Calculus

Problem 7101

Find the growth rate of infections after 10 days using $I(t)=1100 e^{0.038 t}$ . Round to the nearest tenth.

See Solution

Problem 7102

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(4 x+3)^{-1}$ and calculate $f^{\prime}(4)$ .

See Solution

Problem 7103

Find the rate of change of distance from the origin to the point on the curve $y=x^{2}+2$ if $\frac{dx}{dt}=2$ cm/s.

See Solution

Problem 7104

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ given $f(-3)=7$ , $f^{\prime}(-3)=3$ , $g(2)=-3$ , and $g^{\prime}(2)=2$ .

See Solution

Problem 7105

Find the derivative $f'(x)$ for $f(x)=(x^{2}+3x+6)^{4}$ and compute $f'(2)$ .

See Solution

Problem 7106

Find $r'(1)$ for $r(x)=f(g(h(x)))$ given $h(1)=4$ , $g(4)=3$ , $h'(1)=5$ , $g'(4)=4$ , $f'(3)=8$ .

See Solution

Problem 7107

Given functions $f$ and $g$ , find $h^{\prime}(3)$ for $h(x)=f(g(x))$ and $H^{\prime}(1)$ for $H(x)=g(f(x))$ .

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Problem 7108

A star's brightness is modeled by $B(t)=4.0+0.45 \sin \left(\frac{2 \pi t}{4.5}\right)$ . Find $\frac{d B}{d t}$ and at $t=1$ .

See Solution

Problem 7109

Find the velocity and acceleration of a mass on a spring with $s(t)=5 \sin (4 t)$ at $t=4$ seconds. Round to nearest hundredth.

See Solution

Problem 7110

Given $f^{\prime}(0)=-1, f^{\prime}(1)=\frac{3}{4}, f^{\prime}(2)=1, f^{\prime}(4)=-3$ , find $h^{\prime}(0)$ , $h^{\prime}(-2)$ , and $h^{\prime}(2)$ for $h(x)=f\left(x^{2}\right)$ .

See Solution

Problem 7111

Analyze the beer market with demand $Q=140-4p$ and supply $Q=20+2p-42h$ . Find $\mathrm{dp}/\mathrm{dh}$ , $\mathrm{dQ}/\mathrm{dh}$ , and changes in $p$ , $Q$ when $h$ increases by \$2.

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Problem 7112

Find critical points of $f(x)=x^{3}+6 x^{2}-180 x-49$ . Options: $x=-10, x=6$ ; $x=-49$ ; $x=0, -10, 6$ ; $x=0$ ; $x=3, 10, -6$ ; None.

See Solution

Problem 7113

Find the snail's acceleration given its position over days: 0 ft at day 0, 11 ft at day 1, 22 ft at day 2, etc.

See Solution

Problem 7114

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=4 x^{2}+7 x-3$ .

See Solution

Problem 7115

Find the satellite's period if its orbit radius is 5.08 times Earth's radius ( $6.38 \times 10^{6} \mathrm{~m}$ ). Earth's mass is $5.98 \times 10^{24} \mathrm{~kg}$ .

See Solution

Problem 7116

Find the critical numbers of the function $f(x)$ given its derivative $f^{\prime}(x)=\frac{x^{2}-3 x}{(x-5)^{3}(x-6)^{3}}$ .

See Solution

Problem 7117

Find the tangent line equation for $f(x)=3 \cos (x)$ at $x=\frac{4 \pi}{3}$ . Express it as $y=$ .

See Solution

Problem 7118

Find the local extreme points of the rational function $f(x)$ with vertical asymptotes at $x=-6$ and $x=4$ .

See Solution

Problem 7119

Evaluate the Riemann sum $R S(n)$ for $I=\int_{0}^{2} x \, dx$ and find the optimal statement about the error $E(n)$ .

See Solution

Problem 7120

Find the tangent line equation for $y=6 x^{2}-\ln (x)$ at (1,6) and graph it with the function.

See Solution

Problem 7121

Find all critical points of the function $f(x)=(2x^2-3x)^{\frac{1}{3}}$ .

See Solution

Problem 7122

Evaluate the Riemann sum for $I=\int_{0}^{2} x \, dx=2$ using right endpoints. Determine the optimal error $E(n)$ .

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Problem 7123

Find the derivative of $y=\frac{3}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)$ . What is $\frac{d y}{d x}$ ?

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Problem 7124

Find the tangent line equation using implicit differentiation for $x^{2}+x \arctan y=y-1$ at the point $\left(-\frac{\pi}{4}, 1\right)$ .

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Problem 7125

Find the tangent line equation for $f(x)=\frac{1}{2} x \ln x^{4}$ at the point in the interval (-1,0).

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Problem 7126

Approximate $\sqrt{64.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . Find the equation $L(x)=$ . Give answer to 9 significant figures.

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Problem 7127

Find the average rate of change of $f(x)=2 x^{2}+7$ between $x=2$ and $x=4$ . Simplify your answer.

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Problem 7128

Find the tangent line equation using implicit differentiation for $x^{2}+x \arctan y=y-1$ at $\left(-\frac{\pi}{4}, 1\right)$ .

See Solution

Problem 7129

Approximate $\sqrt{36.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=36$ . Find $L(x)=$ .

See Solution

Problem 7130

Find the derivative of $f(x)=3x-4x^{2}$ using the definition of derivative only.

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Problem 7131

Find the tangent line equation for $f(x)=5 \sec (x)$ at $x=\frac{5 \pi}{6}$ . Use $y$ and $x$ .

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Problem 7132

Approximate $\frac{1}{0.104}$ using linear approximation with $f(x)=\frac{1}{x}$ at a nearby "nice" point.

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Problem 7133

Approximate $\sqrt[3]{27.2}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=27$ . Find $m$ and $b$ for $y=mx+b$ .

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Problem 7134

Given values for $f, g, f', g'$ at $x=1,2,3$ , find: a) $h'(3)$ for $h(x)=7x \cdot f(x)$ ; b) $h'(2)$ for $h(x)=f(x)g(x)$ ; c) $h'(1)$ for $h(x)=g(f(x))$ .

See Solution

Problem 7135

Two dinosaurs walk from the same point: one north at $4 \mathrm{mi/hr}$ and the other east at $3 \mathrm{mi/hr}$ . Find the distance increase rate after 3 hours.

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Problem 7136

Find the derivative of the function $h(x)=\frac{\cot (x)}{16-4 x}$ for $0 \leq x \leq \pi$ . What is $h^{\prime}(x)$ ?

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Problem 7137

Solve $\frac{d x}{d t}=2 x$ with $x(0)=1$ . Find $t_{1}$ when $x(t_{1})=2$ .

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Problem 7138

Find the derivative of $h(t) = 18 \csc(t) + i \cot(t)$ . What is $h'(t)$ ?

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Problem 7139

Determine which function pairs $(x_1(t), y_1(t))$ and $(x_2(t), y_2(t))$ solve the ODE: $\dot{x}=2x+y$ , $\dot{y}=3y$ . Functions: $(e^{2t}, 0)$ and $(e^{3t}, e^{3t})$ .

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Problem 7140

Find the derivative of $h(t)=18 \csc (t)+t \cot (t)$ . What is $h^{\prime}(t)=?$

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Problem 7141

Evaluate the integral from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$ of $\left(x^{2}+\ln |\cos x|\right) \sin \frac{x}{2} \, dx$ . Provide a numeric answer.

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Problem 7142

Find the derivative of the function $f(x)=5 x \tan(x)$ .

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Problem 7143

Find the derivative of $f(\theta)=2 \theta \tan (\theta) \sec (\theta)$ . What is $f^{\prime}(\theta)$ ?

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Problem 7144

Check if $x_{1}(t)=t+1$ and $x_{2}(t)=\frac{1+\sqrt{4 t^{2}+1}}{2}$ solve the ODE $\frac{d x}{d t}=\frac{t}{x-1}$ .

See Solution

Problem 7145

Find the temperature change rate:
a. From 6 A.M. to 7 A.M. (temp at 6 A.M. unknown).
b. From 1 P.M. to 2 P.M.

See Solution

Problem 7146

Determine if $x_{1}(t)=t+1$ and $x_{2}(t)=\frac{1+\sqrt{4 t^{2}+1}}{2}$ solve the ODE: $\frac{d x}{d t}=\frac{t}{x-1}$ .

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Problem 7147

Find the derivative of $f(x)=x^{5} e^{x} \cos (x)$ . What is $f^{\prime}(x)=?$

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Problem 7148

Find the tangent line equation for $f(t)=\frac{\sin (t)}{11+11 \cos (t)}$ at $t=\frac{\pi}{3}$ .

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Problem 7149

Analyze the system of differential equations to classify the origin as a saddle, source, or sink for $a=-0.01$ and $d=-2.4$ .

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Problem 7150

Calculate the difference quotient for the function $f(x)=7-2x+5x^{2}$ .

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Problem 7151

Jan pedals on a bike, with foot height $h(t)=-0.18 \sin (t)+0.30$ . Find vertical velocity $v$ at $t=3\pi$ .

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Problem 7152

Find the velocity and acceleration of the spring mass at $t=\frac{\pi}{3}$ s, given $s(t)=300+76 \sin(t)$ .

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Problem 7153

Find the tangent line equation for $y=9 \csc (x)-9 \cot (x)$ at $x=\frac{\pi}{6}$ . Use $y=f(x)$ .

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Problem 7154

Find the tangent line equation for $y=5 \csc (x)-5 \cot (x)$ at $x=\frac{3 \pi}{4}$ .

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Problem 7155

Find the tangent line equation for $f(x)=26 e^{x} \cos ^{2}(x)$ at $x=\frac{\pi}{4}$ .

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Problem 7156

Calculate the integral of $f(x)=6 x^{4}-\frac{2}{3} x^{3}+5 x-2$ .

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Problem 7157

Find $d y$ for $y=\tan(2x+2)$ at $x=4$ with $d x=0.3$ and $d x=0.6$ .

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Problem 7158

A squirrel drops from 100 ft to 45 ft in 5.25 seconds. Find the rate of change in height, rounded to the nearest hundredth.

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Problem 7159

Find the limit: $\lim _{x \rightarrow \infty} f(x)=-4 x^{9}-3 x^{3}+2 x-11$ . True/False for $\lim _{x \rightarrow-\infty} f(x)=\infty$ and $\lim _{x \rightarrow \infty} f(x)=-\infty$ .

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Problem 7160

A dolphin jumped to 3.5 m above water and dove 10 m below in 12.5 seconds. Find the average rate of change.

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Problem 7161

Uprość sumę szeregu $\sum_{k=1}^{n} \frac{1}{k^{2}}$ .

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Problem 7162

Bestimmen Sie die Monotonieintervalle der Funktionen: a) $f(x)=\sqrt{x} \cdot(x-9), x>0$ b) $f(x)=(x+1)^{2} \cdot(x-2)$ c) $f(x)=4 x^{2} \cdot(1-x)^{2}$

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Problem 7163

Fill in the table for $\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}$ using values of $b$ : 0, 0.5, 1.

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Problem 7164

Find the limit: $\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}$ .

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Problem 7165

Find the arc length integral for the curve $x=\sqrt{2y+8}$ from $y=1$ to $y=3$ . Select A or B and fill in the boxes.

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Problem 7166

Find and simplify the integral for the arc length of the curve $x=\sqrt{2y+8}$ from $y=1$ to $y=3$ .

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Problem 7167

Ubung $5^{* * *}$ :
a) Zeigen Sie, dass der Grenzwert der Folge $a_{n}=\left(1+\frac{1}{n}\right)^{n}$ die Zahl $e$ ist.
b) Beweisen Sie, dass $2 e$ nicht der Grenzwert von $a_{n}$ ist.

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Problem 7168

Calculate the value of the function $f(x)=e^{x}$ at $x=9.2$ .

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Problem 7169

Find $f^{\prime}(-1)$ for the function $f(x)=2^{x^{2}-3 x}$ .

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Problem 7170

Gegeben ist die Funktion $f(x) = -x^{3} + 3x^{2}$ . Bestimmen Sie Hoch- und Wendepunkt, Tangentengleichung, Flächeninhalt des Dreiecks und Winkel $\alpha$ .

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Problem 7171

Kayla's utility is $U=q_{1}^{0.6} q_{2}^{0.4}$ . If $p_{1}$ rises from \$15 to \$60, find her compensating variation (CV) and equivalent variation (EV).

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Problem 7172

Find the derivative of the inverse function at 11 for $f(x)=x^{3}+x+1$ : $\left(f^{-1}\right)^{\prime}(11)$ .

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Problem 7173

An arrow shot upward on the moon at $58 \mathrm{~m/s}$ has height $h(t)=58t-0.83t^{2}$ . Find: a. Velocity at $t=a$ ? b. Velocity after 1s? c. When does it hit the moon? d. Impact velocity?

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Problem 7174

Find the average rate of change of $f(x)=-2 x^{2}-5 x+7$ on $[-5,5]$ . Average rate of change $=$

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Problem 7175

Find the one-sided limit: $\lim _{x \rightarrow(\pi / 2)^{+}} \frac{-2}{\cos x}$ . Justify your answer.

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Problem 7176

Find the limit as $n \to \infty$ for the product $\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{2n-1}{2n}$ .

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Problem 7177

Determine the horizontal asymptote of the function $f(x)=\frac{10 x}{7 x^{2}+8}$ .

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Problem 7178

Find the sum of the series: $\sum_{n=1}^{n} \frac{1}{n(n+1)}$ .

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Problem 7179

Find the tangent line equation for $y=(x^{4}-2)^{3}$ at $(1,-1)$ . Answer in slope-intercept form: $y=$ .

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Problem 7180

Untersuchen Sie die Funktion $f(x)=\frac{1}{20} x^{5}-\frac{2}{3} x^{3}+3 x$ auf Extremstellen, Wendepunkte und Verhalten.

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Problem 7181

Find the derivative $\frac{d x}{d p}$ for the equation $x=81 \sqrt{p+9}-350$ .

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Problem 7182

Calculate the integral of $f(x) = -7x^6 - 7x^5 - 8x^4 - 4x^3 + 5x - 10$ and call it $F(x)$ .

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Problem 7183

Bestimme die Ordinatenschnittpunkte der Senkrechten zur Geraden mit mittlerer Steigung zwischen den Extrema von $f(x)=x^{3}-4 x^{2}+4 x$ .

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Problem 7184

Given that $\lim _{x \rightarrow-9^{-}} m(x)=2$ and $\lim _{x \rightarrow-9^{+}} m(x)=2$ , but $m(-9)$ is undefined, does $m$ have a hole or vertical asymptote at $x=-9$ ? Explain.

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Problem 7185

Verständnischeck:
1. Beschreiben Sie, wie man den Flächeninhalt von $f(x)=x^{2}-1$ ohne Hauptsatz bestimmen kann.
2. Berechnen Sie $\int_{-1}^{1}\left(x^{2}-1\right) dx$ .
3. Erklären Sie den Zusammenhang zwischen 1. und 2.

See Solution

Problem 7186

A street light is on a 15 ft pole. Joe, 6 ft tall, walks away from it at 6 ft/s. Find the shadow's tip speed when he's 13 ft away.

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Problem 7187

Find the tangent line equation for $p(z)=\frac{z \tan (z)}{z^{2} \sec (z)+1}+3 e^{z}+1$ at $z=0$ . Is $p$ increasing, decreasing, or neither at $z=0$ ? Why?

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Problem 7188

Bei einer Bergtour ist die Leistung $P(t)=-\frac{1}{3240} t^{3}+\frac{1}{36} t^{2}$ .
a) Finde $W(t)$ .
b) Berechne die Arbeit für 90 Minuten.
c) Bestimme den Zeitpunkt der maximalen Leistung.

See Solution

Problem 7189

Berechnen Sie das Integral $\int_{-1}^{1}(x^{2}-1) \, dx$ .

See Solution

Problem 7190

Differentiate $m(x)=\frac{e^{x}(x^{2}+7 x)}{6 \sec x}$ .

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Problem 7191

Bestimme die Halbwertszeit eines radioaktiven Stoffes, wenn nach 7 Stunden etwa 15\% zerfallen sind.

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Problem 7192

Ordnen Sie die Kärtchen A bis F den passenden Funktionen und deren Ableitungen zu.

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Problem 7193

Finde Werte für a, damit der Graph $G_{f_{a}}$ rechtsgekrümmt ist für: a) $f_{a}(x)=a \cdot(2 x-3)^{4}$ b) $f_{a}(x)=\sqrt{a^{2} x}(x \geq 0)$ c) $f_{a}(x)=\frac{a}{x^{2}}(x \neq 0)$ Gib ein Gegenbeispiel für die falsche Aussage an: Wenn $f^{\prime}$ streng monoton wachsend ist, dann ist auch $f$ streng monoton wachsend.

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Problem 7194

Find the derivative of $h(x)=\frac{7 a x+b}{10 c}$ , where $a, b, c$ are constants. $h^{\prime}(x)=$

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Problem 7195

Bestimmen Sie die Extrem- und Wendepunkte der Funktionen: a) $f(x)=x^{4}-4 x^{3}$ , b) $g(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{3}+3 x^{2}$ , c) $h(x)=x^{6}-1,5 x^{4}$ .

See Solution

Problem 7196

Find the derivative of $f(x)=\frac{(x^{2}+3)^{2/3}(3x-4)^{4}}{\sqrt{x}}$ and compute $f^{\prime}(x) \cdot f^{\prime}(x)$ .

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Problem 7197

Bestimme die Ableitungen der Funktionen $f(x)=-x^{2}+4$ und $g(x)=x^{2}-5x+6$ .

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Problem 7198

Gegeben sind die Funktionen $f(x)=(x+3)^{2}$ , $f(x)=2 \cdot \sin (x)-x^{-\frac{1}{2}}$ , $f(x)=\sqrt{x}-\frac{1}{\sqrt[4]{x}}$ . Korrigieren Sie die fehlerhaften Ableitungen A, B und C.

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Problem 7199

Find the tangent line equation to $y=2 x^{3}-6 x^{2}+2 x-8$ at $x=2$ . Complete the line equation.

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Problem 7200

Bestimmen Sie die Ableitung von $f(x)=(x+1)^{2} \cdot (x-2)$ .

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Calculus

Problem 7101

Find the growth rate of infections after 10 days using I(t)=1100e0.038tI(t)=1100 e^{0.038 t}I(t)=1100e0.038t. Round to the nearest tenth.

Problem 7102

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=(4x+3)−1f(x)=(4 x+3)^{-1}f(x)=(4x+3)−1 and calculate f′(4)f^{\prime}(4)f′(4).

Problem 7103

Find the rate of change of distance from the origin to the point on the curve y=x2+2y=x^{2}+2y=x2+2 if dxdt=2\frac{dx}{dt}=2dtdx​=2 cm/s.

Problem 7104

Find F′(2)F^{\prime}(2)F′(2) for F(x)=f(g(x))F(x)=f(g(x))F(x)=f(g(x)) given f(−3)=7f(-3)=7f(−3)=7, f′(−3)=3f^{\prime}(-3)=3f′(−3)=3, g(2)=−3g(2)=-3g(2)=−3, and g′(2)=2g^{\prime}(2)=2g′(2)=2.

Problem 7105

Find the derivative f′(x)f'(x)f′(x) for f(x)=(x2+3x+6)4f(x)=(x^{2}+3x+6)^{4}f(x)=(x2+3x+6)4 and compute f′(2)f'(2)f′(2).

Problem 7106

Find r′(1)r'(1)r′(1) for r(x)=f(g(h(x)))r(x)=f(g(h(x)))r(x)=f(g(h(x))) given h(1)=4h(1)=4h(1)=4, g(4)=3g(4)=3g(4)=3, h′(1)=5h'(1)=5h′(1)=5, g′(4)=4g'(4)=4g′(4)=4, f′(3)=8f'(3)=8f′(3)=8.

Problem 7107

Given functions fff and ggg, find h′(3)h^{\prime}(3)h′(3) for h(x)=f(g(x))h(x)=f(g(x))h(x)=f(g(x)) and H′(1)H^{\prime}(1)H′(1) for H(x)=g(f(x))H(x)=g(f(x))H(x)=g(f(x)).

Problem 7108

A star's brightness is modeled by B(t)=4.0+0.45sin⁡(2πt4.5)B(t)=4.0+0.45 \sin \left(\frac{2 \pi t}{4.5}\right)B(t)=4.0+0.45sin(4.52πt​). Find dBdt\frac{d B}{d t}dtdB​ and at t=1t=1t=1.

Problem 7109

Find the velocity and acceleration of a mass on a spring with s(t)=5sin⁡(4t)s(t)=5 \sin (4 t)s(t)=5sin(4t) at t=4t=4t=4 seconds. Round to nearest hundredth.

Problem 7110

Problem 7111

Analyze the beer market with demand Q=140−4pQ=140-4pQ=140−4p and supply Q=20+2p−42hQ=20+2p-42hQ=20+2p−42h. Find dp/dh\mathrm{dp}/\mathrm{dh}dp/dh, dQ/dh\mathrm{dQ}/\mathrm{dh}dQ/dh, and changes in ppp, QQQ when hhh increases by \$2.

Problem 7112

Find critical points of f(x)=x3+6x2−180x−49f(x)=x^{3}+6 x^{2}-180 x-49f(x)=x3+6x2−180x−49. Options: x=−10,x=6x=-10, x=6x=−10,x=6; x=−49x=-49x=−49; x=0,−10,6x=0, -10, 6x=0,−10,6; x=0x=0x=0; x=3,10,−6x=3, 10, -6x=3,10,−6; None.

Problem 7113

Find the snail's acceleration given its position over days: 0 ft at day 0, 11 ft at day 1, 22 ft at day 2, etc.

Problem 7114

Find the difference quotient, f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​, for the function f(x)=4x2+7x−3f(x)=4 x^{2}+7 x-3f(x)=4x2+7x−3.

Problem 7115

Find the satellite's period if its orbit radius is 5.08 times Earth's radius (6.38×106 m6.38 \times 10^{6} \mathrm{~m}6.38×106 m). Earth's mass is 5.98×1024 kg5.98 \times 10^{24} \mathrm{~kg}5.98×1024 kg.

Problem 7116

Find the critical numbers of the function f(x)f(x)f(x) given its derivative f′(x)=x2−3x(x−5)3(x−6)3f^{\prime}(x)=\frac{x^{2}-3 x}{(x-5)^{3}(x-6)^{3}}f′(x)=(x−5)3(x−6)3x2−3x​.

Problem 7117

Find the tangent line equation for f(x)=3cos⁡(x)f(x)=3 \cos (x)f(x)=3cos(x) at x=4π3x=\frac{4 \pi}{3}x=34π​. Express it as y=y=y=.

Problem 7118

Find the local extreme points of the rational function f(x)f(x)f(x) with vertical asymptotes at x=−6x=-6x=−6 and x=4x=4x=4.

Problem 7119

Evaluate the Riemann sum RS(n)R S(n)RS(n) for I=∫02x dxI=\int_{0}^{2} x \, dxI=∫02​xdx and find the optimal statement about the error E(n)E(n)E(n).

Problem 7120

Find the tangent line equation for y=6x2−ln⁡(x)y=6 x^{2}-\ln (x)y=6x2−ln(x) at (1,6) and graph it with the function.

Problem 7121

Find all critical points of the function f(x)=(2x2−3x)13f(x)=(2x^2-3x)^{\frac{1}{3}}f(x)=(2x2−3x)31​.

Problem 7122

Evaluate the Riemann sum for I=∫02x dx=2I=\int_{0}^{2} x \, dx=2I=∫02​xdx=2 using right endpoints. Determine the optimal error E(n)E(n)E(n).

Problem 7123

Find the derivative of y=32(12ln⁡x+1x−1+arctan⁡x)y=\frac{3}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)y=23​(21​lnx−1x+1​+arctanx). What is dydx\frac{d y}{d x}dxdy​?

Problem 7124

Find the tangent line equation using implicit differentiation for x2+xarctan⁡y=y−1x^{2}+x \arctan y=y-1x2+xarctany=y−1 at the point (−π4,1)\left(-\frac{\pi}{4}, 1\right)(−4π​,1).

Problem 7125

Find the tangent line equation for f(x)=12xln⁡x4f(x)=\frac{1}{2} x \ln x^{4}f(x)=21​xlnx4 at the point in the interval (-1,0).

Problem 7126

Approximate 64.2\sqrt{64.2}64.2​ using the tangent line of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=64x=64x=64. Find the equation L(x)=L(x)=L(x)=. Give answer to 9 significant figures.

Problem 7127

Find the average rate of change of f(x)=2x2+7f(x)=2 x^{2}+7f(x)=2x2+7 between x=2x=2x=2 and x=4x=4x=4. Simplify your answer.

Problem 7128

Find the tangent line equation using implicit differentiation for x2+xarctan⁡y=y−1x^{2}+x \arctan y=y-1x2+xarctany=y−1 at (−π4,1)\left(-\frac{\pi}{4}, 1\right)(−4π​,1).

Problem 7129

Approximate 36.2\sqrt{36.2}36.2​ using the tangent line of f(x)=xf(x)=\sqrt{x}f(x)=x​ at x=36x=36x=36. Find L(x)=L(x)=L(x)=.

Problem 7130

Find the derivative of f(x)=3x−4x2f(x)=3x-4x^{2}f(x)=3x−4x2 using the definition of derivative only.

Problem 7131

Find the tangent line equation for f(x)=5sec⁡(x)f(x)=5 \sec (x)f(x)=5sec(x) at x=5π6x=\frac{5 \pi}{6}x=65π​. Use yyy and xxx.

Problem 7132

Approximate 10.104\frac{1}{0.104}0.1041​ using linear approximation with f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ at a nearby "nice" point.

Problem 7133

Approximate 27.23\sqrt[3]{27.2}327.2​ using the tangent line of f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ at x=27x=27x=27. Find mmm and bbb for y=mx+by=mx+by=mx+b.

Problem 7134

Given values for f,g,f′,g′f, g, f', g'f,g,f′,g′ at x=1,2,3x=1,2,3x=1,2,3, find: a) h′(3)h'(3)h′(3) for h(x)=7x⋅f(x)h(x)=7x \cdot f(x)h(x)=7x⋅f(x); b) h′(2)h'(2)h′(2) for h(x)=f(x)g(x)h(x)=f(x)g(x)h(x)=f(x)g(x); c) h′(1)h'(1)h′(1) for h(x)=g(f(x))h(x)=g(f(x))h(x)=g(f(x)).

Problem 7135

Two dinosaurs walk from the same point: one north at 4mi/hr4 \mathrm{mi/hr}4mi/hr and the other east at 3mi/hr3 \mathrm{mi/hr}3mi/hr. Find the distance increase rate after 3 hours.

Problem 7136

Find the derivative of the function h(x)=cot⁡(x)16−4xh(x)=\frac{\cot (x)}{16-4 x}h(x)=16−4xcot(x)​ for 0≤x≤π0 \leq x \leq \pi0≤x≤π. What is h′(x)h^{\prime}(x)h′(x)?

Problem 7137

Solve dxdt=2x\frac{d x}{d t}=2 xdtdx​=2x with x(0)=1x(0)=1x(0)=1. Find t1t_{1}t1​ when x(t1)=2x(t_{1})=2x(t1​)=2.

Problem 7138

Find the derivative of h(t)=18csc⁡(t)+icot⁡(t)h(t) = 18 \csc(t) + i \cot(t)h(t)=18csc(t)+icot(t). What is h′(t)h'(t)h′(t)?

Problem 7139

Determine which function pairs (x1(t),y1(t))(x_1(t), y_1(t))(x1​(t),y1​(t)) and (x2(t),y2(t))(x_2(t), y_2(t))(x2​(t),y2​(t)) solve the ODE: x˙=2x+y\dot{x}=2x+yx˙=2x+y, y˙=3y\dot{y}=3yy˙​=3y. Functions: (e2t,0)(e^{2t}, 0)(e2t,0) and (e3t,e3t)(e^{3t}, e^{3t})(e3t,e3t).

Problem 7140

Find the derivative of h(t)=18csc⁡(t)+tcot⁡(t)h(t)=18 \csc (t)+t \cot (t)h(t)=18csc(t)+tcot(t). What is h′(t)=?h^{\prime}(t)=?h′(t)=?

Find the growth rate of infections after 10 days using $I(t)=1100 e^{0.038 t}$ . Round to the nearest tenth.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=(4 x+3)^{-1}$ and calculate $f^{\prime}(4)$ .

Find the rate of change of distance from the origin to the point on the curve $y=x^{2}+2$ if $\frac{dx}{dt}=2$ cm/s.

Find $F^{\prime}(2)$ for $F(x)=f(g(x))$ given $f(-3)=7$ , $f^{\prime}(-3)=3$ , $g(2)=-3$ , and $g^{\prime}(2)=2$ .

Find the derivative $f'(x)$ for $f(x)=(x^{2}+3x+6)^{4}$ and compute $f'(2)$ .

Find $r'(1)$ for $r(x)=f(g(h(x)))$ given $h(1)=4$ , $g(4)=3$ , $h'(1)=5$ , $g'(4)=4$ , $f'(3)=8$ .

Given functions $f$ and $g$ , find $h^{\prime}(3)$ for $h(x)=f(g(x))$ and $H^{\prime}(1)$ for $H(x)=g(f(x))$ .

A star's brightness is modeled by $B(t)=4.0+0.45 \sin \left(\frac{2 \pi t}{4.5}\right)$ . Find $\frac{d B}{d t}$ and at $t=1$ .

Find the velocity and acceleration of a mass on a spring with $s(t)=5 \sin (4 t)$ at $t=4$ seconds. Round to nearest hundredth.

Analyze the beer market with demand $Q=140-4p$ and supply $Q=20+2p-42h$ . Find $\mathrm{dp}/\mathrm{dh}$ , $\mathrm{dQ}/\mathrm{dh}$ , and changes in $p$ , $Q$ when $h$ increases by \$2.

Find critical points of $f(x)=x^{3}+6 x^{2}-180 x-49$ . Options: $x=-10, x=6$ ; $x=-49$ ; $x=0, -10, 6$ ; $x=0$ ; $x=3, 10, -6$ ; None.

Find the difference quotient, $\frac{f(x+h)-f(x)}{h}$ , for the function $f(x)=4 x^{2}+7 x-3$ .

Find the satellite's period if its orbit radius is 5.08 times Earth's radius ( $6.38 \times 10^{6} \mathrm{~m}$ ). Earth's mass is $5.98 \times 10^{24} \mathrm{~kg}$ .

Find the critical numbers of the function $f(x)$ given its derivative $f^{\prime}(x)=\frac{x^{2}-3 x}{(x-5)^{3}(x-6)^{3}}$ .

Find the tangent line equation for $f(x)=3 \cos (x)$ at $x=\frac{4 \pi}{3}$ . Express it as $y=$ .

Find the local extreme points of the rational function $f(x)$ with vertical asymptotes at $x=-6$ and $x=4$ .

Evaluate the Riemann sum $R S(n)$ for $I=\int_{0}^{2} x \, dx$ and find the optimal statement about the error $E(n)$ .

Find the tangent line equation for $y=6 x^{2}-\ln (x)$ at (1,6) and graph it with the function.

Find all critical points of the function $f(x)=(2x^2-3x)^{\frac{1}{3}}$ .

Evaluate the Riemann sum for $I=\int_{0}^{2} x \, dx=2$ using right endpoints. Determine the optimal error $E(n)$ .

Find the derivative of $y=\frac{3}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)$ . What is $\frac{d y}{d x}$ ?

Find the tangent line equation using implicit differentiation for $x^{2}+x \arctan y=y-1$ at the point $\left(-\frac{\pi}{4}, 1\right)$ .

Find the tangent line equation for $f(x)=\frac{1}{2} x \ln x^{4}$ at the point in the interval (-1,0).

Approximate $\sqrt{64.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=64$ . Find the equation $L(x)=$ . Give answer to 9 significant figures.

Find the average rate of change of $f(x)=2 x^{2}+7$ between $x=2$ and $x=4$ . Simplify your answer.

Find the tangent line equation using implicit differentiation for $x^{2}+x \arctan y=y-1$ at $\left(-\frac{\pi}{4}, 1\right)$ .

Approximate $\sqrt{36.2}$ using the tangent line of $f(x)=\sqrt{x}$ at $x=36$ . Find $L(x)=$ .

Find the derivative of $f(x)=3x-4x^{2}$ using the definition of derivative only.

Find the tangent line equation for $f(x)=5 \sec (x)$ at $x=\frac{5 \pi}{6}$ . Use $y$ and $x$ .

Approximate $\frac{1}{0.104}$ using linear approximation with $f(x)=\frac{1}{x}$ at a nearby "nice" point.

Approximate $\sqrt[3]{27.2}$ using the tangent line of $f(x)=\sqrt[3]{x}$ at $x=27$ . Find $m$ and $b$ for $y=mx+b$ .

Given values for $f, g, f', g'$ at $x=1,2,3$ , find: a) $h'(3)$ for $h(x)=7x \cdot f(x)$ ; b) $h'(2)$ for $h(x)=f(x)g(x)$ ; c) $h'(1)$ for $h(x)=g(f(x))$ .

Two dinosaurs walk from the same point: one north at $4 \mathrm{mi/hr}$ and the other east at $3 \mathrm{mi/hr}$ . Find the distance increase rate after 3 hours.

Find the derivative of the function $h(x)=\frac{\cot (x)}{16-4 x}$ for $0 \leq x \leq \pi$ . What is $h^{\prime}(x)$ ?

Solve $\frac{d x}{d t}=2 x$ with $x(0)=1$ . Find $t_{1}$ when $x(t_{1})=2$ .

Find the derivative of $h(t) = 18 \csc(t) + i \cot(t)$ . What is $h'(t)$ ?

Determine which function pairs $(x_1(t), y_1(t))$ and $(x_2(t), y_2(t))$ solve the ODE: $\dot{x}=2x+y$ , $\dot{y}=3y$ . Functions: $(e^{2t}, 0)$ and $(e^{3t}, e^{3t})$ .

Find the derivative of $h(t)=18 \csc (t)+t \cot (t)$ . What is $h^{\prime}(t)=?$

Evaluate the integral from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$ of $\left(x^{2}+\ln |\cos x|\right) \sin \frac{x}{2} \, dx$ . Provide a numeric answer.

Find the derivative of the function $f(x)=5 x \tan(x)$ .

Find the derivative of $f(\theta)=2 \theta \tan (\theta) \sec (\theta)$ . What is $f^{\prime}(\theta)$ ?

Check if $x_{1}(t)=t+1$ and $x_{2}(t)=\frac{1+\sqrt{4 t^{2}+1}}{2}$ solve the ODE $\frac{d x}{d t}=\frac{t}{x-1}$ .

Find the temperature change rate:
a. From 6 A.M. to 7 A.M. (temp at 6 A.M. unknown).
b. From 1 P.M. to 2 P.M.

Determine if $x_{1}(t)=t+1$ and $x_{2}(t)=\frac{1+\sqrt{4 t^{2}+1}}{2}$ solve the ODE: $\frac{d x}{d t}=\frac{t}{x-1}$ .

Find the derivative of $f(x)=x^{5} e^{x} \cos (x)$ . What is $f^{\prime}(x)=?$

Find the tangent line equation for $f(t)=\frac{\sin (t)}{11+11 \cos (t)}$ at $t=\frac{\pi}{3}$ .

Analyze the system of differential equations to classify the origin as a saddle, source, or sink for $a=-0.01$ and $d=-2.4$ .

Calculate the difference quotient for the function $f(x)=7-2x+5x^{2}$ .

Jan pedals on a bike, with foot height $h(t)=-0.18 \sin (t)+0.30$ . Find vertical velocity $v$ at $t=3\pi$ .

Find the velocity and acceleration of the spring mass at $t=\frac{\pi}{3}$ s, given $s(t)=300+76 \sin(t)$ .

Find the tangent line equation for $y=9 \csc (x)-9 \cot (x)$ at $x=\frac{\pi}{6}$ . Use $y=f(x)$ .

Find the tangent line equation for $y=5 \csc (x)-5 \cot (x)$ at $x=\frac{3 \pi}{4}$ .

Find the tangent line equation for $f(x)=26 e^{x} \cos ^{2}(x)$ at $x=\frac{\pi}{4}$ .

Calculate the integral of $f(x)=6 x^{4}-\frac{2}{3} x^{3}+5 x-2$ .

Find $d y$ for $y=\tan(2x+2)$ at $x=4$ with $d x=0.3$ and $d x=0.6$ .

Uprość sumę szeregu $\sum_{k=1}^{n} \frac{1}{k^{2}}$ .

Bestimmen Sie die Monotonieintervalle der Funktionen: a) $f(x)=\sqrt{x} \cdot(x-9), x>0$ b) $f(x)=(x+1)^{2} \cdot(x-2)$ c) $f(x)=4 x^{2} \cdot(1-x)^{2}$

Fill in the table for $\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}$ using values of $b$ : 0, 0.5, 1.

Find the limit: $\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}$ .

Find the arc length integral for the curve $x=\sqrt{2y+8}$ from $y=1$ to $y=3$ . Select A or B and fill in the boxes.

Find and simplify the integral for the arc length of the curve $x=\sqrt{2y+8}$ from $y=1$ to $y=3$ .

Ubung $5^{* * *}$ :
a) Zeigen Sie, dass der Grenzwert der Folge $a_{n}=\left(1+\frac{1}{n}\right)^{n}$ die Zahl $e$ ist.
b) Beweisen Sie, dass $2 e$ nicht der Grenzwert von $a_{n}$ ist.

Calculate the value of the function $f(x)=e^{x}$ at $x=9.2$ .

Find $f^{\prime}(-1)$ for the function $f(x)=2^{x^{2}-3 x}$ .

Gegeben ist die Funktion $f(x) = -x^{3} + 3x^{2}$ . Bestimmen Sie Hoch- und Wendepunkt, Tangentengleichung, Flächeninhalt des Dreiecks und Winkel $\alpha$ .

Kayla's utility is $U=q_{1}^{0.6} q_{2}^{0.4}$ . If $p_{1}$ rises from \$15 to \$60, find her compensating variation (CV) and equivalent variation (EV).

Find the derivative of the inverse function at 11 for $f(x)=x^{3}+x+1$ : $\left(f^{-1}\right)^{\prime}(11)$ .

An arrow shot upward on the moon at $58 \mathrm{~m/s}$ has height $h(t)=58t-0.83t^{2}$ . Find: a. Velocity at $t=a$ ? b. Velocity after 1s? c. When does it hit the moon? d. Impact velocity?

Find the average rate of change of $f(x)=-2 x^{2}-5 x+7$ on $[-5,5]$ . Average rate of change $=$