Calculus

Problem 13101

Determine where the function $f(x)=-\frac{x^{4}}{4}+x^{3}-x^{2}$ is increasing or decreasing.

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Problem 13102

Find and simplify the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x^{2} \tan x}{\sec x}$ .

See Solution

Problem 13103

Find the radius expansion rate of a circular area wetting at $10 \mathrm{~mm}^{2} / \mathrm{s}$ when radius is $112 \mathrm{~mm}$ .

See Solution

Problem 13104

Find $f^{\prime}(x)$ for $f(x)=(4 x+3)^{-1}$ and determine $f^{\prime}(4) = \square \sigma^{6}$ .

See Solution

Problem 13105

Find the rate of area increase of a metal plate when its radius is $46 \mathrm{~cm}$ and increases at $0.05 \mathrm{~cm/min}$ .

See Solution

Problem 13106

Find the degree 2 Taylor polynomial of $f(x)=6x \ln x$ centered at $x=2$ .

See Solution

Problem 13107

Find the concavity intervals and inflection points for the function $g(t)=3 t^{5}+15 t^{4}+20 t^{3}+70$ .

See Solution

Problem 13108

Find the linearization $L(x)$ at $x=1$ for the function $f(x)=3x^3-2x-1$ .

See Solution

Problem 13109

Find the area represented by the integral $\int_{-6}^{0}\left(2+\sqrt{36-x^{2}}\right) d x$ .

See Solution

Problem 13110

Find the degree 3 Taylor polynomial $T_{3}$ at $x=0$ for $f(x)=\frac{1}{(4 x+1)^{1/2}}$ .

See Solution

Problem 13111

Find the Taylor series at $x=0$ for $f(x)=\cos(5x)$ .

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Problem 13112

Evaluate these antiderivatives: (a) $\int\left(8 x^{3}-5 x^{1 / 3}-\frac{1}{x^{2 / 3}}+e^{2 x}\right) d x$ (b) $\int(x-3)^{1 / 4} x d x$ (c) $\int \cos \left(x^{2}+1\right) x d x$ (d) $\int \frac{\sin \sqrt{t}}{\sqrt{t}} d t$ . Show all steps for full credit.

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Problem 13113

Prove the inequality $|\sin a - \sin b| \leq |a - b|$ using the Mean Value Theorem for all $a$ and $b$ .

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Problem 13114

Prove $|\sin a - \sin b| \leq |a - b|$ using the Mean Value Theorem for any values of $a$ and $b$ .

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Problem 13115

Find the derivative of $g(t) = \ln(12 t^3)$ .

See Solution

Problem 13116

Arjun walks south at $1.5 \mathrm{~m/s}$ and Maya east at $2 \mathrm{~m/s}$ . Find the distance change rate after 2 mins.

See Solution

Problem 13117

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{4}+y^{5}=-7$ .

See Solution

Problem 13118

Find the derivative of $f(x)=\sqrt[3]{e^{x}+9 x^{2}}$ .

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Problem 13119

Find the limit: $\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}$ .

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Problem 13120

Find the slope of the tangent line to the curve $r=\sin \theta+2 \cos \theta$ at $\theta=\frac{\pi}{2}$ .

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Problem 13121

A tennis ball's motion is given by $x=32.0 t$ and $y=42.0 t-4.90 t^{2}$ . Find velocity and acceleration at $t=6.00 \mathrm{~s}$ .

See Solution

Problem 13122

Find the limit: $\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}$

See Solution

Problem 13123

Gravel is dumped at $24 \mathrm{~m}^{3} / \mathrm{h}$ forming a cone. Find height increase rate when height is $5 \mathrm{~m}$ .

See Solution

Problem 13124

A 1.7 m tall person walks away from a 5.95 m pole at 1 m/s. Find the speed of their shadow's tip when 14 m from the pole.

See Solution

Problem 13125

Find $\int_{6}^{4} f(s) d s$ if $\int_{4}^{6} f(x) d x=\frac{11}{3}$ .

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Problem 13126

Find the bounds $a$ and $b$ for the integral if $\int_{a}^{b} f(x) dx = \int_{-5}^{2} f(x) dx + \int_{2}^{4} f(x) dx - \int_{-5}^{-1} f(x) dx$ .

See Solution

Problem 13127

Untersuche, ob $F(x)=0,1 x^{6}-5$ oder $G(x)=\frac{3}{30} x^{6}$ die Stammfunktion von $h(x)=\frac{3}{5} x^{5}$ ist.

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Problem 13128

Find the point elasticity of demand for $q=\frac{68}{p}+\ln(126-p^{3})$ at $p=5$ . Then, estimate the impact of a 2% price drop on quantity sold and revenue.

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Problem 13129

Zeichne den Graphen von $c(t)=t^{3}-17 t^{2}+63 t+81$ für $0 \leq t \leq 9$ . Bestimme Maximal- und Endkonzentration.

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Problem 13130

Calculate the antiderivative of $f(x)=-3 x^{2}+\frac{1}{2} x^{3}$ .

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Problem 13131

Find the limit: $\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$

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Problem 13132

A light ray enters a raindrop, reflecting internally. Use Snell's Law $\sin \beta=0.75 \sin \theta$ to find angles.
a. Find $\beta$ when $\theta=0.8$ . b. Find $\psi$ when $\theta=0.8$ and show $\psi=4 \beta-2 \theta$ . Express $\psi$ as a function of $\theta$ . c. As $\theta$ varies from 0 to $\frac{\pi}{2}$ , find the max $\psi$ and its relation to rainbows.
For extreme colors, find max $\psi$ for $k=0.7513$ (red) and $k=0.7435$ (violet). The rainbow's width is about 2 degrees.

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Problem 13133

Differentiate $f(x)=\frac{x^{3}-7 x+9 x^{2}}{7 x}$ with respect to $x$ .

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Problem 13134

Find $\frac{d y}{d x}$ for $y=3-\frac{1}{5 \sqrt[3]{x^{5}}}+4 x$ .

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Problem 13135

Ein Zug muss vor einem $1000 \mathrm{~m}$ Signal anhalten. Bestimme, wann der Zug stoppt und ob er das Signal erreicht.

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Problem 13136

Bestimmen Sie die Ableitung von $f$ bei $x_{0}=2$ mit dem Differenzenquotienten für kleine $h$ für folgende Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=\frac{2}{x}$ , c) $f(x)=2 x^{2}-3$ , d) $f(x)=x^{4}$ , e) $f(x)=x^{3}$ , f) $f(x)=4 x-x^{2}$ , g) $f(x)=\sqrt{x}$ , h) $f(x)=5$ .

See Solution

Problem 13137

Berechne die Ableitung von $f$ bei $x_{0}=2$ mit dem Differenzenquotienten für kleine $h$ für die Funktionen: a) $f(x)=x^{2}$ , b) $f(x)=\frac{2}{x}$ , c) $f(x)=2 x^{2}-3$ , d) $f(x)=x^{4}$ , e) $f(x)=x^{3}$ , f) $f(x)=4 x-x^{2}$ , g) $f(x)=\sqrt{x}$ , h) $f(x)=5$ .

See Solution

Problem 13138

Bestimme die Ableitung der Funktion $f$ bei $x_{0}=2$ mit dem Differenzenquotienten für kleine $h$ für die folgenden Funktionen: a) $f(x)=\frac{x^{2}}{3}$ b) $f(x)=\frac{2}{x}$ c) $f(x)=2 x^{2}-3$ d) $f(x)=x^{4}$ e) $f(x)=x^{3}$ f) $f(x)=4 x-x^{2}$ g) $f(x)=\sqrt{x}$ h) $f(x)=5$

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Problem 13139

Find the derivative of $f\left(\tan^{-1} x\right)$ if $f'(x) = \tan x$ . What is the result?

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Problem 13140

Finde die Stelle, an der die Steigung der Funktion $f(x)=x^{5}$ gleich 80 ist. Gib nur eine Lösung an.

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Problem 13141

Find the derivative of $\cos^{-1} x$ with respect to $\sqrt{1-x^{2}}$ . Choose from: (A) $\frac{1}{\sqrt{1-x^{2}}}$ , (B) $\sqrt{1-x^{2}}$ , (C) $\frac{1}{2 x}$ , (D) $\frac{1}{x}$ .

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Problem 13142

Finde den Hochpunkt der Funktion $f(x)=2 x \cdot e^{-0,5 x}$ , die Tangentengleichung bei $x_0=1$ und zeige, dass $F(x)=-4(x+2) \cdot e^{-0,5 x}$ eine Stammfunktion ist.

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Problem 13143

Find the value of $x$ where the tangent to $y=x^{3}-6 x^{2}+9 x+4$ has maximum slope for $0 \leq x \leq 5$ .

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Problem 13144

Berechne die Steigung der Funktionen: a) $f(x)=\frac{3}{x}$ bei $x=-2$ , b) $f(x)=2 \cdot \sqrt{x}$ bei $x=3$ .

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Problem 13145

Evaluate the double integral $\iint_{D} 8 y x^{3} dA$ where $D=\{(x, y) \mid -1 \leq y \leq 2, -1 \leq x\}$ .

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Problem 13146

Find $\int e^{x}(\cos x-\sin x) d x$ . Choose from: (a) $e^{x} \cos x+c$ , (b) $e^{x} \sin x+c$ , (c) $-e^{x} \cos x+c$ , (d) $-e^{x} \sin x+c$ .

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Problem 13147

Assertion-Reason Questions:
For Q19, given $f(x)$ with $\frac{d}{d x}(f(x))=(x-1)^{3}(x-3)^{2}$ , is $A$ : min at $x=1$ true? $R$ : min condition true?

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Problem 13148

Evaluate the double integral $\iint_{D} 8 y x^{3} d A$ for the region $D=\{(x, y) \mid -1 \leq y \leq 2, -1 \leq x \leq 1+y^{2}\}$ .

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Problem 13149

Convert the integral $\iint_{D}(3xy^{2}-2) dA$ from Cartesian to polar coordinates, where $D$ is the unit circle.

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Problem 13150

Bestimme die Ableitung von $f(x)=\frac{3}{5} x^{5}$ mit der Faktorregel.

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Problem 13151

Evaluate the double integral: $\int_{-1}^{2}\left[\int_{-1}^{1+y^{2}} 8 y x^{3} d x\right] d y$ .

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Problem 13152

Find the limits of $g(x)=\ln(3-x)$ as $x \to -\infty$ and as $x \to 3^-$ (from the left).

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Problem 13153

Analyze the end behavior of $f(x)=-1+5(1.02)^{x}$ as $x \to \infty$ and $x \to -\infty$ using limits.

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Problem 13154

Calculate the area between the lines $f(x) = 8x + 5$ and $g(x) = 8x - 5$ .

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Problem 13155

Compounding Interest Problem Taylor invests \$1 at 100% interest. Calculate her amount after 1 year for different compounding periods. Also, if \$300 compounds continuously at 6% for 5 years, find the final amount.

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Problem 13156

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

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Problem 13157

Find the derivative $h^{\prime}(2)$ if $h(x)=\frac{f(x)}{2x+1}$ .

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Problem 13158

Find the derivative of the function $y=3 e^{x} \cdot \cos x$ . What is $\frac{d y}{d x}$ ?

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Problem 13159

Calculate the integral $\int_{-4}^{9} \sqrt{16-x^{2}} \, dx$ .

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Problem 13160

Berechnen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ bei $x_{0}=2$ , b) $f(x)=2 x^{2}$ bei $x_{0}=1$ , c) $f(x)=-x^{2}$ bei $x_{0}=2$ .

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Problem 13161

Find the derivative of the function $y=\frac{3 x^{5}}{5}+\frac{2}{x^{2}}-\sqrt[3]{x^{5}}$ . What is $\frac{d y}{d x}$ ?

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Problem 13162

Find the tangent line equation for $f(x)=\ln(x^{2}+1)-3\sin(e^{x})$ at $x=2$ .

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Problem 13163

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ by implicit differentiation of $5 y^{2}-x y+3 x^{2}=2$ .

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Problem 13164

Find the derivative of the function $g(a)=\sqrt{a} \cdot(1-a)$ , which is $g^{\prime}(a)=\frac{1-3 a}{2 \sqrt{a}}$ .

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Problem 13165

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt{\frac{4 x^{3}}{5 x-1}}$ .

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Problem 13166

Evaluate (A) $\lim _{x \rightarrow 3} \frac{2 \ln x-2 \ln 3}{x-3}$ and (B) $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-\sqrt{9}}{h}$ as derivatives.

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Problem 13167

Find values of $b$ and $c$ for which $f(x)$ is continuous but not differentiable at $x=4$ :
$f(x)=\left\{\begin{array}{l} x^{3 / 2}+5, x<4 \\ x^{2}-b x+c, x \geqslant 4 \end{array}\right.$

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Problem 13168

Find the total cost to produce 1500 yards of ribbon using $C^{\prime}(x)=-0.00001 x^{2}-0.02 x+56$ with 5 subintervals.

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Problem 13169

Approximate the area under $f(x)=0.01 x^{4}-1.69 x^{2}+99$ from $x=8$ to $x=12$ using 4 left endpoints.

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Problem 13170

Find the hourly growth rate of a bacteria population that grows from 1000 to 1153 in 4 hours. Express as a percentage.

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Problem 13171

Find the half-life of a radioactive substance with a decay rate of $2.3\%$ per day. Round to the nearest hundredth.

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Problem 13172

Alan coloca una botella de agua en una hielera. La temperatura $C(t)$ en °C es $C(t)=2+20 e^{-0.041 t}$ . ¿Cuántos minutos debe dejarla para que alcance 16 °C? Redondea a la décima.

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Problem 13173

Find the limit of $f(x)=2x^3 + x^2 + x - 1$ as $x \to -\infty$ and $x \to \infty$ . Sketch the graph.

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Problem 13174

Find the hourly growth rate of a bacteria population that grows from 2400 to 2800 in 4 hours. Express as a percentage: $\square \%$

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Problem 13175

How long does it take for a bacteria population to double with a growth rate of $7.1\%$ per hour? Round to the nearest hundredth.

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Problem 13176

Finde eine Stammfunktion von $f(x)=(3+4x)^{4}$ .

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Problem 13177

Finde die Stammfunktion von $f(x)=(1-x)^{6}$ .

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Problem 13178

Finde die Stammfunktion von $f(x)=(2 x+3)^{-3}$ .

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Problem 13179

Find $d y$ for the function $y = x \sqrt{17 - x^3}$ .

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Problem 13180

Find the derivative of $f(x)=\frac{3-x}{x}$ and explain how to get $f^{\prime}(x)=\frac{-3 x}{x^{2}}$ .

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Problem 13181

Alicia's utility is $U(x_{1}, x_{2})=x_{1}^{3}x_{2}$ . Find MRS, plot indifference curve at (1,1), and find optimal bundle with prices \$40 and \$20.

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Problem 13182

Ein Ballon mit Radius $r$ hat den Oberflächeninhalt $O(r)=4 \pi r^{2}$ .
1) Finde die mittlere Änderungsrate von $O$ in $[r; z]$ und berechne sie für $[10; 20]$ und $[20; 30]$ . Welches Intervall hat die größere Änderungsrate? 2) Bestimme $O^{\prime}(r)$ und berechne $O^{\prime}(1)$ und $O^{\prime}(3)$ . Wo ändert sich $O$ stärker? 3) Ist $O^{\prime}(r)$ linear, quadratisch oder keines von beidem?

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Problem 13183

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

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Problem 13184

Bestimme die Stammfunktion $F$ von $f(x)=6 x^{2}-\frac{5}{x^{2}}$ für $x>0$ , sodass $F(1)=5$ .

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Problem 13185

Find the limit: $\lim _{n \rightarrow \infty} n\left(\frac{1}{n^{2}+1}+\frac{1}{n^{2}+2}+\cdots+\frac{1}{n^{2}+n}\right)$

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Problem 13186

Der Graph von $f'$ hat einen Tiefpunkt. Was bedeutet das für $f''$ und die Steigungen der Ableitungen?

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Problem 13187

Find the rate of increase of the surface area (in $\mathrm{cm}^{2} / \mathrm{min}$ ) when the radius is $5 \mathrm{~cm}$ , given it increases at $2 \mathrm{~cm} / \mathrm{min}$ .

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Problem 13188

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

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Problem 13189

Calculate the integral of the function: $\int(2 a x^{4}+6^{2} x) \, dx$ .

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Problem 13190

Find the point of diminishing returns for productivity modeled by $P(t)=-3 t^{3}+49.5 t^{2}+6 t$ in a 10-hour shift.

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Problem 13191

Find the local maximum of $f(x)=4+9x+36x^{-1}$ and confirm the local minimum is at $x=2$ with value $\sigma$ .

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Problem 13192

Find the derivative of $(2.5x^{2}-125) e^{0.4x-2}$ .

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Problem 13193

Find where the function $f(x)=3x+8x^{-1}$ is undefined, its critical numbers, and intervals for increasing, decreasing, concave up, and concave down.

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Problem 13194

Find the antiderivative $S(t)$ if $S^{\prime}(t)=-10 t^{\frac{2}{3}}$ .

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Problem 13195

Finde die Extremstellen der Funktion $h(u)=(1,2-u)^{2}+\left(3,5-(-0,5 u^{2}+4,5)\right)^{2}$ durch Differenzieren.

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Problem 13196

Find the population of a city 16 years after incorporation if $\frac{d N}{d t}=800+300 \sqrt{t}$ and initial population is 5000.

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Problem 13197

Finde die Extremstellen der Funktion $f(x) = (4+4x) \cdot e^{2x}$ .

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Problem 13198

Find $\frac{d y}{d x}$ using the chain rule for: (a) $y=6 u-9$ , $u=\frac{x^{4}}{2}$ ; (b) $y=\sqrt{u}$ , $u=\sin (x)$ ; (c) $y=3 u^{3}$ , $u=8 x-1$ ; (d) $y=\frac{1}{\sqrt[3]{u}}$ , $u=\cos (x)$ .

See Solution

Problem 13199

Determine if L'Hospital's Rule can be applied to these limits. Write "YES" with indeterminate form or "NO" if not applicable.
Limits:
1. $\lim _{x \rightarrow 0} \frac{x^{2}}{x+2}$
2. $\lim _{x \rightarrow 0} \frac{\tan (3 x)}{x}$
3. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$
4. $\lim _{x \rightarrow \infty} \frac{\sin x}{x}$
5. $\lim _{x \rightarrow \infty} \frac{2^{x}}{x^{2}}$
6. $\lim _{x \rightarrow \infty} \frac{3 x^{2}+4}{x^{2}}$
7. $\lim _{x \rightarrow \frac{\pi}{2}}\left(x-\frac{\pi}{2}\right)(\tan x)$

See Solution

Problem 13200

Compute the derivatives of these functions: (a) $p(t)=\sqrt{3-t}$ , (b) $s(t)=\sin\left(\frac{3\pi}{2}t\right)+\cos\left(\frac{3\pi}{2}t\right)$ , (c) $h(x)=x^{2}\sin(x^{4})+\frac{x^{2}}{\cos(x^{4})}$ , (d) $q(s)=(4s+3)^{4}(s+1)^{-3}$ , (e) $m(x)=\cos(\sin(x))$ .

See Solution

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Calculus

Problem 13101

Determine where the function f(x)=−x44+x3−x2f(x)=-\frac{x^{4}}{4}+x^{3}-x^{2}f(x)=−4x4​+x3−x2 is increasing or decreasing.

Problem 13102

Find and simplify the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=2x2tan⁡xsec⁡xf(x)=\frac{2 x^{2} \tan x}{\sec x}f(x)=secx2x2tanx​.

Problem 13103

Find the radius expansion rate of a circular area wetting at 10 mm2/s10 \mathrm{~mm}^{2} / \mathrm{s}10 mm2/s when radius is 112 mm112 \mathrm{~mm}112 mm.

Problem 13104

Find f′(x)f^{\prime}(x)f′(x) for f(x)=(4x+3)−1f(x)=(4 x+3)^{-1}f(x)=(4x+3)−1 and determine f′(4)=□σ6f^{\prime}(4) = \square \sigma^{6}f′(4)=□σ6.

Problem 13105

Find the rate of area increase of a metal plate when its radius is 46 cm46 \mathrm{~cm}46 cm and increases at 0.05 cm/min0.05 \mathrm{~cm/min}0.05 cm/min.

Problem 13106

Find the degree 2 Taylor polynomial of f(x)=6xln⁡xf(x)=6x \ln xf(x)=6xlnx centered at x=2x=2x=2.

Problem 13107

Find the concavity intervals and inflection points for the function g(t)=3t5+15t4+20t3+70g(t)=3 t^{5}+15 t^{4}+20 t^{3}+70g(t)=3t5+15t4+20t3+70.

Problem 13108

Find the linearization L(x)L(x)L(x) at x=1x=1x=1 for the function f(x)=3x3−2x−1f(x)=3x^3-2x-1f(x)=3x3−2x−1.

Problem 13109

Find the area represented by the integral ∫−60(2+36−x2)dx\int_{-6}^{0}\left(2+\sqrt{36-x^{2}}\right) d x∫−60​(2+36−x2​)dx.

Problem 13110

Find the degree 3 Taylor polynomial T3T_{3}T3​ at x=0x=0x=0 for f(x)=1(4x+1)1/2f(x)=\frac{1}{(4 x+1)^{1/2}}f(x)=(4x+1)1/21​.

Problem 13111

Find the Taylor series at x=0x=0x=0 for f(x)=cos⁡(5x)f(x)=\cos(5x)f(x)=cos(5x).

Problem 13112

Problem 13113

Prove the inequality ∣sin⁡a−sin⁡b∣≤∣a−b∣|\sin a - \sin b| \leq |a - b|∣sina−sinb∣≤∣a−b∣ using the Mean Value Theorem for all aaa and bbb.

Problem 13114

Prove ∣sin⁡a−sin⁡b∣≤∣a−b∣|\sin a - \sin b| \leq |a - b|∣sina−sinb∣≤∣a−b∣ using the Mean Value Theorem for any values of aaa and bbb.

Problem 13115

Find the derivative of g(t)=ln⁡(12t3)g(t) = \ln(12 t^3)g(t)=ln(12t3).

Problem 13116

Arjun walks south at 1.5 m/s1.5 \mathrm{~m/s}1.5 m/s and Maya east at 2 m/s2 \mathrm{~m/s}2 m/s. Find the distance change rate after 2 mins.

Problem 13117

Find dydx\frac{d y}{d x}dxdy​ using implicit differentiation for the equation x4+y5=−7x^{4}+y^{5}=-7x4+y5=−7.

Problem 13118

Find the derivative of f(x)=ex+9x23f(x)=\sqrt[3]{e^{x}+9 x^{2}}f(x)=3ex+9x2​.

Problem 13119

Find the limit: lim⁡h→0(−2+h)2−4h\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}limh→0​h(−2+h)2−4​.

Problem 13120

Find the slope of the tangent line to the curve r=sin⁡θ+2cos⁡θr=\sin \theta+2 \cos \thetar=sinθ+2cosθ at θ=π2\theta=\frac{\pi}{2}θ=2π​.

Problem 13121

A tennis ball's motion is given by x=32.0tx=32.0 tx=32.0t and y=42.0t−4.90t2y=42.0 t-4.90 t^{2}y=42.0t−4.90t2. Find velocity and acceleration at t=6.00 st=6.00 \mathrm{~s}t=6.00 s.

Problem 13122

Find the limit: lim⁡s→2525−s5−s\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}lims→25​5−s​25−s​

Problem 13123

Gravel is dumped at 24 m3/h24 \mathrm{~m}^{3} / \mathrm{h}24 m3/h forming a cone. Find height increase rate when height is 5 m5 \mathrm{~m}5 m.

Problem 13124

A 1.7 m tall person walks away from a 5.95 m pole at 1 m/s. Find the speed of their shadow's tip when 14 m from the pole.

Problem 13125

Find ∫64f(s)ds\int_{6}^{4} f(s) d s∫64​f(s)ds if ∫46f(x)dx=113\int_{4}^{6} f(x) d x=\frac{11}{3}∫46​f(x)dx=311​.

Problem 13126

Find the bounds aaa and bbb for the integral if ∫abf(x)dx=∫−52f(x)dx+∫24f(x)dx−∫−5−1f(x)dx\int_{a}^{b} f(x) dx = \int_{-5}^{2} f(x) dx + \int_{2}^{4} f(x) dx - \int_{-5}^{-1} f(x) dx∫ab​f(x)dx=∫−52​f(x)dx+∫24​f(x)dx−∫−5−1​f(x)dx.

Problem 13127

Untersuche, ob F(x)=0,1x6−5F(x)=0,1 x^{6}-5F(x)=0,1x6−5 oder G(x)=330x6G(x)=\frac{3}{30} x^{6}G(x)=303​x6 die Stammfunktion von h(x)=35x5h(x)=\frac{3}{5} x^{5}h(x)=53​x5 ist.

Problem 13128

Find the point elasticity of demand for q=68p+ln⁡(126−p3)q=\frac{68}{p}+\ln(126-p^{3})q=p68​+ln(126−p3) at p=5p=5p=5. Then, estimate the impact of a 2% price drop on quantity sold and revenue.

Problem 13129

Zeichne den Graphen von c(t)=t3−17t2+63t+81c(t)=t^{3}-17 t^{2}+63 t+81c(t)=t3−17t2+63t+81 für 0≤t≤90 \leq t \leq 90≤t≤9. Bestimme Maximal- und Endkonzentration.

Problem 13130

Calculate the antiderivative of f(x)=−3x2+12x3f(x)=-3 x^{2}+\frac{1}{2} x^{3}f(x)=−3x2+21​x3.

Problem 13131

Find the limit: lim⁡h→0f(x0+h)−f(x0)h\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}limh→0​hf(x0​+h)−f(x0​)​

Problem 13132

Problem 13133

Differentiate f(x)=x3−7x+9x27xf(x)=\frac{x^{3}-7 x+9 x^{2}}{7 x}f(x)=7xx3−7x+9x2​ with respect to xxx.

Problem 13134

Find dydx\frac{d y}{d x}dxdy​ for y=3−15x53+4xy=3-\frac{1}{5 \sqrt[3]{x^{5}}}+4 xy=3−53x5​1​+4x.

Problem 13135

Ein Zug muss vor einem 1000 m1000 \mathrm{~m}1000 m Signal anhalten. Bestimme, wann der Zug stoppt und ob er das Signal erreicht.

Problem 13136

Problem 13137

Problem 13138

Problem 13139

Find the derivative of f(tan⁡−1x)f\left(\tan^{-1} x\right)f(tan−1x) if f′(x)=tan⁡xf'(x) = \tan xf′(x)=tanx. What is the result?

Problem 13140

Finde die Stelle, an der die Steigung der Funktion f(x)=x5f(x)=x^{5}f(x)=x5 gleich 80 ist. Gib nur eine Lösung an.

Problem 13141

Find the derivative of cos⁡−1x\cos^{-1} xcos−1x with respect to 1−x2\sqrt{1-x^{2}}1−x2​. Choose from: (A) 11−x2\frac{1}{\sqrt{1-x^{2}}}1−x2​1​, (B) 1−x2\sqrt{1-x^{2}}1−x2​, (C) 12x\frac{1}{2 x}2x1​, (D) 1x\frac{1}{x}x1​.

Problem 13142

Finde den Hochpunkt der Funktion f(x)=2x⋅e−0,5xf(x)=2 x \cdot e^{-0,5 x}f(x)=2x⋅e−0,5x, die Tangentengleichung bei x0=1x_0=1x0​=1 und zeige, dass F(x)=−4(x+2)⋅e−0,5xF(x)=-4(x+2) \cdot e^{-0,5 x}F(x)=−4(x+2)⋅e−0,5x eine Stammfunktion ist.

Determine where the function $f(x)=-\frac{x^{4}}{4}+x^{3}-x^{2}$ is increasing or decreasing.

Find and simplify the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{2 x^{2} \tan x}{\sec x}$ .

Find the radius expansion rate of a circular area wetting at $10 \mathrm{~mm}^{2} / \mathrm{s}$ when radius is $112 \mathrm{~mm}$ .

Find $f^{\prime}(x)$ for $f(x)=(4 x+3)^{-1}$ and determine $f^{\prime}(4) = \square \sigma^{6}$ .

Find the rate of area increase of a metal plate when its radius is $46 \mathrm{~cm}$ and increases at $0.05 \mathrm{~cm/min}$ .

Find the degree 2 Taylor polynomial of $f(x)=6x \ln x$ centered at $x=2$ .

Find the concavity intervals and inflection points for the function $g(t)=3 t^{5}+15 t^{4}+20 t^{3}+70$ .

Find the linearization $L(x)$ at $x=1$ for the function $f(x)=3x^3-2x-1$ .

Find the area represented by the integral $\int_{-6}^{0}\left(2+\sqrt{36-x^{2}}\right) d x$ .

Find the degree 3 Taylor polynomial $T_{3}$ at $x=0$ for $f(x)=\frac{1}{(4 x+1)^{1/2}}$ .

Find the Taylor series at $x=0$ for $f(x)=\cos(5x)$ .

Prove the inequality $|\sin a - \sin b| \leq |a - b|$ using the Mean Value Theorem for all $a$ and $b$ .

Prove $|\sin a - \sin b| \leq |a - b|$ using the Mean Value Theorem for any values of $a$ and $b$ .

Find the derivative of $g(t) = \ln(12 t^3)$ .

Arjun walks south at $1.5 \mathrm{~m/s}$ and Maya east at $2 \mathrm{~m/s}$ . Find the distance change rate after 2 mins.

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x^{4}+y^{5}=-7$ .

Find the derivative of $f(x)=\sqrt[3]{e^{x}+9 x^{2}}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{(-2+h)^{2}-4}{h}$ .

Find the slope of the tangent line to the curve $r=\sin \theta+2 \cos \theta$ at $\theta=\frac{\pi}{2}$ .

A tennis ball's motion is given by $x=32.0 t$ and $y=42.0 t-4.90 t^{2}$ . Find velocity and acceleration at $t=6.00 \mathrm{~s}$ .

Find the limit: $\lim _{s \rightarrow 25} \frac{25-s}{5-\sqrt{s}}$

Gravel is dumped at $24 \mathrm{~m}^{3} / \mathrm{h}$ forming a cone. Find height increase rate when height is $5 \mathrm{~m}$ .

Find $\int_{6}^{4} f(s) d s$ if $\int_{4}^{6} f(x) d x=\frac{11}{3}$ .

Find the bounds $a$ and $b$ for the integral if $\int_{a}^{b} f(x) dx = \int_{-5}^{2} f(x) dx + \int_{2}^{4} f(x) dx - \int_{-5}^{-1} f(x) dx$ .

Untersuche, ob $F(x)=0,1 x^{6}-5$ oder $G(x)=\frac{3}{30} x^{6}$ die Stammfunktion von $h(x)=\frac{3}{5} x^{5}$ ist.

Find the point elasticity of demand for $q=\frac{68}{p}+\ln(126-p^{3})$ at $p=5$ . Then, estimate the impact of a 2% price drop on quantity sold and revenue.

Zeichne den Graphen von $c(t)=t^{3}-17 t^{2}+63 t+81$ für $0 \leq t \leq 9$ . Bestimme Maximal- und Endkonzentration.

Calculate the antiderivative of $f(x)=-3 x^{2}+\frac{1}{2} x^{3}$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$

Differentiate $f(x)=\frac{x^{3}-7 x+9 x^{2}}{7 x}$ with respect to $x$ .

Find $\frac{d y}{d x}$ for $y=3-\frac{1}{5 \sqrt[3]{x^{5}}}+4 x$ .

Ein Zug muss vor einem $1000 \mathrm{~m}$ Signal anhalten. Bestimme, wann der Zug stoppt und ob er das Signal erreicht.

Find the derivative of $f\left(\tan^{-1} x\right)$ if $f'(x) = \tan x$ . What is the result?

Finde die Stelle, an der die Steigung der Funktion $f(x)=x^{5}$ gleich 80 ist. Gib nur eine Lösung an.

Find the derivative of $\cos^{-1} x$ with respect to $\sqrt{1-x^{2}}$ . Choose from: (A) $\frac{1}{\sqrt{1-x^{2}}}$ , (B) $\sqrt{1-x^{2}}$ , (C) $\frac{1}{2 x}$ , (D) $\frac{1}{x}$ .

Finde den Hochpunkt der Funktion $f(x)=2 x \cdot e^{-0,5 x}$ , die Tangentengleichung bei $x_0=1$ und zeige, dass $F(x)=-4(x+2) \cdot e^{-0,5 x}$ eine Stammfunktion ist.

Find the value of $x$ where the tangent to $y=x^{3}-6 x^{2}+9 x+4$ has maximum slope for $0 \leq x \leq 5$ .

Berechne die Steigung der Funktionen: a) $f(x)=\frac{3}{x}$ bei $x=-2$ , b) $f(x)=2 \cdot \sqrt{x}$ bei $x=3$ .

Evaluate the double integral $\iint_{D} 8 y x^{3} dA$ where $D=\{(x, y) \mid -1 \leq y \leq 2, -1 \leq x\}$ .

Find $\int e^{x}(\cos x-\sin x) d x$ . Choose from: (a) $e^{x} \cos x+c$ , (b) $e^{x} \sin x+c$ , (c) $-e^{x} \cos x+c$ , (d) $-e^{x} \sin x+c$ .

Assertion-Reason Questions:
For Q19, given $f(x)$ with $\frac{d}{d x}(f(x))=(x-1)^{3}(x-3)^{2}$ , is $A$ : min at $x=1$ true? $R$ : min condition true?

Evaluate the double integral $\iint_{D} 8 y x^{3} d A$ for the region $D=\{(x, y) \mid -1 \leq y \leq 2, -1 \leq x \leq 1+y^{2}\}$ .

Convert the integral $\iint_{D}(3xy^{2}-2) dA$ from Cartesian to polar coordinates, where $D$ is the unit circle.

Bestimme die Ableitung von $f(x)=\frac{3}{5} x^{5}$ mit der Faktorregel.

Evaluate the double integral: $\int_{-1}^{2}\left[\int_{-1}^{1+y^{2}} 8 y x^{3} d x\right] d y$ .

Find the limits of $g(x)=\ln(3-x)$ as $x \to -\infty$ and as $x \to 3^-$ (from the left).

Analyze the end behavior of $f(x)=-1+5(1.02)^{x}$ as $x \to \infty$ and $x \to -\infty$ using limits.

Calculate the area between the lines $f(x) = 8x + 5$ and $g(x) = 8x - 5$ .

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

Find the derivative $h^{\prime}(2)$ if $h(x)=\frac{f(x)}{2x+1}$ .

Find the derivative of the function $y=3 e^{x} \cdot \cos x$ . What is $\frac{d y}{d x}$ ?

Calculate the integral $\int_{-4}^{9} \sqrt{16-x^{2}} \, dx$ .

Berechnen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ bei $x_{0}=2$ , b) $f(x)=2 x^{2}$ bei $x_{0}=1$ , c) $f(x)=-x^{2}$ bei $x_{0}=2$ .

Find the derivative of the function $y=\frac{3 x^{5}}{5}+\frac{2}{x^{2}}-\sqrt[3]{x^{5}}$ . What is $\frac{d y}{d x}$ ?

Find the tangent line equation for $f(x)=\ln(x^{2}+1)-3\sin(e^{x})$ at $x=2$ .

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ by implicit differentiation of $5 y^{2}-x y+3 x^{2}=2$ .

Find the derivative of the function $g(a)=\sqrt{a} \cdot(1-a)$ , which is $g^{\prime}(a)=\frac{1-3 a}{2 \sqrt{a}}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\sqrt{\frac{4 x^{3}}{5 x-1}}$ .

Evaluate (A) $\lim _{x \rightarrow 3} \frac{2 \ln x-2 \ln 3}{x-3}$ and (B) $\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-\sqrt{9}}{h}$ as derivatives.

Find values of $b$ and $c$ for which $f(x)$ is continuous but not differentiable at $x=4$ :
$f(x)=\left\{\begin{array}{l} x^{3 / 2}+5, x<4 \\ x^{2}-b x+c, x \geqslant 4 \end{array}\right.$

Find the total cost to produce 1500 yards of ribbon using $C^{\prime}(x)=-0.00001 x^{2}-0.02 x+56$ with 5 subintervals.

Approximate the area under $f(x)=0.01 x^{4}-1.69 x^{2}+99$ from $x=8$ to $x=12$ using 4 left endpoints.

Find the half-life of a radioactive substance with a decay rate of $2.3\%$ per day. Round to the nearest hundredth.

Alan coloca una botella de agua en una hielera. La temperatura $C(t)$ en °C es $C(t)=2+20 e^{-0.041 t}$ . ¿Cuántos minutos debe dejarla para que alcance 16 °C? Redondea a la décima.

Find the limit of $f(x)=2x^3 + x^2 + x - 1$ as $x \to -\infty$ and $x \to \infty$ . Sketch the graph.

Find the hourly growth rate of a bacteria population that grows from 2400 to 2800 in 4 hours. Express as a percentage: $\square \%$