Calculus

Problem 5101

Find the derivative of $h(x)=(x^{2}+4)(2x^{2}-x+1)$ .

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Problem 5102

A potato is launched from a 50-ft building with an initial speed of $75 \, \text{ft/s}$ . Find its velocity when it hits the ground.

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Problem 5103

Differentiate $z=\frac{A}{y^{6}}+B e^{y}$ with respect to $y$ .

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Problem 5104

Connor invested \$77,000 at 4.5\% interest compounded continuously. How much will he have after 12 years?

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Problem 5105

How much will Serenity have after 12 years if she invests \$3,100 at a 2.9% continuous interest rate?

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Problem 5106

Find the derivative of $h(x)=(4x^{2}+3x)(5x^{2}-2)$ .

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Problem 5107

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

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Problem 5108

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

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Problem 5109

Gegeben ist die Funktion $f(x) = \frac{1}{x} \cos(x)$ . Bestimmen Sie Definitionsmenge, Nullstellen, Grenzwert und zeichnen Sie Graphen.

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Problem 5110

Find the revenue function $R(x)$ from $p=10-0.002x$ , then calculate $R'(x)$ and its values at $x=8000$ and $x=9000$ .

See Solution

Problem 5111

Berechne die verbleibende Menge an Cs 137 nach 2, 5, 10 und 33 Jahren, wenn 2,1\% jährlich zerfallen, aus 500 g.

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Problem 5112

Find the critical numbers for the function $f(x)=\frac{2}{3} x^{3}-3 x^{2}-80 x-10$ .

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Problem 5113

Is it true or false that if $f^{\prime}(x)>0$ on an interval, then $f$ is positive there? Explain your answer. Options: A, B, C, D.

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Problem 5114

Is it true or false that derivatives alternate between positive and negative across regions separated by critical numbers?

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Problem 5115

Find the critical numbers, increasing intervals, and decreasing intervals for $f(x)=2.5+4.2 x-1.1 x^{2}$ .

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Problem 5116

Find the marginal profit at $x=6$ for the profit function $P(x)=-x^{2}+12x-32$ .

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Problem 5117

Identify functions that can be evaluated without the Chain Rule: $z=\sqrt{\sin t}$ , $y=\tan x \cos x$ , $T(v)=(3 v+7)^{2}$ , $S(x)=3 \tan ^{2} x$ , $A(x)=\tan \left(3 x^{2}\right)$ , $f(x)=\tan (\cos x)$ .

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Problem 5118

Trouvez les limites à gauche et à droite de la fonction en -2 et 3. Écrivez inf pour $\infty$ et -inf pour $-\infty$ .
$\lim _{x \rightarrow-2^{-}} f=$ $\lim _{x \rightarrow-2^{+}} f=$ $\lim _{x \rightarrow 3^{-}} f=$ $\lim _{x \rightarrow 3^{+}} f=$

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Problem 5119

Find the average rate of change of the function $h(x)=-x^{2}-4x+7$ on the interval $-7 \leq x \leq 0$ .

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Problem 5120

Find the derivative of $f(x)=\sin(5x^{2}-x)$ using the Chain Rule.

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Problem 5121

Find the derivatives: a) For $h(x)=f(3x)$ , find $h'(2)$ . b) For $y=f(x^2+2)$ , find $y'|_{x=1}$ . c) For $G(x)=\cos(f(x))$ , find $G'(\pi)$ .

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Problem 5122

Déterminez la convergence de la suite $a_{n}=\frac{3 n^{4}}{\sqrt{5 n^{8}+4 n^{5}}}$ . Limite ou diverge ?

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Problem 5123

Trouvez $a$ pour que $\lim _{x \rightarrow 2} f(x)$ existe, avec $f(x)=\begin{cases} e^{x-a}-9 & \text{si } x \geq 2 \\ x^{2}+5 & \text{si } x<2 \end{cases}$ .

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Problem 5124

Find the derivative using the Power Rule: $\left.\frac{d}{d t} t^{-3}\right|_{t=4}=$

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Problem 5125

Find the units of $S=2^t$ , the average rate of change in 5 months, and the growth rate at $t=5$ using an interval of 0.001.

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Problem 5126

Find the slope $m$ of the tangent line to $y=f(x)$ at $(-5, f(-5))$ where $f(x) = 6x^2 - 6$ .

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Problem 5127

Find the derivative of $h(t)=9 \sqrt{t}-\frac{4}{\sqrt{t}}$ . Express in exact form: $h^{\prime}(t)=$

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Problem 5128

Find the derivative of $P(V)=\frac{3}{V}$ at $V=-5$ . What is $\left.\frac{d P}{d V}\right|_{V=-5}=$ ?

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Problem 5129

Find points on the graph of $f(x)=27x-x^{3}$ where the tangent line is horizontal. Provide as $(x, y)$ pairs.

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Problem 5130

Find $f^{\prime}(1)$ for $f(x)=5^{x^{2}+2}$ . Answer in the form "number/n natural log".

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Problem 5131

Find all $x$ where tangent lines to $y=x^{4}$ and $y=x^{5}$ have the same slope.
$x=$

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Problem 5132

Find $x$ where the slope of the tangent to $y=5 x^{2}-21 x+7$ is steeper than that of $y=\frac{1}{3} x$ (interval form).

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Problem 5133

Find the average rate of change of $j(x)=3x^{3}$ from $x=1$ to $x=1+h$ .

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Problem 5134

Find the derivative $\frac{dy}{dx}$ for $\sqrt{x}+\sqrt{y}=100$ using implicit differentiation.

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Problem 5135

Find $f^{\prime}(x)$ using the limit definition for $f(x) = x^{2} + 23x$ .

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Problem 5136

Find $x$ where the slope of the tangent to $y=5x^2-21x+7$ is steeper than that of $y=\frac{1}{3}x^3$ . Answer as an interval $(a, b)$ .

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Problem 5137

Find the rate of change of a circle's area with respect to radius $r$ . What is it when $r=6$ ?

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Problem 5138

Find $f^{\prime}(x)$ using the limit definition for $f(x)=(17 x)^{-1/2}+5$ .

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Problem 5139

Find $\frac{d K}{d t}$ for $K=\frac{A^{5} B^{3}}{4}$ , with $B$ varying and $A$ constant.

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Problem 5140

Find the surface area change rate of a sphere with respect to radius $r$ . What is it when $r=12$ ?

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Problem 5141

Given $K=\frac{A^{2} B^{3}}{3}$ , find $\frac{d K}{d t}$ when $A$ and $B$ vary with respect to $t$ .

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Problem 5142

Find the values of $x$ ( $-8, -2, 0.3, 5$ ) where the limit of $f$ exists, given the limits at $-2$ and $3$ .

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Problem 5143

Find the rate of change of a circle's area with respect to radius $r$ when $r=6$ . What is the rate?

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Problem 5144

Find the horizontal asymptote of the function $f(x)=\frac{1-5 x-2 x^{2}}{3 x^{2}+7}$ for $x>0$ . Choices: A) $y=-\frac{2}{3}$ B) $y=\frac{1}{3}$ C) $y=\frac{2}{3}$ .

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Problem 5145

Find the derivative of $f(x)=\frac{4 a}{3 b x^{2}}$ in terms of $x$ , $a$ , and $b$ . $f^{\prime}(x)=$

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Problem 5146

Find the derivative using the Power Rule: $\left.\frac{d}{d t} t^{-3}\right|_{t=2}=$

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Problem 5147

Find the derivative using the Power Rule: $\frac{d}{d t} t^{-8/5}$ at $t=1$ . What is the result?

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Problem 5148

Find the horizontal asymptote of the function $f(x)=\frac{3^{x}+2}{e^{2 x}+1}$ for $x>0$ . Options: (A) $y=0$ , (B) $y=\frac{3}{e^{2}}$ , (C) $y=1$ , (D) none.

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Problem 5149

Use the table to solve these:
a) Find $\left.\frac{d}{d x}[0 f(x)]\right|_{x=5}=$ b) Find $\left.\frac{d}{d x}[f(x)+g(x)]\right|_{x=3}=$ c) Find the tangent line equation $y=f(x)-g(x)$ at $x=7$ .

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Problem 5150

Find the horizontal asymptote of the function $f(x)=\frac{2-x+3 x^{2}+5 x^{3}-7 x^{4}}{x^{4}-2 x^{3}-5 x^{2}+2 x-3}$ for $x>0$ . Options: (A) $y=-7$ , (B) $y=2$ , (C) $y=7$ , (D) none.

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Problem 5151

Find $f^{\prime}(1)$ for $f(x)=4 \sqrt[3]{x}$ and the tangent line's equation at $x=1$ . Express as $y=f(x)$ .

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Problem 5152

Find the derivative of $h(t)=3 \sqrt{t}-\frac{4}{\sqrt{t}}$ . Use exact numbers and symbolic notation.

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Problem 5153

Find $f^{\prime}$ and the tangent line equation at $x=5$ for $f(x)=12 x^{-3}$ . Express as $y=f(x)$ .

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Problem 5154

Find the volume $V$ of the solid formed by rotating the area between $y=3\sqrt{16-x^2}$ , $y=0$ , $x=2$ , $x=3$ around the $x$ -axis.

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Problem 5155

Find all values of $x$ where the tangent lines to $y=x^{7}$ and $y=x^{8}$ have the same slope.

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Problem 5156

Find the derivative of the function $y=5 x^{2} e^{3 x}$ .

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Problem 5157

Find $\frac{d K}{d t}$ for $K=\frac{A^{5} B^{4}}{2}$ where $A$ varies with $t$ and $B$ is constant.

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Problem 5158

Find the derivative $\frac{dy}{dx}$ for the equation $x^{3}+y^{3}=5$ using implicit differentiation.

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Problem 5159

Plot $f(x)=\left|x-\cos \left(x-\frac{\pi}{2}\right)\right|$ using a graphing tool and analyze its behavior at $x=0$ . Is $f$ nondifferentiable or differentiable? Is the tangent line vertical, nonexistent, or horizontal?

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Problem 5160

Find the derivative of $f(x)=\frac{7 a}{3 b x^{2}}$ in terms of $x$ , $a$ , and $b$ . $f^{\prime}(x)=$

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Problem 5161

Find true statements about the function $f(x)=2x^{3}+x^{2}+6x+8$ from its derivative.

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Problem 5162

Find the marginal profit of the function $P(x)=x^{3}-4 x^{2}+10 x+7$ at $x=6$ .

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Problem 5163

Find: a) $\left.\frac{d}{d x}[-3 f(x)]\right|_{x=3}$ , b) $\left.\frac{d}{d x}[f(x)+g(x)]\right|_{x=6}$ , c) Tangent line equation for $y=f(x)-g(x)$ at $x=2$ .

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Problem 5164

Find the number of CDs, $x$ , that maximizes revenue given $R(x)=50x-\frac{x^2}{6}$ .

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Problem 5165

Find the fourth derivative of the function $f(x)=\cos (\sin (x))$ at $x=0$ , denoted as $f^{(4)}(0)$ .

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Problem 5166

Find the 4th derivative of $f(x)=\cos(\sin(x))$ at $x=0$ using the Taylor series.

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Problem 5167

Differentiate $y = \frac{3x}{\sin x + \cos x}$ .

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Problem 5168

Find $f^{\prime}(x)$ using logarithmic differentiation for $f(x)=\left(\frac{(3x^{3}-1)\sqrt{5x^{3}-4}}{2x+2}\right)5$ .

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Problem 5169

Find the derivative of the function $\frac{3 x+10}{x^{2}+3}$ .

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Problem 5170

Find the derivative of $f(x)=\frac{x^{9}(x-6)^{9}}{(x^{2}+7)^{7}}$ using logarithmic differentiation. $f^{\prime}(x)=$

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Problem 5171

Find the derivative of $y=(7+\sin (x))^{x}$ using Logarithmic Differentiation: $\frac{d y}{d x}=$

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Problem 5172

Find the displacement of a vehicle with velocity $v(t) = 24 + 5t - t^2$ after 5 seconds using overestimate $U_{5}$ and underestimate $L_{5}$ .

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Problem 5173

Find the global extrema of $f(x)=\cos(2x)-x$ for $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

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Problem 5174

Find $f^{\prime}(x)$ for $f(x)=(6 x-3)^{4} \cdot(7 x^{2}+3)^{4}$ using logarithmic differentiation.

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Problem 5175

Nennen Sie drei verschiedene Funktionen, die für $x \rightarrow+\infty$ keinen Grenzwert haben.

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Problem 5176

Bestimmen Sie die Tangente und Normale von $f$ bei $P(-1, f(-1))$ und berechnen Sie den Steigungswinkel. a) $f(x)=x^{5}-3 x^{3}$ b) $f(x)=\sin (\pi \cdot x)$ c) $f(x)=\sqrt{2 x+4}$ d) $f(x)=\frac{1}{x^{2}}$

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Problem 5177

Limits and asymptotes:
1. Find $\lim_{x \to -1} f(x)$ , $\lim_{x \to 0^+} f(x)$ , $f(2)$ , and $\lim_{x \to 3} f(x)$ .
2. Identify vertical asymptotes of $f(x)=1-\frac{1}{x^2}$ .
3. Determine $\lim_{x \to 2^+} \frac{x}{2-x}$ .
4. Which function has no vertical asymptote at $x=\pi$ ?

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Problem 5178

Bestimmen Sie die Tangente und Normale an $f$ bei $P(-1, f(-1))$ und berechnen Sie den Steigungswinkel für: a) $f(x)=x^{5}-3 x^{3}$ , b) $f(x)=\sin(\pi x)$ , c) $f(x)=\sqrt{2x+4}$ , d) $f(x)=\frac{1}{x^{2}}$ .

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Problem 5179

Bestimmen Sie die Stammfunktionen für: a. $f_{1}(x)=x^{3}$ , b. $f_{2}(x)=x^{14}$ .

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Problem 5180

Finde alle Stammfunktionen für die Funktion $f_{1}(x)=x^{3}$ .

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Problem 5181

Beweisen Sie, dass wenn $\mathrm{F}$ eine Stammfunktion von $\mathrm{f}$ ist, dann $G(x)=F(x)+c$ auch eine Stammfunktion ist.

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Problem 5182

Bestimmen Sie Tangenten- und Normalengleichungen sowie den Steigungswinkel der Tangente für $f$ bei $P(-1, f(-1))$ . a) $f(x)=x^{5}-3 x^{3}$ b) $f(x)=\sin (\pi \cdot x)$ c) $f(x)=\sqrt{2 x+4}$ d) $f(x)=\frac{1}{x^{2}}$

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Problem 5183

Bestimmen Sie das Integral $\int x^{3} d x$ und überprüfen Sie das Ergebnis durch Ableitung.

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Problem 5184

Finden Sie die Stammfunktionen für die folgenden Funktionen: a. $f_{1}(x)=x^{3}$ , b. $f_{2}(x)=x^{14}$ , c. $f_{3}(x)=14$ , d. $f_{4}(x)=x$ , e. $f_{5}(x)=2 x^{3}$ , f. $f_{6}(x)=-x^{3}$ , g. $f_{7}(x)=x^{-2}$ .

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Problem 5185

Bestimmen Sie die Integrale $\int x^{3} dx$ und $\int a \cdot x^{3} dx$ und überprüfen Sie die Ergebnisse durch Ableitung.

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Problem 5186

Bestimmen Sie die Integrale und überprüfen Sie die Ergebnisse durch Ableitung: a. $\int x^{3} dx$ , b. $\int a \cdot x^{3} dx$ , c. $\int a \cdot x^{n} dx$ .

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Problem 5187

Bestimme die Grenzwerte von $f(x)=\frac{1}{4}(x+1)^{2}(x-2,5)$ für $x \to \infty$ und $x \to -\infty$ .

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Problem 5188

Bestimmen Sie die Ableitung von $f_{a}(x)$ für die Funktionen a) $f_{a}(x)=\sin (a x) \cdot a x^{2}$ , b) $f_{a}(x)=(a x)^{4} \cdot \sqrt{a x}$ , c) $f_{a}(x)=\left(x^{2}+a\right) \cdot \sqrt{x}$ .

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Problem 5189

Find the derivative of $f(x)=\frac{2}{4x-1}$ .

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Problem 5190

Calculate the integral: $\int e^{x} \cdot e^{x+2} \, dx$

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Problem 5191

j) Find the integral of $e^{x} \cdot e^{x+2}$ with respect to $x$ . k) Calculate the integral of $\frac{4}{e^{x}}$ with respect to $x$ .

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Problem 5192

Calculate the integral: $\int\left(2 x+\frac{1}{x}\right) \cdot x \, dx$

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Problem 5193

Calculate the integral: $\int n \cdot x^{2 n-1} d x$

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Problem 5194

Evaluate the integrals: 1) $\int \frac{4}{e^{x}} dx$ and 2) $\int(\sin x + 2 \cos x) dx$ .

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Problem 5195

Finde alle Werte für a, sodass der Graph $G_{f_{a}}$ rechtsgekrümmt ist für die Funktionen: a) $f_{a}(x)=a \cdot(2 x-3)^{4}$ b) $f_{a}(x)=\sqrt{a^{2} x}(x \geq 0)$ c) $f_{a}(x)=\frac{a}{x^{2}}(x \neq 0)$

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Problem 5196

Bestimme die Steigung des Graphen von $f(x)=\frac{1}{9}(3x+2)^{3}$ bei $P(2, f(2))$ .

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Problem 5197

Find the slope of the tangent line for $f(x)=\frac{1}{2} x^{2}+3 x-\frac{1}{2}$ at $x=2$ .

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Problem 5198

Find $x$ values for horizontal tangents of $f(x)=3x^2-6x+4$ . Select "None" if there are none.

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Problem 5199

Find all $x$ values for horizontal tangent lines of the function $f(x)=-3 x^{2}+12 x-7$ . Select "None" if none exist.

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Problem 5200

Find the limit as $x$ approaches -5 for the expression $\frac{x^{2}+3x-10}{x+5}$ .

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Calculus

Problem 5101

Find the derivative of h(x)=(x2+4)(2x2−x+1)h(x)=(x^{2}+4)(2x^{2}-x+1)h(x)=(x2+4)(2x2−x+1).

Problem 5102

A potato is launched from a 50-ft building with an initial speed of 75 ft/s75 \, \text{ft/s}75ft/s. Find its velocity when it hits the ground.

Problem 5103

Differentiate z=Ay6+Beyz=\frac{A}{y^{6}}+B e^{y}z=y6A​+Bey with respect to yyy.

Problem 5104

Connor invested \$77,000 at 4.5\% interest compounded continuously. How much will he have after 12 years?

Problem 5105

How much will Serenity have after 12 years if she invests \$3,100 at a 2.9% continuous interest rate?

Problem 5106

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

Problem 5107

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

Problem 5108

Find the derivative of f(x)=(x2+2x)⋅x3f(x)=(x^{2}+2x) \cdot \sqrt[3]{x}f(x)=(x2+2x)⋅3x​.

Problem 5109

Gegeben ist die Funktion f(x)=1xcos⁡(x)f(x) = \frac{1}{x} \cos(x)f(x)=x1​cos(x). Bestimmen Sie Definitionsmenge, Nullstellen, Grenzwert und zeichnen Sie Graphen.

Problem 5110

Find the revenue function R(x)R(x)R(x) from p=10−0.002xp=10-0.002xp=10−0.002x, then calculate R′(x)R'(x)R′(x) and its values at x=8000x=8000x=8000 and x=9000x=9000x=9000.

Problem 5111

Berechne die verbleibende Menge an Cs 137 nach 2, 5, 10 und 33 Jahren, wenn 2,1\% jährlich zerfallen, aus 500 g.

Problem 5112

Find the critical numbers for the function f(x)=23x3−3x2−80x−10f(x)=\frac{2}{3} x^{3}-3 x^{2}-80 x-10f(x)=32​x3−3x2−80x−10.

Problem 5113

Is it true or false that if f′(x)>0f^{\prime}(x)>0f′(x)>0 on an interval, then fff is positive there? Explain your answer. Options: A, B, C, D.

Problem 5114

Is it true or false that derivatives alternate between positive and negative across regions separated by critical numbers?

Problem 5115

Find the critical numbers, increasing intervals, and decreasing intervals for f(x)=2.5+4.2x−1.1x2f(x)=2.5+4.2 x-1.1 x^{2}f(x)=2.5+4.2x−1.1x2.

Problem 5116

Find the marginal profit at x=6x=6x=6 for the profit function P(x)=−x2+12x−32P(x)=-x^{2}+12x-32P(x)=−x2+12x−32.

Problem 5117

Problem 5118

Problem 5119

Find the average rate of change of the function h(x)=−x2−4x+7h(x)=-x^{2}-4x+7h(x)=−x2−4x+7 on the interval −7≤x≤0-7 \leq x \leq 0−7≤x≤0.

Problem 5120

Find the derivative of f(x)=sin⁡(5x2−x)f(x)=\sin(5x^{2}-x)f(x)=sin(5x2−x) using the Chain Rule.

Problem 5121

Find the derivatives: a) For h(x)=f(3x)h(x)=f(3x)h(x)=f(3x), find h′(2)h'(2)h′(2). b) For y=f(x2+2)y=f(x^2+2)y=f(x2+2), find y′∣x=1y'|_{x=1}y′∣x=1​. c) For G(x)=cos⁡(f(x))G(x)=\cos(f(x))G(x)=cos(f(x)), find G′(π)G'(\pi)G′(π).

Problem 5122

Déterminez la convergence de la suite an=3n45n8+4n5a_{n}=\frac{3 n^{4}}{\sqrt{5 n^{8}+4 n^{5}}}an​=5n8+4n5​3n4​. Limite ou diverge ?

Problem 5123

Trouvez aaa pour que lim⁡x→2f(x)\lim _{x \rightarrow 2} f(x)limx→2​f(x) existe, avec f(x)={ex−a−9si x≥2x2+5si x<2f(x)=\begin{cases} e^{x-a}-9 & \text{si } x \geq 2 \\ x^{2}+5 & \text{si } x<2 \end{cases}f(x)={ex−a−9x2+5​si x≥2si x<2​.

Problem 5124

Find the derivative using the Power Rule: ddtt−3∣t=4=\left.\frac{d}{d t} t^{-3}\right|_{t=4}=dtd​t−3∣∣​t=4​=

Problem 5125

Find the units of S=2tS=2^tS=2t, the average rate of change in 5 months, and the growth rate at t=5t=5t=5 using an interval of 0.001.

Problem 5126

Find the slope mmm of the tangent line to y=f(x)y=f(x)y=f(x) at (−5,f(−5))(-5, f(-5))(−5,f(−5)) where f(x)=6x2−6f(x) = 6x^2 - 6f(x)=6x2−6.

Problem 5127

Find the derivative of h(t)=9t−4th(t)=9 \sqrt{t}-\frac{4}{\sqrt{t}}h(t)=9t​−t​4​. Express in exact form: h′(t)=h^{\prime}(t)=h′(t)=

Problem 5128

Find the derivative of P(V)=3VP(V)=\frac{3}{V}P(V)=V3​ at V=−5V=-5V=−5. What is dPdV∣V=−5=\left.\frac{d P}{d V}\right|_{V=-5}=dVdP​∣∣​V=−5​=?

Problem 5129

Find points on the graph of f(x)=27x−x3f(x)=27x-x^{3}f(x)=27x−x3 where the tangent line is horizontal. Provide as (x,y)(x, y)(x,y) pairs.

Problem 5130

Find f′(1)f^{\prime}(1)f′(1) for f(x)=5x2+2f(x)=5^{x^{2}+2}f(x)=5x2+2. Answer in the form "number/n natural log".

Problem 5131

Find all xxx where tangent lines to y=x4y=x^{4}y=x4 and y=x5y=x^{5}y=x5 have the same slope. x= x= x=

Problem 5132

Find xxx where the slope of the tangent to y=5x2−21x+7y=5 x^{2}-21 x+7y=5x2−21x+7 is steeper than that of y=13xy=\frac{1}{3} xy=31​x (interval form).

Problem 5133

Find the average rate of change of j(x)=3x3j(x)=3x^{3}j(x)=3x3 from x=1x=1x=1 to x=1+hx=1+hx=1+h.

Problem 5134

Find the derivative dydx\frac{dy}{dx}dxdy​ for x+y=100\sqrt{x}+\sqrt{y}=100x​+y​=100 using implicit differentiation.

Problem 5135

Find f′(x)f^{\prime}(x)f′(x) using the limit definition for f(x)=x2+23xf(x) = x^{2} + 23xf(x)=x2+23x.

Problem 5136

Find xxx where the slope of the tangent to y=5x2−21x+7y=5x^2-21x+7y=5x2−21x+7 is steeper than that of y=13x3y=\frac{1}{3}x^3y=31​x3. Answer as an interval (a,b)(a, b)(a,b).

Problem 5137

Find the rate of change of a circle's area with respect to radius rrr. What is it when r=6r=6r=6?

Problem 5138

Find f′(x)f^{\prime}(x)f′(x) using the limit definition for f(x)=(17x)−1/2+5f(x)=(17 x)^{-1/2}+5f(x)=(17x)−1/2+5.

Problem 5139

Find dKdt\frac{d K}{d t}dtdK​ for K=A5B34K=\frac{A^{5} B^{3}}{4}K=4A5B3​, with BBB varying and AAA constant.

Problem 5140

Find the surface area change rate of a sphere with respect to radius rrr. What is it when r=12r=12r=12?

Problem 5141

Find the derivative of $h(x)=(x^{2}+4)(2x^{2}-x+1)$ .

A potato is launched from a 50-ft building with an initial speed of $75 \, \text{ft/s}$ . Find its velocity when it hits the ground.

Differentiate $z=\frac{A}{y^{6}}+B e^{y}$ with respect to $y$ .

Find the derivative of $h(x)=(4x^{2}+3x)(5x^{2}-2)$ .

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

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

Gegeben ist die Funktion $f(x) = \frac{1}{x} \cos(x)$ . Bestimmen Sie Definitionsmenge, Nullstellen, Grenzwert und zeichnen Sie Graphen.

Find the revenue function $R(x)$ from $p=10-0.002x$ , then calculate $R'(x)$ and its values at $x=8000$ and $x=9000$ .

Find the critical numbers for the function $f(x)=\frac{2}{3} x^{3}-3 x^{2}-80 x-10$ .

Is it true or false that if $f^{\prime}(x)>0$ on an interval, then $f$ is positive there? Explain your answer. Options: A, B, C, D.

Find the critical numbers, increasing intervals, and decreasing intervals for $f(x)=2.5+4.2 x-1.1 x^{2}$ .

Find the marginal profit at $x=6$ for the profit function $P(x)=-x^{2}+12x-32$ .

Find the average rate of change of the function $h(x)=-x^{2}-4x+7$ on the interval $-7 \leq x \leq 0$ .

Find the derivative of $f(x)=\sin(5x^{2}-x)$ using the Chain Rule.

Find the derivatives: a) For $h(x)=f(3x)$ , find $h'(2)$ . b) For $y=f(x^2+2)$ , find $y'|_{x=1}$ . c) For $G(x)=\cos(f(x))$ , find $G'(\pi)$ .

Déterminez la convergence de la suite $a_{n}=\frac{3 n^{4}}{\sqrt{5 n^{8}+4 n^{5}}}$ . Limite ou diverge ?

Trouvez $a$ pour que $\lim _{x \rightarrow 2} f(x)$ existe, avec $f(x)=\begin{cases} e^{x-a}-9 & \text{si } x \geq 2 \\ x^{2}+5 & \text{si } x<2 \end{cases}$ .

Find the derivative using the Power Rule: $\left.\frac{d}{d t} t^{-3}\right|_{t=4}=$

Find the units of $S=2^t$ , the average rate of change in 5 months, and the growth rate at $t=5$ using an interval of 0.001.

Find the slope $m$ of the tangent line to $y=f(x)$ at $(-5, f(-5))$ where $f(x) = 6x^2 - 6$ .

Find the derivative of $h(t)=9 \sqrt{t}-\frac{4}{\sqrt{t}}$ . Express in exact form: $h^{\prime}(t)=$

Find the derivative of $P(V)=\frac{3}{V}$ at $V=-5$ . What is $\left.\frac{d P}{d V}\right|_{V=-5}=$ ?

Find points on the graph of $f(x)=27x-x^{3}$ where the tangent line is horizontal. Provide as $(x, y)$ pairs.

Find $f^{\prime}(1)$ for $f(x)=5^{x^{2}+2}$ . Answer in the form "number/n natural log".

Find all $x$ where tangent lines to $y=x^{4}$ and $y=x^{5}$ have the same slope.
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Find $x$ where the slope of the tangent to $y=5 x^{2}-21 x+7$ is steeper than that of $y=\frac{1}{3} x$ (interval form).

Find the average rate of change of $j(x)=3x^{3}$ from $x=1$ to $x=1+h$ .

Find the derivative $\frac{dy}{dx}$ for $\sqrt{x}+\sqrt{y}=100$ using implicit differentiation.

Find $f^{\prime}(x)$ using the limit definition for $f(x) = x^{2} + 23x$ .

Find $x$ where the slope of the tangent to $y=5x^2-21x+7$ is steeper than that of $y=\frac{1}{3}x^3$ . Answer as an interval $(a, b)$ .

Find the rate of change of a circle's area with respect to radius $r$ . What is it when $r=6$ ?

Find $f^{\prime}(x)$ using the limit definition for $f(x)=(17 x)^{-1/2}+5$ .

Find $\frac{d K}{d t}$ for $K=\frac{A^{5} B^{3}}{4}$ , with $B$ varying and $A$ constant.

Find the surface area change rate of a sphere with respect to radius $r$ . What is it when $r=12$ ?

Given $K=\frac{A^{2} B^{3}}{3}$ , find $\frac{d K}{d t}$ when $A$ and $B$ vary with respect to $t$ .

Find the values of $x$ ( $-8, -2, 0.3, 5$ ) where the limit of $f$ exists, given the limits at $-2$ and $3$ .

Find the rate of change of a circle's area with respect to radius $r$ when $r=6$ . What is the rate?

Find the horizontal asymptote of the function $f(x)=\frac{1-5 x-2 x^{2}}{3 x^{2}+7}$ for $x>0$ . Choices: A) $y=-\frac{2}{3}$ B) $y=\frac{1}{3}$ C) $y=\frac{2}{3}$ .

Find the derivative of $f(x)=\frac{4 a}{3 b x^{2}}$ in terms of $x$ , $a$ , and $b$ . $f^{\prime}(x)=$

Find the derivative using the Power Rule: $\left.\frac{d}{d t} t^{-3}\right|_{t=2}=$

Find the derivative using the Power Rule: $\frac{d}{d t} t^{-8/5}$ at $t=1$ . What is the result?

Find the horizontal asymptote of the function $f(x)=\frac{3^{x}+2}{e^{2 x}+1}$ for $x>0$ . Options: (A) $y=0$ , (B) $y=\frac{3}{e^{2}}$ , (C) $y=1$ , (D) none.

Find the horizontal asymptote of the function $f(x)=\frac{2-x+3 x^{2}+5 x^{3}-7 x^{4}}{x^{4}-2 x^{3}-5 x^{2}+2 x-3}$ for $x>0$ . Options: (A) $y=-7$ , (B) $y=2$ , (C) $y=7$ , (D) none.

Find $f^{\prime}(1)$ for $f(x)=4 \sqrt[3]{x}$ and the tangent line's equation at $x=1$ . Express as $y=f(x)$ .

Find the derivative of $h(t)=3 \sqrt{t}-\frac{4}{\sqrt{t}}$ . Use exact numbers and symbolic notation.

Find $f^{\prime}$ and the tangent line equation at $x=5$ for $f(x)=12 x^{-3}$ . Express as $y=f(x)$ .

Find the volume $V$ of the solid formed by rotating the area between $y=3\sqrt{16-x^2}$ , $y=0$ , $x=2$ , $x=3$ around the $x$ -axis.

Find all values of $x$ where the tangent lines to $y=x^{7}$ and $y=x^{8}$ have the same slope.

Find the derivative of the function $y=5 x^{2} e^{3 x}$ .

Find $\frac{d K}{d t}$ for $K=\frac{A^{5} B^{4}}{2}$ where $A$ varies with $t$ and $B$ is constant.

Find the derivative $\frac{dy}{dx}$ for the equation $x^{3}+y^{3}=5$ using implicit differentiation.

Plot $f(x)=\left|x-\cos \left(x-\frac{\pi}{2}\right)\right|$ using a graphing tool and analyze its behavior at $x=0$ . Is $f$ nondifferentiable or differentiable? Is the tangent line vertical, nonexistent, or horizontal?

Find the derivative of $f(x)=\frac{7 a}{3 b x^{2}}$ in terms of $x$ , $a$ , and $b$ . $f^{\prime}(x)=$

Find true statements about the function $f(x)=2x^{3}+x^{2}+6x+8$ from its derivative.

Find the marginal profit of the function $P(x)=x^{3}-4 x^{2}+10 x+7$ at $x=6$ .

Find: a) $\left.\frac{d}{d x}[-3 f(x)]\right|_{x=3}$ , b) $\left.\frac{d}{d x}[f(x)+g(x)]\right|_{x=6}$ , c) Tangent line equation for $y=f(x)-g(x)$ at $x=2$ .

Find the number of CDs, $x$ , that maximizes revenue given $R(x)=50x-\frac{x^2}{6}$ .

Find the fourth derivative of the function $f(x)=\cos (\sin (x))$ at $x=0$ , denoted as $f^{(4)}(0)$ .

Find the 4th derivative of $f(x)=\cos(\sin(x))$ at $x=0$ using the Taylor series.

Differentiate $y = \frac{3x}{\sin x + \cos x}$ .

Find $f^{\prime}(x)$ using logarithmic differentiation for $f(x)=\left(\frac{(3x^{3}-1)\sqrt{5x^{3}-4}}{2x+2}\right)5$ .

Find the derivative of the function $\frac{3 x+10}{x^{2}+3}$ .

Find the derivative of $f(x)=\frac{x^{9}(x-6)^{9}}{(x^{2}+7)^{7}}$ using logarithmic differentiation. $f^{\prime}(x)=$

Find the derivative of $y=(7+\sin (x))^{x}$ using Logarithmic Differentiation: $\frac{d y}{d x}=$

Find the displacement of a vehicle with velocity $v(t) = 24 + 5t - t^2$ after 5 seconds using overestimate $U_{5}$ and underestimate $L_{5}$ .

Find the global extrema of $f(x)=\cos(2x)-x$ for $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

Find $f^{\prime}(x)$ for $f(x)=(6 x-3)^{4} \cdot(7 x^{2}+3)^{4}$ using logarithmic differentiation.