Calculus

Problem 8101

Show that if $f(x)=\tan(x)$ , then $f'(x)=\sec^2(x)$ using the definition of the derivative directly.

See Solution

Problem 8102

Bestimmen Sie die Bedingungen für Wendepunkte und notieren Sie alternative hinreichende Bedingungen.

See Solution

Problem 8103

For $f(x)=(3 x-1)(5 x+3)$ , compare the derivative using the product rule vs. expanding and using the power rule.

See Solution

Problem 8104

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{6 x^{3}-1}{x}$ .

See Solution

Problem 8105

Find the derivative of $y=\sin x \cos x$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 8106

A rock dropped from a bridge takes 5 seconds to hit water. What is its velocity before impact? A) $5 \mathrm{~m/s}$ B) $2 \mathrm{~m/s}$ C) $50 \mathrm{~m/s}$ D) $125 \mathrm{~m/s}$

See Solution

Problem 8107

Find the derivative of $y=(2t^2-1)(3t+4)$ . Which of the following is correct? $y' = 18t^2 + 16t - 4$ $y' = 16t^2 + 18t - 3$ $y' = 8t^2 + 16t - 3$ $y' = 18t^2 + 16t - 3$ $y' = 15t^2 + 20t - 5$

See Solution

Problem 8108

A rock falls from rest. How far has it fallen when its speed is 39.2 m/s? (Neglect friction.) A) $19.6 \mathrm{~m}$ B) $44.1 \mathrm{~m}$ C) $78.3 \mathrm{~m}$ D) $123 \mathrm{~m}$

See Solution

Problem 8109

Prove the mean residual-life function: $\mathrm{e}(\mathrm{x})=\frac{\int_{x}^{\infty} S(t) d t}{S(x)}$ , with $S(t)$ as the survival function.

See Solution

Problem 8110

Find the rate of change of the area of a rectangle with length $6t + 5$ and height $\sqrt{t}$ over time $t$ .

See Solution

Problem 8111

Gegeben ist die Funktion $f(x)=(x-1)e^{-0,5x}$ . Finde die Tangente an der Nullstelle und prüfe ihre Eigenschaften.

See Solution

Problem 8112

Gegeben ist die Funktion $f(x)=(x-1) \cdot e^{-0,5 x}$ . Bestimme die Tangentengleichung $\mathrm{g}$ an der Nullstelle $\mathrm{x}_{0}$ .

See Solution

Problem 8113

Find the derivative of $y=-7 \sin (\sin (\sin x))$ .

See Solution

Problem 8114

Differentiate the function $f(x)=\ln(e^{7x}+6)$ . Find $f^{\prime}(x)$ .

See Solution

Problem 8115

Untersuchen Sie das Verhalten der Funktion $\frac{1}{x}$ für $x \to +\infty$ und $x \to -\infty$ .

See Solution

Problem 8116

Untersuchen Sie das Verhalten von $2^{x-2}$ für $x \to \infty$ .

See Solution

Problem 8117

Differentiate the function $g(x)=\ln(x e^{-8x})$ . Find $g^{\prime}(x)$ .

See Solution

Problem 8118

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{-11 x}{\sin (x)+\cos (x)}$ at $x=-\pi$ .

See Solution

Problem 8119

Geben Sie Gegenbeispiele für die falschen Aussagen a), b) und c) über die Monotonie und Krümmung von $f$ .

See Solution

Problem 8120

Untersuche das Verhalten der Funktionen für $x \rightarrow-\infty$ und $x \rightarrow+\infty$ : a) $f(x)=-2 x^{5}-4 x^{2}+3$ b) $f(x)=3 x^{4}+2 x^{2}+x+10$ c) $f(x)=\frac{1}{x^{2}}$ d) $f(x)=3^{x-3}$

See Solution

Problem 8121

Find the missing expression to complete: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{8}+7}=e^{\mathrm{x}^{8}+7} ?$

See Solution

Problem 8122

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{x}{e^{x^{2}}}$ .

See Solution

Problem 8123

Find the derivative $y'$ for $y=(8+7x-3x^2)e^x$ and simplify.

See Solution

Problem 8124

Berechnen Sie die dritte Ableitung von $k(x)=250 x \cdot e^{-0.5 x}+20$ .

See Solution

Problem 8125

Find the rectangle with the largest area under the parabola $y=8-x^{2}$ , with its base on the $x$ -axis.

See Solution

Problem 8126

Find the tangent line equation for $f(x)=19 e^{x}+8 x$ at $x=0$ . What is $y=$ ?

See Solution

Problem 8127

Nach einem Brand steigt die PFT-Konzentration im See. Berechne mit $k(x)=250 x \cdot e^{-0,5 x}+20$ verschiedene Werte.

See Solution

Problem 8128

Use the Intermediate Value Theorem to show there's a root for $x^{4}+x-4=0$ in $(1,2)$ . Calculate $f(1)$ and $f(2)$ .

See Solution

Problem 8129

Find derivatives from the given data:
a) If $h(x)=f(3x)$ , then $h'(2)=15$ . b) If $y=f(x^2+2)$ , find $y'|_{x=1}$ .

See Solution

Problem 8130

Determine the end behavior of the function $f(x) = -162 + 18x^{2} - 6x^{3} + 54x$ .

See Solution

Problem 8131

Find the derivative $f^{\prime}(a)$ for the function $f(t)=\frac{2 t+4}{t+3}$ .

See Solution

Problem 8132

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

See Solution

Problem 8133

Find the average rate of change of $f(x)=2x^{2}+5$ between $x=-2$ and $x=0$ . Simplify your answer.

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Problem 8134

Calculate the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

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Problem 8135

Find the rate of change of concentration $C(t)=4.35 e^{-t}$ after 2 and 4 hours, and graph $C$ .

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Problem 8136

Find the $x$ -value where $f$ is discontinuous and its continuity type: right, left, or neither.
$f(x)=\left\{\begin{array}{ll} 2+x^{2} & \text { if } x \leq 0 \\ 8-x & \text { if } 0<x \leq 8 \\ (x-8)^{2} & \text { if } x>8 \end{array}\right.$

See Solution

Problem 8137

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}$ using l'Hospital's Rule or another method.

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Problem 8138

Find the limit using I'Hospital's Rule if needed: $\lim _{x \rightarrow 0} \frac{x 4^{x}}{4^{x}-1}$

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Problem 8139

Find the limit: $\lim _{t \rightarrow 0} \frac{e^{9 t}-1}{\sin (t)}$ using I'Hospital's Rule or a simpler method.

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Problem 8140

Find the tangent line equation for the curve $y=\frac{9 x}{(x+1)^{2}}$ at the point $(0,0)$ .

See Solution

Problem 8141

Find the limit using l'Hospital's Rule or another method: $\lim _{x \rightarrow 4} \frac{\ln \left(\frac{x}{4}\right)}{4-x}$

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Problem 8142

Find the limit using l'Hospital's Rule if needed: $\lim _{x \rightarrow 0^{+}}\left(\frac{7}{x}-\frac{7}{\tan (x)}\right)$

See Solution

Problem 8143

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

See Solution

Problem 8144

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence using Taylor series near $x=0$ .

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Problem 8145

A diver jumps from a $3 \mathrm{~m}$ board at $5.4 \mathrm{~m/s}$ . Sketch position, velocity, and acceleration vs. time graphs.

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Problem 8146

Analyze the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} dx$ for convergence or divergence near $x=0$ .

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Problem 8147

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \rightarrow+\infty$ to check integral convergence: $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ .

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Problem 8148

Solve the differential equation by separation of variables: $e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x-y}$ .

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Problem 8149

Solve the differential equation $\frac{d y}{d x} - (\sin x) y = 2 \sin x$ .

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Problem 8150

Find the derivative of $f(x)=3 x^{3}-4 x$ using the limit definition: $f^{\prime}(x) =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 8151

Calculate the integral $\int_{1}^{3}\left(\frac{1}{x^{2}}-1\right) d x$ .

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Problem 8152

Berechne mit $\mathrm{J}_{0}(\mathrm{x})=\frac{1}{3} \mathrm{x}^{3}$ die Integrale: a) $\int_{0}^{6} \mathrm{t}^{2} \mathrm{dt}$ , b) $\int_{0}^{1,5} x^{2} dx$ , c) $\int_{2}^{3} x^{2} dx$ , d) $\int_{4}^{7} z^{2} dz$ , e) $\int_{8}^{8} x^{2} dx$ .

See Solution

Problem 8153

Berechnen Sie die Ableitung von $f$ an $x_{0}$ mit dem Differenzenquotienten für $h$ nahe 0 für die folgenden Funktionen: a) $f(x)=x^{3}$ , $x_{0}=3$ ; b) $f(x)=x^{4}-2x^{3}$ , $x_{0}=1,5$ ; c) $f(x)=2^{x}$ , $x_{0}=-1$ ; d) $f(x)=0,7x-1$ , $x_{0}=0$ ; e) $f(x)=\frac{1}{x}$ , $x_{0}=2$ ; f) $f(x)=\sqrt{x}$ , $x_{0}=4$ .

See Solution

Problem 8154

Find the value of $a$ (not an integer) such that $\int_{a}^{a^{2}} (a x^{2}-7 a^{2} x-\frac{7 a^{3}}{3}) \mathrm{d} x=0$ .

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Problem 8155

Vervollständigen Sie die Ableitung von $f(x)$ in den folgenden Aufgaben.
a) $f(x)=x^{3}(4x+3)$ , $f'(x)=\square(4x+3)+x^{3}\cdot$ b) $f(x)=2x e^{x}$ , $f'(x)=\square e^{x}+2x$ c) $f(x)=(x^{2}+1)\sqrt{x}$ , $f'(x)=\square\sqrt{x}+(x^{2}+1)$ d) $f(x)=\frac{1}{x}e^{x}$ , $f'(x)=\cdot e^{x}+\frac{1}{x}\cdot$ e) $f(x)=\frac{1}{x^{2}}(x^{2}-2x)$ , $f'(x)=\square(x^{2}-2x)+\frac{1}{x^{2}}$ f) $f(x)=(2x^{3}-4x^{2})x^{-1}$ , $f'(x)=\square x^{-1}+(2x^{3}-4x^{2})$

See Solution

Problem 8156

Berechne den Flächeninhalt unter $f(x) = (x+1) \cdot \cos(x)$ und über der $x$ -Achse im Intervall $[-2, 5]$ .

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Problem 8157

Berechne die Ableitung von $f(x)=\frac{2}{\sqrt{x}}+\frac{\sqrt{x}}{2}$ .

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Problem 8158

Find $f^{\prime}\left(\frac{-1}{2}\right), f^{\prime}(-1), f^{\prime}(-2)$ , and $f^{\prime}(-3)$ for $f(x)= x^3$ . Round to one decimal place.

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Problem 8159

A rare insect's population after transfer is $P(t)=\frac{50+25 t}{2+0.01 t}$ . Find the $y$ -intercept and $\lim_{t \rightarrow \infty} P(t)$ .

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Problem 8160

Find the first and second derivatives $y'$ and $y''$ of the function $y=(4+\sqrt{x})^{3}$ .

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Problem 8161

Find $y=(4+\sqrt{x})^{3}$ and its derivatives $y' = 34 + \frac{1}{x} 2 \frac{1}{2} x^{-\frac{1}{2}}$ , $y''=1$ .

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Problem 8162

Rex throws a vase up at $26.2 \mathrm{~m/s}$ . Find the maximum height reached when $V_{f}=0 \mathrm{~m/s}$ .

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Problem 8163

Find the derivative of $f(x)=4+8x-6x^{2}$ using the definition. Also, state the domains of $f(x)$ and $f'(x)$ in interval notation.

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Problem 8164

Find the derivative of $g(x)=\left(2 x^{2}+7\right)^{18}\left(3 x^{4}+4\right)^{14}$ . What is $g^{\prime}(x)$ ?

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Problem 8165

Find the derivative of $g(x)=\frac{1}{9+\sqrt{x}}$ and state the domains of $g(x)$ and $g'(x)$ in interval notation.

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Problem 8166

Gegeben ist $f(t)=\frac{1}{2} t^{3}-2 t$ . a) Skizzieren Sie $f$ . b) Finden Sie $F$ mit $F(4)=0$ und zeigen Sie $F(-4)=0$ . c) Bestimmen Sie die Konstante zwischen $F$ und $F_{0}(x)=\int_{0}^{x}(\frac{1}{2} t^{3}-2 t) \mathrm{d} x$ . d) Bestimmen Sie die Extrempunkte von $F_{0}$ .

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Problem 8167

Zwei Funktionen $F(x)=\sqrt{x+1}-2$ und $G(x)=\frac{x}{1+\sqrt{x+1}}$ sind gegeben. Zeigen Sie, dass sie Stammfunktionen einer Funktion $f$ sind und geben Sie sie als $\int_{a}^{x} f(t) \mathrm{d} t$ an. Was unterscheidet die Integrale geometrisch?

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Problem 8168

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

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Problem 8169

Find the derivative of the function $f(x)=7 \sin (\cos x)$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 8170

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

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Problem 8171

Differentiate the function $f(x)=\ln(4x^{2}+8x+1)$ without simplifying the derivative. Find $f^{\prime}(x)=$ .

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Problem 8172

Find the limit: $\lim _{x \rightarrow-2} \frac{x(x-1)(x+2)^{2}}{(x+1)(x-1)(x+2)^{3}}$ .

See Solution

Problem 8173

Differentiate the function $G(x)=x^{4} \ln (\operatorname{abs}(7 x))$ . Find $G^{\prime}(x)=$ .

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Problem 8174

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

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Problem 8175

Find the derivative of $y=\arcsin(3x)$ : $\frac{dy}{dx}=\square$

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Problem 8176

Find $f^{\prime}(x)$ for $f(x)=5 x \arcsin(x)$ and calculate $f^{\prime}(0.5)$ , rounding to the nearest hundredth.

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Problem 8177

Differentiate $f(x)=\arccos(4x)$ without approximation. Find $f'(x)=$ .

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Problem 8178

Find the derivative of $\ln(2x + 4)$ .

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Problem 8179

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

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Problem 8180

Find the derivative $\frac{d y}{d x}$ for $y=\frac{\ln (5 x+8)}{9 e^{4 x}}$ .

See Solution

Problem 8181

Find the derivative of the function $f(x)=\ln(x^{2}-4x+8)$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 8182

Find the derivative of $2 \ln (x)$ with respect to $x$ .

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Problem 8183

Find the derivative of $g(t)=\ln(12 t^{3})$ . What is $g^{\prime}(t)=?$

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Problem 8184

Find $\frac{d A}{d R}$ for the area function $A(R) = \pi R^{2}$ .

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Problem 8185

Find $f'(x)$ for $f(x)=4x \arcsin(x)$ and calculate $f'(0.5)$ , rounding to the nearest hundredth.

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Problem 8186

Find $\Delta G^{\circ}$ at $75^{\circ}C$ for $A(aq) + B(aq) \leftrightarrow C(aq) + D(aq)$ with $K=24.0$ at $25^{\circ}C$ and $K=37.0$ at $50^{\circ}C$ . Use $\Delta G^{\circ} = -RT \ln(K)$ and the van't Hoff equation.

See Solution

Problem 8187

Differentiate the function $f(x)=x^{3} \ln (a b s(15 x))$ without simplifying the derivative. $f^{\prime}(x)=$

See Solution

Problem 8188

Differentiate the function $f(x)=x^{3} \ln (\operatorname{abs}(15 x))$ . Find $f^{\prime}(x)$ .

See Solution

Problem 8189

Find the derivative $f'(x)$ for $f(x)=4x \arcsin(x)$ and compute $f'(0.7)$ rounded to the nearest hundredth.

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Problem 8190

Find the velocity of an object at $t=0$ and $t=5$ given $s(t)=t^{3}-t+1$ .

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Problem 8191

Find the derivative of $f(t)=\sqrt[3]{t}$ at $a \neq 0$ : $f^{\prime}(a)=$

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Problem 8192

Find the first four derivatives of the function $f(x)=2 x^{2}-x^{3}$ .

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Problem 8193

Find the average rate of change of $f(x)=x^{2}$ for these intervals: (a) $[0,2]$ , (b) $[2,4]$ , (c) $[4,6]$ . Explain the graph's trend.

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Problem 8194

Show that $f(x)=|x-4|$ is not differentiable at 4 by evaluating the right and left limits as $x$ approaches 4.

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Problem 8195

Find the formula for $f^{\prime}$ of $f(x)= |x-4|$ and sketch its graph, noting it's not differentiable at $x=4$ .

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Problem 8196

Show that $f(x)=|x-4|$ is not differentiable at 4 by comparing the limits from both sides. Find $f'$ .

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Problem 8197

An object is launched upwards with an initial speed of $39.2 \mathrm{~m/s}$ . Given $s(t)=-4.9t^2+39.2t$ , find:
(a) velocity at time $t$ (b) time to reach max height (c) max height (d) acceleration at time $t$ (e) total time in air (f) impact velocity and speed (g) total distance traveled

See Solution

Problem 8198

Find the derivative of the function defined by $x y^{2}+x^{2} y^{5}-3 x^{3}=6$ with respect to $x$ .

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Problem 8199

Find the average rate of change of $f(x)=-2 x^{2}-7 x+1$ from $x=1$ to $x=4$ . Calculate: Average rate of change $=$

See Solution

Problem 8200

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

See Solution

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Calculus

Problem 8101

Show that if f(x)=tan⁡(x)f(x)=\tan(x)f(x)=tan(x), then f′(x)=sec⁡2(x)f'(x)=\sec^2(x)f′(x)=sec2(x) using the definition of the derivative directly.

Problem 8102

Bestimmen Sie die Bedingungen für Wendepunkte und notieren Sie alternative hinreichende Bedingungen.

Problem 8103

For f(x)=(3x−1)(5x+3)f(x)=(3 x-1)(5 x+3)f(x)=(3x−1)(5x+3), compare the derivative using the product rule vs. expanding and using the power rule.

Problem 8104

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=6x3−1xy=\frac{6 x^{3}-1}{x}y=x6x3−1​.

Problem 8105

Find the derivative of y=sin⁡xcos⁡xy=\sin x \cos xy=sinxcosx. What is dydx\frac{d y}{d x}dxdy​?

Problem 8106

A rock dropped from a bridge takes 5 seconds to hit water. What is its velocity before impact? A) 5 m/s5 \mathrm{~m/s}5 m/s B) 2 m/s2 \mathrm{~m/s}2 m/s C) 50 m/s50 \mathrm{~m/s}50 m/s D) 125 m/s125 \mathrm{~m/s}125 m/s

Problem 8107

Problem 8108

A rock falls from rest. How far has it fallen when its speed is 39.2 m/s? (Neglect friction.) A) 19.6 m19.6 \mathrm{~m}19.6 m B) 44.1 m44.1 \mathrm{~m}44.1 m C) 78.3 m78.3 \mathrm{~m}78.3 m D) 123 m123 \mathrm{~m}123 m

Problem 8109

Prove the mean residual-life function: e(x)=∫x∞S(t)dtS(x)\mathrm{e}(\mathrm{x})=\frac{\int_{x}^{\infty} S(t) d t}{S(x)}e(x)=S(x)∫x∞​S(t)dt​, with S(t)S(t)S(t) as the survival function.

Problem 8110

Find the rate of change of the area of a rectangle with length 6t+56t + 56t+5 and height t\sqrt{t}t​ over time ttt.

Problem 8111

Gegeben ist die Funktion f(x)=(x−1)e−0,5xf(x)=(x-1)e^{-0,5x}f(x)=(x−1)e−0,5x. Finde die Tangente an der Nullstelle und prüfe ihre Eigenschaften.

Problem 8112

Gegeben ist die Funktion f(x)=(x−1)⋅e−0,5xf(x)=(x-1) \cdot e^{-0,5 x}f(x)=(x−1)⋅e−0,5x. Bestimme die Tangentengleichung g\mathrm{g}g an der Nullstelle x0\mathrm{x}_{0}x0​.

Problem 8113

Find the derivative of y=−7sin⁡(sin⁡(sin⁡x))y=-7 \sin (\sin (\sin x))y=−7sin(sin(sinx)).

Problem 8114

Differentiate the function f(x)=ln⁡(e7x+6)f(x)=\ln(e^{7x}+6)f(x)=ln(e7x+6). Find f′(x)f^{\prime}(x)f′(x).

Problem 8115

Untersuchen Sie das Verhalten der Funktion 1x\frac{1}{x}x1​ für x→+∞x \to +\inftyx→+∞ und x→−∞x \to -\inftyx→−∞.

Problem 8116

Untersuchen Sie das Verhalten von 2x−22^{x-2}2x−2 für x→∞x \to \inftyx→∞.

Problem 8117

Differentiate the function g(x)=ln⁡(xe−8x)g(x)=\ln(x e^{-8x})g(x)=ln(xe−8x). Find g′(x)g^{\prime}(x)g′(x).

Problem 8118

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=−11xsin⁡(x)+cos⁡(x)f(x)=\frac{-11 x}{\sin (x)+\cos (x)}f(x)=sin(x)+cos(x)−11x​ at x=−πx=-\pix=−π.

Problem 8119

Geben Sie Gegenbeispiele für die falschen Aussagen a), b) und c) über die Monotonie und Krümmung von fff.

Problem 8120

Problem 8121

Find the missing expression to complete: ddxex8+7=ex8+7?\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{8}+7}=e^{\mathrm{x}^{8}+7} ?dxd​ex8+7=ex8+7?

Problem 8122

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=xex2f(x)=\frac{x}{e^{x^{2}}}f(x)=ex2x​.

Problem 8123

Find the derivative y′y'y′ for y=(8+7x−3x2)exy=(8+7x-3x^2)e^xy=(8+7x−3x2)ex and simplify.

Problem 8124

Berechnen Sie die dritte Ableitung von k(x)=250x⋅e−0.5x+20k(x)=250 x \cdot e^{-0.5 x}+20k(x)=250x⋅e−0.5x+20.

Problem 8125

Find the rectangle with the largest area under the parabola y=8−x2y=8-x^{2}y=8−x2, with its base on the xxx-axis.

Problem 8126

Find the tangent line equation for f(x)=19ex+8xf(x)=19 e^{x}+8 xf(x)=19ex+8x at x=0x=0x=0. What is y=y=y=?

Problem 8127

Nach einem Brand steigt die PFT-Konzentration im See. Berechne mit k(x)=250x⋅e−0,5x+20k(x)=250 x \cdot e^{-0,5 x}+20k(x)=250x⋅e−0,5x+20 verschiedene Werte.

Problem 8128

Use the Intermediate Value Theorem to show there's a root for x4+x−4=0x^{4}+x-4=0x4+x−4=0 in (1,2)(1,2)(1,2). Calculate f(1)f(1)f(1) and f(2)f(2)f(2).

Problem 8129

Find derivatives from the given data: a) If h(x)=f(3x)h(x)=f(3x)h(x)=f(3x), then h′(2)=15h'(2)=15h′(2)=15. b) If y=f(x2+2)y=f(x^2+2)y=f(x2+2), find y′∣x=1y'|_{x=1}y′∣x=1​.

Problem 8130

Determine the end behavior of the function f(x)=−162+18x2−6x3+54xf(x) = -162 + 18x^{2} - 6x^{3} + 54xf(x)=−162+18x2−6x3+54x.

Problem 8131

Find the derivative f′(a)f^{\prime}(a)f′(a) for the function f(t)=2t+4t+3f(t)=\frac{2 t+4}{t+3}f(t)=t+32t+4​.

Problem 8132

Find the derivative of the function f(x)=ex2f(x)=e^{x^{2}}f(x)=ex2.

Problem 8133

Find the average rate of change of f(x)=2x2+5f(x)=2x^{2}+5f(x)=2x2+5 between x=−2x=-2x=−2 and x=0x=0x=0. Simplify your answer.

Problem 8134

Calculate the limit: lim⁡t→∞t+t24t−t2\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}t→∞lim​4t−t2t​+t2​. Enter ∞\infty∞, −∞-\infty−∞, or DNE if it doesn't exist.

Problem 8135

Find the rate of change of concentration C(t)=4.35e−tC(t)=4.35 e^{-t}C(t)=4.35e−t after 2 and 4 hours, and graph CCC.

Problem 8136

Problem 8137

Find the limit: lim⁡x→0tan⁡(5x)sin⁡(8x)\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}x→0lim​sin(8x)tan(5x)​ using l'Hospital's Rule or another method.

Problem 8138

Find the limit using I'Hospital's Rule if needed: lim⁡x→0x4x4x−1\lim _{x \rightarrow 0} \frac{x 4^{x}}{4^{x}-1}x→0lim​4x−1x4x​

Problem 8139

Find the limit: lim⁡t→0e9t−1sin⁡(t)\lim _{t \rightarrow 0} \frac{e^{9 t}-1}{\sin (t)}t→0lim​sin(t)e9t−1​ using I'Hospital's Rule or a simpler method.

Problem 8140

Find the tangent line equation for the curve y=9x(x+1)2y=\frac{9 x}{(x+1)^{2}}y=(x+1)29x​ at the point (0,0)(0,0)(0,0).

Problem 8141

Find the limit using l'Hospital's Rule or another method: lim⁡x→4ln⁡(x4)4−x\lim _{x \rightarrow 4} \frac{\ln \left(\frac{x}{4}\right)}{4-x}x→4lim​4−xln(4x​)​

Show that if $f(x)=\tan(x)$ , then $f'(x)=\sec^2(x)$ using the definition of the derivative directly.

For $f(x)=(3 x-1)(5 x+3)$ , compare the derivative using the product rule vs. expanding and using the power rule.

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{6 x^{3}-1}{x}$ .

Find the derivative of $y=\sin x \cos x$ . What is $\frac{d y}{d x}$ ?

A rock dropped from a bridge takes 5 seconds to hit water. What is its velocity before impact? A) $5 \mathrm{~m/s}$ B) $2 \mathrm{~m/s}$ C) $50 \mathrm{~m/s}$ D) $125 \mathrm{~m/s}$

A rock falls from rest. How far has it fallen when its speed is 39.2 m/s? (Neglect friction.) A) $19.6 \mathrm{~m}$ B) $44.1 \mathrm{~m}$ C) $78.3 \mathrm{~m}$ D) $123 \mathrm{~m}$

Prove the mean residual-life function: $\mathrm{e}(\mathrm{x})=\frac{\int_{x}^{\infty} S(t) d t}{S(x)}$ , with $S(t)$ as the survival function.

Find the rate of change of the area of a rectangle with length $6t + 5$ and height $\sqrt{t}$ over time $t$ .

Gegeben ist die Funktion $f(x)=(x-1)e^{-0,5x}$ . Finde die Tangente an der Nullstelle und prüfe ihre Eigenschaften.

Gegeben ist die Funktion $f(x)=(x-1) \cdot e^{-0,5 x}$ . Bestimme die Tangentengleichung $\mathrm{g}$ an der Nullstelle $\mathrm{x}_{0}$ .

Find the derivative of $y=-7 \sin (\sin (\sin x))$ .

Differentiate the function $f(x)=\ln(e^{7x}+6)$ . Find $f^{\prime}(x)$ .

Untersuchen Sie das Verhalten der Funktion $\frac{1}{x}$ für $x \to +\infty$ und $x \to -\infty$ .

Untersuchen Sie das Verhalten von $2^{x-2}$ für $x \to \infty$ .

Differentiate the function $g(x)=\ln(x e^{-8x})$ . Find $g^{\prime}(x)$ .

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{-11 x}{\sin (x)+\cos (x)}$ at $x=-\pi$ .

Geben Sie Gegenbeispiele für die falschen Aussagen a), b) und c) über die Monotonie und Krümmung von $f$ .

Find the missing expression to complete: $\frac{\mathrm{d}}{\mathrm{dx}} e^{\mathrm{x}^{8}+7}=e^{\mathrm{x}^{8}+7} ?$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=\frac{x}{e^{x^{2}}}$ .

Find the derivative $y'$ for $y=(8+7x-3x^2)e^x$ and simplify.

Berechnen Sie die dritte Ableitung von $k(x)=250 x \cdot e^{-0.5 x}+20$ .

Find the rectangle with the largest area under the parabola $y=8-x^{2}$ , with its base on the $x$ -axis.

Find the tangent line equation for $f(x)=19 e^{x}+8 x$ at $x=0$ . What is $y=$ ?

Nach einem Brand steigt die PFT-Konzentration im See. Berechne mit $k(x)=250 x \cdot e^{-0,5 x}+20$ verschiedene Werte.

Use the Intermediate Value Theorem to show there's a root for $x^{4}+x-4=0$ in $(1,2)$ . Calculate $f(1)$ and $f(2)$ .

Find derivatives from the given data:
a) If $h(x)=f(3x)$ , then $h'(2)=15$ . b) If $y=f(x^2+2)$ , find $y'|_{x=1}$ .

Determine the end behavior of the function $f(x) = -162 + 18x^{2} - 6x^{3} + 54x$ .

Find the derivative $f^{\prime}(a)$ for the function $f(t)=\frac{2 t+4}{t+3}$ .

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

Find the average rate of change of $f(x)=2x^{2}+5$ between $x=-2$ and $x=0$ . Simplify your answer.

Calculate the limit: $\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{4 t-t^{2}}$ . Enter $\infty$ , $-\infty$ , or DNE if it doesn't exist.

Find the rate of change of concentration $C(t)=4.35 e^{-t}$ after 2 and 4 hours, and graph $C$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\tan (5 x)}{\sin (8 x)}$ using l'Hospital's Rule or another method.

Find the limit using I'Hospital's Rule if needed: $\lim _{x \rightarrow 0} \frac{x 4^{x}}{4^{x}-1}$

Find the limit: $\lim _{t \rightarrow 0} \frac{e^{9 t}-1}{\sin (t)}$ using I'Hospital's Rule or a simpler method.

Find the tangent line equation for the curve $y=\frac{9 x}{(x+1)^{2}}$ at the point $(0,0)$ .

Find the limit using l'Hospital's Rule or another method: $\lim _{x \rightarrow 4} \frac{\ln \left(\frac{x}{4}\right)}{4-x}$

Find the limit using l'Hospital's Rule if needed: $\lim _{x \rightarrow 0^{+}}\left(\frac{7}{x}-\frac{7}{\tan (x)}\right)$

Evaluate the integral $\int_{0}^{1} \frac{e^{-x}}{x} dx$ for convergence using Taylor series near $x=0$ .

A diver jumps from a $3 \mathrm{~m}$ board at $5.4 \mathrm{~m/s}$ . Sketch position, velocity, and acceleration vs. time graphs.

Analyze the integral $\int_{0}^{1} \frac{\cos ^{2} x}{\sqrt{x}} dx$ for convergence or divergence near $x=0$ .

Find the leading term of $\frac{\sqrt[3]{x+3}}{x^{3}}$ as $x \rightarrow+\infty$ to check integral convergence: $\int_{1}^{+\infty} \frac{\sqrt[3]{x+3}}{x^{3}} dx$ .

Solve the differential equation by separation of variables: $e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x-y}$ .

Solve the differential equation $\frac{d y}{d x} - (\sin x) y = 2 \sin x$ .

Find the derivative of $f(x)=3 x^{3}-4 x$ using the limit definition: $f^{\prime}(x) =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ .

Calculate the integral $\int_{1}^{3}\left(\frac{1}{x^{2}}-1\right) d x$ .

Find the value of $a$ (not an integer) such that $\int_{a}^{a^{2}} (a x^{2}-7 a^{2} x-\frac{7 a^{3}}{3}) \mathrm{d} x=0$ .

Berechne den Flächeninhalt unter $f(x) = (x+1) \cdot \cos(x)$ und über der $x$ -Achse im Intervall $[-2, 5]$ .

Berechne die Ableitung von $f(x)=\frac{2}{\sqrt{x}}+\frac{\sqrt{x}}{2}$ .

Find $f^{\prime}\left(\frac{-1}{2}\right), f^{\prime}(-1), f^{\prime}(-2)$ , and $f^{\prime}(-3)$ for $f(x)= x^3$ . Round to one decimal place.

A rare insect's population after transfer is $P(t)=\frac{50+25 t}{2+0.01 t}$ . Find the $y$ -intercept and $\lim_{t \rightarrow \infty} P(t)$ .

Find the first and second derivatives $y'$ and $y''$ of the function $y=(4+\sqrt{x})^{3}$ .

Find $y=(4+\sqrt{x})^{3}$ and its derivatives $y' = 34 + \frac{1}{x} 2 \frac{1}{2} x^{-\frac{1}{2}}$ , $y''=1$ .

Rex throws a vase up at $26.2 \mathrm{~m/s}$ . Find the maximum height reached when $V_{f}=0 \mathrm{~m/s}$ .

Find the derivative of $f(x)=4+8x-6x^{2}$ using the definition. Also, state the domains of $f(x)$ and $f'(x)$ in interval notation.

Find the derivative of $g(x)=\left(2 x^{2}+7\right)^{18}\left(3 x^{4}+4\right)^{14}$ . What is $g^{\prime}(x)$ ?

Find the derivative of $g(x)=\frac{1}{9+\sqrt{x}}$ and state the domains of $g(x)$ and $g'(x)$ in interval notation.

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

Find the derivative of the function $f(x)=7 \sin (\cos x)$ , denoted as $f^{\prime}(x)$ .

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

Differentiate the function $f(x)=\ln(4x^{2}+8x+1)$ without simplifying the derivative. Find $f^{\prime}(x)=$ .

Find the limit: $\lim _{x \rightarrow-2} \frac{x(x-1)(x+2)^{2}}{(x+1)(x-1)(x+2)^{3}}$ .

Differentiate the function $G(x)=x^{4} \ln (\operatorname{abs}(7 x))$ . Find $G^{\prime}(x)=$ .

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