)=107 K^{1/3}$ and $K(t)=0.5 t^2 + 300 t + 501$ .

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Problem 6181

Find the derivative of $h(x)=(3.1 x-3)^{2}-\frac{1}{(3.1 x-3)^{2}}$ . What is $h^{\prime}(x)$ ?

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Problem 6182

Find the derivative $h^{\prime}(x)$ for the function $h(x)=\frac{f(x)}{x^{15}}$ .

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Problem 6183

Find the rate of change of $f(t)=\frac{88 t}{99 t-88}$ after 1 hour. Round to the nearest integer.

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Problem 6184

Estimate the cost of manufacturing 71 bicycles if $C(70)=9000$ and $C^{\prime}(70)=95$ . Cost is approximately \$\square.

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Problem 6185

Estimate the cost of manufacturing 61 bicycles if $C(60)=4000$ and $C^{\prime}(60)=45$ . The cost is approximately \$ \square.

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Problem 6186

Find the marginal cost for the function $C(x)=15,000+30x+\frac{1,000}{x}$ at production level $x=100$ . Units: dollars.

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Problem 6187

Find $x$ where the derivative $f^{\prime}(x)=0$ for $f(x)=\frac{x}{x^{2}+49}$ . What are the values of $x$ ?

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Problem 6188

Find the marginal cost, revenue, and profit from $C(x)=3x$ and $R(x)=8x-0.001x^2$ . Where is marginal profit zero? $x=$

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Problem 6189

Solve the equation: $\frac{d y}{d x} \cos x + \frac{y}{\cos x} = \frac{2}{\cos x}$ .

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Problem 6190

Find $\frac{d x}{d t}$ at $x=-3$ given $y=-3 x^{2}-4$ and $\frac{d y}{d t}=-1$ . What is $\frac{d x}{d t}$ ?

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Problem 6191

Airing $x$ commercials costs $C(x)=20+5,000x+0.04x^{2}$ .
(a) Find $C^{\prime}(x)$ and estimate cost increase at $x=4$ .
(b) Find $\bar{C}(x)$ and evaluate $\bar{C}(4)$ .

See Solution

Problem 6192

Find the marginal cost function $C'(x)$ for $C(x)=20+5,000x+0.04x^2$ . Estimate cost increase at $x=4$ and compare with exact cost of fifth commercial. Find average cost function $\bar{C}$ and evaluate $\bar{C}(4)$ . What does this tell you about the average cost?

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Problem 6193

A snowball's radius decreases at $0.1 \mathrm{~cm} / \mathrm{min}$ . Find the volume decrease rate when radius is $17 \mathrm{~cm}$ . Use $V=\frac{4}{3} \pi r^{3}$ .

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Problem 6194

Cost to produce $x$ servings: $C(x)=350+0.10 x+0.002 x^{2}$ . Revenue from $x$ servings: $R(x)= 1.1x$ .
(a) Find $R^{\prime}(x)$ and $P^{\prime}(x)$ .
(b) Compute revenue, profit, marginal revenue, and profit for 200 servings (in \$). Interpret results.
(c) When is marginal profit zero? Interpret.

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Problem 6195

Evaluate the integral from 1 to 2: $\int_{1}^{2} x^{5} \sqrt{x^{3}-1} \, dx$

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Problem 6196

Calculate the average rate of change of $f(x)=5 x^{2}-3$ from $x=1$ to $x=b$ . Express your answer in terms of $b$ .

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Problem 6197

Find the max rate of change of $f(x, y)=\ln(x^{2}+y^{2})$ at $(5,4)$ and the direction (unit vector) it occurs.

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Problem 6198

Find the rate of change of $f(x, y)=4xy+y^{2}$ at $(4,1)$ in the direction $\vec{v}=-2i-2j$ . Also, determine the direction and maximum rate of change.

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Problem 6199

Find the rate at which the distance between Joe and Dave's cars is decreasing when Joe is $0.4 \mathrm{~km}$ and Dave is $0.3 \mathrm{~km}$ from the intersection.

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Problem 6200

At 1:00 p.m., ship A is 80 km south of ship B. A sails north at 30 km/h and B east at 40 km/h. Find distance change rate at 3:00 p.m.

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Calculus

Problem 6101

Find the derivative of f(x)=12e13x+2f(x)=\frac{1}{2} e^{\frac{1}{3} x+2}f(x)=21​e31​x+2 and simplify it.

Problem 6102

Gegeben ist die Funktion f(x)=19(3x+2)3f(x)=\frac{1}{9}(3 x+2)^{3}f(x)=91​(3x+2)3. Bestimme die Steigung in P(2∣f(2))P(2 \mid f(2))P(2∣f(2)), waagerechte Tangenten und Punkte mit 45∘45^{\circ}45∘ Steigungswinkel.

Problem 6103

Problem 6104

Problem 6105

Problem 6106

Find the total displacement of an object from a mountain-shaped velocity vs. time graph.

Problem 6107

Zeichnen Sie im Intervall [−2;2][-2 ; 2][−2;2] den Graphen einer Funktion fff für folgende Integrale: a) 000, b) 222, c) −4-4−4, d) π\piπ.

Problem 6108

Find the tangent lines at x=2x=2x=2 for: a. y=f(x)+g(x)y=f(x)+g(x)y=f(x)+g(x), b. y=f(x)−3g(x)y=f(x)-3g(x)y=f(x)−3g(x), c. y=2f(x)y=2f(x)y=2f(x) given tangents.

Problem 6109

Find the tangent line equations at x=4x=4x=4 for: a. y=f(x)+g(x)y=f(x)+g(x)y=f(x)+g(x) b. y=f(x)−2g(x)y=f(x)-2g(x)y=f(x)−2g(x) c. y=3f(x)y=3f(x)y=3f(x) Given: f′(4)=3f'(4)=3f′(4)=3 and g′(4)=3g'(4)=3g′(4)=3.

Problem 6110

Problem 6111

Problem 6112

Zeigen Sie, dass für b>0b > 0b>0 gilt: ∫0bx3dx=14b4\int_{0}^{b} x^{3} dx = \frac{1}{4} b^{4}∫0b​x3dx=41​b4. Hinweis: 03+13+…+(n−1)3=14(n−1)2n20^{3}+1^{3}+\ldots+(n-1)^{3}=\frac{1}{4}(n-1)^{2}n^{2}03+13+…+(n−1)3=41​(n−1)2n2.

Problem 6113

The student claims the derivative of x2x^{2}x2 is 2x′2 x^{\prime}2x′. Identify the mistake and provide the correct derivative.

Problem 6114

Check if Rolle's theorem applies to g(x)=8x4−3x2g(x)=8 x^{4}-3 x^{2}g(x)=8x4−3x2 on [−3,3][-3,3][−3,3] and find values ccc that satisfy it.

Problem 6115

Check if Rolle's Theorem applies to f(x)=2sin⁡(2x)f(x)=2 \sin (2 x)f(x)=2sin(2x) on [0,2π][0,2 \pi][0,2π] and find all ccc satisfying it.

Problem 6116

Check if f(x)=2x−2xf(x)=2 \sqrt{x}-2 xf(x)=2x​−2x meets the Mean Value Theorem on [4,25][4,25][4,25] and find ccc values.

Problem 6117

Check if Rolle's theorem applies to g(x)=8x4−3x2g(x)=8 x^{4}-3 x^{2}g(x)=8x4−3x2 on [−3,3][-3,3][−3,3] and find values ccc that satisfy it.

Problem 6118

Leite die Funktion f(x)=23x3+3x2+4xf(x)=\frac{2}{3} x^{3}+3 x^{2}+4 xf(x)=32​x3+3x2+4x ab.

Problem 6119

Find local extrema of Q(x)=(x−2)(x−4)(x−5)+7Q(x)=(x-2)(x-4)(x-5)+7Q(x)=(x−2)(x−4)(x−5)+7 by deriving, setting to zero, and analyzing intervals. Graph Q(x)Q(x)Q(x).

Problem 6120

Bestimmen Sie den Gesamtinhalt AAA zwischen f(x)=16x4−32x2f(x)=\frac{1}{6} x^{4}-\frac{3}{2} x^{2}f(x)=61​x4−23​x2 und der x-Achse im Intervall I=[−3;3,5]I=[-3 ; 3,5]I=[−3;3,5].

Problem 6121

Rabbit population on an island is given by P(t)=140t−0.3t4+1000P(t)=140t-0.3t^{4}+1000P(t)=140t−0.3t4+1000. Find max population and when it disappears.

Problem 6122

Bestimme den Gesamtinhalt AAA zwischen f(x)=x3−3xf(x) = x^3 - 3xf(x)=x3−3x und der x-Achse im Intervall I=[−2,2.5]I = [-2, 2.5]I=[−2,2.5].

Problem 6123

Berechnen Sie die Steigung der Funktion an den angegebenen Stellen: a) f(x)=x2,x0=2f(x)=x^{2}, x_{0}=2f(x)=x2,x0​=2 b) f(x)=−3x2,x0=1f(x)=-3 x^{2}, x_{0}=1f(x)=−3x2,x0​=1 d) f(x)=x,x0=2f(x)=\sqrt{x}, x_{0}=2f(x)=x​,x0​=2 e) f(x)=2x2+3x,x0=3f(x)=2 x^{2}+3 x, x_{0}=3f(x)=2x2+3x,x0​=3.

Problem 6124

Berechne die Ableitung von f(x)=xf(x)=\sqrt{x}f(x)=x​ bei x0=2x_{0}=2x0​=2.

Problem 6125

Gegeben ist die Funktion f(x)=x3−2x2−x+2f(x)=x^3-2x^2-x+2f(x)=x3−2x2−x+2. Finde die Nullstellen, berechne die Fläche unter fff und das Integral von −1-1−1 bis 222.

Problem 6126

Calculate the integral: ∫(ax2+6x) dx\int (a x^{2} + 6 x) \, dx∫(ax2+6x)dx.

Problem 6127

Problem 6128

Find dydx\frac{d y}{d x}dxdy​ given the equation xy=x+x2−1x y = x + x^{2} - 1xy=x+x2−1.

Problem 6129

Find the limit: lim⁡x→∞(x3−3x23−x)\lim _{x \rightarrow \infty}\left(\sqrt[3]{x^{3}-3 x^{2}}-x\right)limx→∞​(3x3−3x2​−x).

Problem 6130

Question 1: Given Y=ZK+BNY=Z K+B NY=ZK+BN, find output per worker yyy in terms of capital per worker kkk, and determine the marginal and average product of kkk. Also, derive the dynamic equation of kkk considering growth rate nnn and depreciation rate ddd.

Problem 6131

Find f′(a)f^{\prime}(a)f′(a) for f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ using the limit definition. Show f′(0)f^{\prime}(0)f′(0) doesn't exist.

Problem 6132

Find the slope of the graph of f(x)=−6−5x2f(x)=-6-5 x^{2}f(x)=−6−5x2 at the point (4,−86)(4,-86)(4,−86). If it doesn't exist, enter 'DNE'.

Problem 6133

Derive the dynamic equation of capital kkk from the production function Y=ZK+BNY=Z K+B NY=ZK+BN with growth rates nnn and ddd.

Problem 6134

Find f′(a)f^{\prime}(a)f′(a) for f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ using f′(a)=lim⁡h→0f(a+h)−f(a)hf^{\prime}(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}f′(a)=limh→0​hf(a+h)−f(a)​, a≠0a \neq 0a=0.

Problem 6135

Find the slope of f(x)=−6−5x2f(x)=-6-5 x^{2}f(x)=−6−5x2 at the point (4,−86)(4,-86)(4,−86). If no slope, enter 'DNE'.

Problem 6136

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

Problem 6137

Find the tangent line equation for f(x)=xf(x) = \sqrt{x}f(x)=x​ at (1,1)(1,1)(1,1) and graph both using a utility.

Problem 6138

Find the equations of the tangent lines to f(x)=x2f(x)=x^{2}f(x)=x2 that pass through (1,−3)(1,-3)(1,−3). Negative slope: y=y=y=, Positive slope: y=y=y=.

Problem 6139

Find the derivative drdθ\frac{d r}{d \theta}dθdr​ for the function r=sin⁡θtan⁡θr=\sin \theta \tan \thetar=sinθtanθ.

Problem 6140

Given the production function Y=ZK+BNY=Z K+B NY=ZK+BN, find output per worker yyy as a function of capital per worker kkk, and determine the marginal and average product of kkk. Also, derive the dynamic equation of kkk and state conditions for steady state and endogenous growth.

Problem 6141

Given the production function Y=ZK+BNY=Z K+B NY=ZK+BN, find output per worker yyy in terms of kkk, and determine the marginal and average products. Then derive the dynamic equation for kkk and state conditions for steady state and endogenous growth.

Problem 6142

Given the function Y=ZK+BNY=Z K+B NY=ZK+BN, find output per worker yyy as a function of capital per worker kkk. Determine marginal and average products of kkk. Then, derive the motion equation of kkk and discuss conditions for steady state vs. endogenous growth.

Problem 6143

Find the derivative of $f(x)=\frac{1}{2} e^{\frac{1}{3} x+2}$ and simplify it.

Gegeben ist die Funktion $f(x)=\frac{1}{9}(3 x+2)^{3}$ . Bestimme die Steigung in $P(2 \mid f(2))$ , waagerechte Tangenten und Punkte mit $45^{\circ}$ Steigungswinkel.

Zeichnen Sie im Intervall $[-2 ; 2]$ den Graphen einer Funktion $f$ für folgende Integrale: a) $0$ , b) $2$ , c) $-4$ , d) $\pi$ .

Find the tangent lines at $x=2$ for: a. $y=f(x)+g(x)$ , b. $y=f(x)-3g(x)$ , c. $y=2f(x)$ given tangents.

Find the tangent line equations at $x=4$ for: a. $y=f(x)+g(x)$ b. $y=f(x)-2g(x)$ c. $y=3f(x)$ Given: $f'(4)=3$ and $g'(4)=3$ .

Zeigen Sie, dass für $b > 0$ gilt: $\int_{0}^{b} x^{3} dx = \frac{1}{4} b^{4}$ . Hinweis: $0^{3}+1^{3}+\ldots+(n-1)^{3}=\frac{1}{4}(n-1)^{2}n^{2}$ .

The student claims the derivative of $x^{2}$ is $2 x^{\prime}$ . Identify the mistake and provide the correct derivative.

Check if Rolle's theorem applies to $g(x)=8 x^{4}-3 x^{2}$ on $[-3,3]$ and find values $c$ that satisfy it.

Check if Rolle's Theorem applies to $f(x)=2 \sin (2 x)$ on $[0,2 \pi]$ and find all $c$ satisfying it.

Check if $f(x)=2 \sqrt{x}-2 x$ meets the Mean Value Theorem on $[4,25]$ and find $c$ values.

Check if Rolle's theorem applies to $g(x)=8 x^{4}-3 x^{2}$ on $[-3,3]$ and find values $c$ that satisfy it.

Leite die Funktion $f(x)=\frac{2}{3} x^{3}+3 x^{2}+4 x$ ab.

Find local extrema of $Q(x)=(x-2)(x-4)(x-5)+7$ by deriving, setting to zero, and analyzing intervals. Graph $Q(x)$ .

Bestimmen Sie den Gesamtinhalt $A$ zwischen $f(x)=\frac{1}{6} x^{4}-\frac{3}{2} x^{2}$ und der x-Achse im Intervall $I=[-3 ; 3,5]$ .

Rabbit population on an island is given by $P(t)=140t-0.3t^{4}+1000$ . Find max population and when it disappears.

Bestimme den Gesamtinhalt $A$ zwischen $f(x) = x^3 - 3x$ und der x-Achse im Intervall $I = [-2, 2.5]$ .

Berechnen Sie die Steigung der Funktion an den angegebenen Stellen: a) $f(x)=x^{2}, x_{0}=2$ b) $f(x)=-3 x^{2}, x_{0}=1$ d) $f(x)=\sqrt{x}, x_{0}=2$ e) $f(x)=2 x^{2}+3 x, x_{0}=3$ .

Berechne die Ableitung von $f(x)=\sqrt{x}$ bei $x_{0}=2$ .

Gegeben ist die Funktion $f(x)=x^3-2x^2-x+2$ . Finde die Nullstellen, berechne die Fläche unter $f$ und das Integral von $-1$ bis $2$ .

Calculate the integral: $\int (a x^{2} + 6 x) \, dx$ .

Find $\frac{d y}{d x}$ given the equation $x y = x + x^{2} - 1$ .

Find the limit: $\lim _{x \rightarrow \infty}\left(\sqrt[3]{x^{3}-3 x^{2}}-x\right)$ .

Question 1: Given $Y=Z K+B N$ , find output per worker $y$ in terms of capital per worker $k$ , and determine the marginal and average product of $k$ . Also, derive the dynamic equation of $k$ considering growth rate $n$ and depreciation rate $d$ .

Find $f^{\prime}(a)$ for $f(x)=\sqrt[3]{x}$ using the limit definition. Show $f^{\prime}(0)$ doesn't exist.

Find the slope of the graph of $f(x)=-6-5 x^{2}$ at the point $(4,-86)$ . If it doesn't exist, enter 'DNE'.

Derive the dynamic equation of capital $k$ from the production function $Y=Z K+B N$ with growth rates $n$ and $d$ .

Find $f^{\prime}(a)$ for $f(x)=\sqrt[3]{x}$ using $f^{\prime}(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$ , $a \neq 0$ .

Find the slope of $f(x)=-6-5 x^{2}$ at the point $(4,-86)$ . If no slope, enter 'DNE'.

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

Find the tangent line equation for $f(x) = \sqrt{x}$ at $(1,1)$ and graph both using a utility.

Find the equations of the tangent lines to $f(x)=x^{2}$ that pass through $(1,-3)$ . Negative slope: $y=$ , Positive slope: $y=$ .

Find the derivative $\frac{d r}{d \theta}$ for the function $r=\sin \theta \tan \theta$ .

Given the production function $Y=Z K+B N$ , find output per worker $y$ as a function of capital per worker $k$ , and determine the marginal and average product of $k$ . Also, derive the dynamic equation of $k$ and state conditions for steady state and endogenous growth.

Given the production function $Y=Z K+B N$ , find output per worker $y$ in terms of $k$ , and determine the marginal and average products. Then derive the dynamic equation for $k$ and state conditions for steady state and endogenous growth.

Given the function $Y=Z K+B N$ , find output per worker $y$ as a function of capital per worker $k$ . Determine marginal and average products of $k$ . Then, derive the motion equation of $k$ and discuss conditions for steady state vs. endogenous growth.

A ball falls from $O$ to $P$ (20 m down) and bounces on a $50^{\circ}$ incline. Find its speed at impact and time to $Q$ .

Find the derivative $y'$ , given the function $y=x^{2}$ at the point $(2,4)$ .

Find the derivative of the function $y=\frac{6}{7 x^{4}}$ and simplify your answer.

Estimate the slope of the tangent line to $y=x^{2/3}$ at (1,1) and verify it analytically. Find $y'$ .

Find the tangent line equation for the function $f(x)=-8 x^{4}+11 x^{2}-3$ at the point $(1,0)$ . $y=$

Find the derivative of the function $f(x)=\sqrt{x}-10 \sqrt[3]{x}$ . What is $f^{\prime}(x)$ ?

Find the derivative of $f(x) = \frac{2x^3 + 6x^2}{x}$ .

Find the slope of $f(x)=3(2-x)^{2}$ at the point (7,75) using a graphing utility. Calculate $f^{\prime}(7)$ .

Find the integral of $x^{3} e^{x}$ with respect to $x$ .

Find the integral of $e^{2 x} \sin (3 x)$ with respect to $x$ .

Find the integral of $\sin^{4} x$ with respect to $x$ : $\int \sin^{4} x \, dx$ .

Calculate the integral $\int \sin (4 x) \cos (5 x) d x$ .

Evaluate the integral: $\int \frac{e^{-x}}{e^{x}-5+4 e^{-x}} d x$

Calculate the integral $\int \frac{4 x^{2}-3 x+2}{4 x^{2}-4 x+3} d x$ .

Evaluate the integral: $\int \frac{x}{\sqrt{3-2 x-x^{2}}} d x$

Determine if the following integrals are convergent or divergent: (i) $\int_{0}^{+\infty} e^{-x^{2}} d x$ , (ii) $\int_{0}^{+\infty} \frac{4}{\sqrt{x}(9+x)} d x$ .

Prove that $f(x)=\int_{0}^{1} \frac{1}{1+(t x)^{3}} dt$ is continuous at zero: show $\lim_{x \to 0^{+}} f(x)=f(0)$ .

Calculate $I_{0}$ , show $(2n+1)I_{n}=\sqrt{2}-2nI_{n-1}$ , and find $I_{1}$ and $I_{2}$ .

Find the limit: $\lim _{x \rightarrow 0^{+}}(1+\sin (4 x))^{\cot x}$ .

Find the derivative of $f(t) = \ln(t^4 + 7)$ . What is $f'(t)$ ?

Find the derivative $d y / d x$ for $y=e^{x} \sqrt{x^{2}+1}$ . What is $\frac{d y}{d x}=$ ?

Find the derivative of the function $f(x)=6 \tan^{-1}(6 e^{x})$ . What is $f^{\prime}(x)$ ?

Find the derivative of $q(r) = \frac{4 r}{2 r + 3}$ . Simplify first if helpful. What is $q^{\prime}(r)$ ?

Find the derivative of $f(x)=\pi^{\cos x}$ . What is $f^{\prime}(x)$ ?

Determine if I'Hôspital's rule applies to evaluate the limit: $\lim _{x \rightarrow-1} \frac{x^{2}+6 x+5}{x^{2}+x}$ .

Determine if l'Hôpital's rule applies to evaluate the limit: $\lim _{x \rightarrow-\infty} \frac{9 x^{2}+10 x-1}{8 x^{2}-5 x}$

Find the derivative of $h(t)=400-16 t^{2}$ at $(3,256)$ and explain the falling speed of the bundle.

Evaluate $\lim _{x \rightarrow 1} \frac{x^{2}-2 x+1}{x^{2}-x}$ using l'Hospital's rule if applicable.

Find the derivative of the function $f(t)=\frac{1}{t^{3}}(\cot t+5)$ . What is $f^{\prime}(t)$ ?

Find the rate of growth of the area $A(t)=\pi(70+6t)^{2}$ one hour after spotting the oil slick. Express as a derivative.

Find the slope of the tangent to $g(x) = 3x^4$ at the point $(-2, 48)$ using $g^{\prime}(-2)$ .