Calculus

Problem 15101

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

See Solution

Problem 15102

Find the marginal revenue from the demand function $p=\frac{15}{\ln (q+1)}$ . Calculate $\frac{d r}{d q}=\square$ .

See Solution

Problem 15103

Differentiate the function $y=(9 x)^{x}$ .

See Solution

Problem 15104

Find the slope of the tangent line at $x=-2$ for $h(x)=\frac{(3x-4)^{2}}{x}$ .

See Solution

Problem 15105

Find the critical numbers of the function $f(x)=x^{2} \sqrt{4-x}$ on the interval $[1,4]$ .

See Solution

Problem 15106

Find the derivative of the function $h(x)=7^{x^{7}}$ . What is $h^{\prime}(x)=\square$ ?

See Solution

Problem 15107

Find the rate of change of pressure $P(x)=10^{4} e^{-0.00012 x}$ at $x=1020$ m and $x=1240$ m, rounded to two decimals.

See Solution

Problem 15108

Find the slope of the tangent line to $h(x)=f(2x^{2}-x)$ at $x=-1$ for $f(x)=\frac{1}{2}x+3.5$ .

See Solution

Problem 15109

Find the marginal-cost function from the average cost $\bar{c}=\frac{3250 e^{\frac{q}{650}}}{q}$ for $q=325$ and $q=650$ .

See Solution

Problem 15110

Find the number of tractors $x$ that minimizes the cost given by $C(x)=11000-60x+0.1x^{2}$ .

See Solution

Problem 15111

Find the rate of change of price $p$ with respect to quantity $q$ for $p=e^{-0.004 q}$ at $q=400$ .

See Solution

Problem 15112

A cylinder's radius increases at 7 ft/min. Volume is constant at 554 ft³. Find height change rate when radius is $8$ ft. Use $V=\pi r^{2} h$ . Round to three decimal places.

See Solution

Problem 15113

Find the general antiderivative of $f(x)=33 x^{2}$ and verify by differentiation. What is $\int f(x) d x =$ ?

See Solution

Problem 15114

Find the antiderivative of $f(x)=5 x^{-3/5}$ and verify by differentiating. $\int f(x) dx =$

See Solution

Problem 15115

Find the marginal-cost function for $\bar{c}=\frac{9500 e^{\frac{q}{950}}}{q}$ at $q=475$ and $q=950$ .

See Solution

Problem 15116

Find the antiderivative of $f(x)=29 x+17 x^{-2}$ and verify by differentiating. Use $C$ for the constant. $\int f(x) dx =$

See Solution

Problem 15117

Find $f^{\prime}$ and $f$ given $f^{\prime \prime}(x)=\cos(x)$ , $f^{\prime}(\frac{\pi}{2})=15$ , $f(\frac{\pi}{2})=6$ .

See Solution

Problem 15118

Show that the function $f(x)=2x^2+3x+5$ takes a specific value using the Intermediate Value Theorem in a given interval.

See Solution

Problem 15119

Determine where the function $f(x)=\frac{x+5}{x^{2}-3x-40}$ is increasing or decreasing.

See Solution

Problem 15120

Calculate the limit of $\frac{2x+h}{1}+\frac{h^{2}}{x^{2}h^{2}+xh^{3}}$ as $h$ approaches 0.

See Solution

Problem 15121

Find the marginal-cost function and its values for $\bar{c}=\frac{6500 e^{\frac{q}{650}}}{q}$ at $q=325$ and $q=650$ .

See Solution

Problem 15122

A hammer falls for 7 seconds under gravity $-9.8 \mathrm{~m/s}^{2}$ . Find the distance fallen, $s(7)$ . Round to one decimal place.

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Problem 15123

A $3600-\mathrm{kg}$ rocket's velocity $v(m)$ changes as mass $m$ decreases. Find $v(2809)$ given $\frac{d v}{d m}=-40 \mathrm{~m}^{-1 / 2}$ and $v(3600)=0$ . $v(2809) =$ $\mathrm{m} / \mathrm{s}$

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Problem 15124

Calculate the limit: $\lim _{h \rightarrow 0} \frac{x^{2}+2 x h+h^{2}+\frac{1}{x+h}-x^{2}-\frac{1}{x}}{h}$ .

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Problem 15125

For the demand equation $p=1050-3 q$ , show demand is elastic and total revenue increases for $0<q<175$ , and inelastic with decreasing revenue for $175<q<350$ . Find $\eta$ using $\eta=\frac{p}{q} \cdot \frac{d q}{d p}$ and simplify.

See Solution

Problem 15126

Evaluate $f(x)=2 x^{2}+3 x+5$ using the Intermediate Value Theorem on the interval $[1,4]$ .

See Solution

Problem 15127

Find the limit: $\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)}{h}$ .

See Solution

Problem 15128

Differentiate $y=11 \ln \left(x^{2} \sqrt[3]{4 x+5}\right)$ .

See Solution

Problem 15129

Find the limit: $\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)^{2}}{h}$

See Solution

Problem 15130

Find constants $c_{1}$ and $c_{2}$ so that $F(x)=c_{1} x e^{-x}+c_{2} e^{-x}$ is an antiderivative of $f(x)=5 x e^{-x}$ .

See Solution

Problem 15131

Find the marginal-cost function and its values for $q=450$ and $q=900$ given $\bar{c}=\frac{9000 e^{\frac{q}{900}}}{q}$ .

See Solution

Problem 15132

Evaluate the integral $\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{\sqrt{x^{2}+y^{2}}}^{2}(x^{2}+y^{2}) \, dz \, dy \, dx$ . Use cylindrical coordinates.

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Problem 15133

Find the derivative of $f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})$ using the product rule.

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Problem 15134

Find the mass of a 2138-gram radioactive sample after 2 hours with a decay rate of $11\%$ per hour. Round to the nearest tenth.

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Problem 15135

Find the derivative of $f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})$ .

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Problem 15136

Find the indefinite integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ .

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Problem 15137

Evaluate the integral: $\int_{1}^{16} \int_{1}^{4}\left(\frac{x}{y}+\frac{y}{x}\right) d y d x$

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Problem 15138

Find the limit: $\lim _{x \rightarrow-1^{+}}\left[3-\log _{2}(x+1)\right]$ .

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Problem 15139

Calculez la dérivée de $f(x)=e^{-\sqrt{x}}+\ln(x^{3}+1)$ (30 points). Justifiez votre réponse.

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Problem 15140

Find the saddle point of the function where $f' = 6(x-2)(x+1)$ .

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Problem 15141

Find the derivative of $5 x^{2}-e^{4 y^{2}}=-6$ with respect to $x$ .

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Problem 15142

How long does it take a marble to fall 1420 feet using $h(t)=-16 t^{2}+1420$ ? Ignore air resistance.

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Problem 15143

Find the derivative of $\ln x e^{3 y} = 2 y^{2}$ with respect to $x$ .

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Problem 15144

Find the partial derivative of $x \sin 2y = y \cos 2x$ with respect to $x$ at $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ .

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Problem 15145

Find the consumer surplus when the demand function is $p(x)=11000-0.048 x^{2}$ and price is set at 8000.

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Problem 15146

Find the linearization $L(x)$ of $f(x)=\ln(x^2)$ at $x=e$ , estimate $f(3)$ , and explain the approximation's validity.

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Problem 15147

Find the consumers' surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

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Problem 15148

Find the marginal-cost function and its values for $\bar{c}=\frac{8000 e^{\frac{q}{800}}}{q}$ at $q=400$ and $q=800$ .

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Problem 15149

Determine the restrictions of $f(x)=\frac{x-4}{(x-1)^{2}(x+1)^{3}}$ for asymptotes and similar functions near them.

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Problem 15150

Use the Ratio Test on the series $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$ to determine convergence or divergence. Show your work.

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Problem 15151

Find the average cost per ton for mining 1000 tons, and the marginal cost at this production level using $C(x)=4200+5.40 x-0.001 x^{2}+0.000002 x^{3}$ .

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Problem 15152

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$ .

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Problem 15153

Determine if the series converge or diverge using the Ratio Test or another method. Show all work.
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$
2. $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$

See Solution

Problem 15154

Determine convergence/divergence of these series using the Ratio Test or another method. Show all work.
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$
2. $\sum_{n=0}^{\infty} \frac{(-3)^{n}}{n!}$
3. $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$

See Solution

Problem 15155

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5+n}$ .

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Problem 15156

Calculate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5+n}$ .

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Problem 15157

Determine convergence or divergence of these series using the Ratio Test or an alternate method. Show your work.
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$
2. $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$
3. $\sum_{n=0}^{\infty} \frac{(-3)^{n}}{n!}$

See Solution

Problem 15158

Calculate the sum: $\sum_{n=1}^{\infty} e^{-n}(n+2)!$

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Problem 15159

Determine the long term behavior of $f(x) = \frac{5x}{x + 9}$ and identify any horizontal asymptote.

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Problem 15160

Given the demand equation $q=\frac{55}{p}+\ln \left(9-p^{3}\right)$ , find:
(a) Elasticity at $p=2$ and classify it. (b) Estimate quantity change with a 2% price drop from \$2.00 to \$1.96. (c) Will revenue increase or decrease? Explain.

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Problem 15161

The cost function for mining $x$ tons is $C(x) = 4200 + 5.40x - 0.001x^2 + 0.000002x^3$ .
(a) Find average cost for 1000 tons. (b) Find marginal cost for 1000 tons. (c) Explain average cost change if production exceeds 1000 tons.

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Problem 15162

Find the long-term behavior of $f(x)=\frac{2 x^{2}+7 x-2}{6 x^{3}+5}$ . Does it have a horizontal asymptote? What is its equation?

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Problem 15163

Find the long-term behavior of $f(x)=\frac{5 x^{4}+3 x^{2}-8 x}{x^{3}+2}$ . Does it have a horizontal asymptote?

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Problem 15164

Analyze the function $f(x)=\frac{5 x^{4}+3 x^{2}-8 x}{x^{3}+2}$ for long term behavior and check for horizontal asymptotes.

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Problem 15165

Determine convergence or divergence of these series using the Ratio Test or an alternate method:
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$
2. $\sum_{n=0}^{\infty} \frac{(-3)^{n}}{n!}$
3. $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5+n}$
4. $\sum_{n=1}^{\infty} e^{-n}(n+2)$
Show all work.

See Solution

Problem 15166

Find the long-term behavior of $f(x)=\frac{(2 x-3)(x-7)}{x(x+2)(x-2)^{2}}$ and state if there's a horizontal asymptote.

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Problem 15167

Expand $f(z)=\frac{1}{z(1-z)^{2}}$ in a Laurent series for $|z|>1$ with terms $a_{k}(z-z_{0})^{k}$ for $-5 \leq k \leq 5$ .

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Problem 15168

Find $f'(x)$ if $f(x)=\int_{-2}^{x^{3}} t^{4} dt$ .

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Problem 15169

Approximate the area under the curve $y=x^{3}$ from $x=0$ to $x=2$ using 4 Right Endpoint subdivisions.

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Problem 15170

Find the displacement and total distance of an object moving with velocity $v(t)=t^{2}+2t-3$ from $t=0$ to $t=7$ .

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Problem 15171

Evaluate the definite integrals of the function: $\int_{0}^{2} f(x) dx = -3$ and $\int_{2}^{7} f(x) dx = 8$ .

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Problem 15172

Which term completes: Near a function's vertical asymptotes, values become very large positive or negative. A. undefined B. large C. rational D. small

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Problem 15173

Analyze the end behavior of $f(x)=\left(\frac{2}{3}\right)^{x}-2$ . What happens as $x \to \pm \infty$ ?

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Problem 15174

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

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Problem 15175

Find the derivative of $(t^{2}-4)^{2}+(2 t-14)^{2}+(2 t-13)^{2}$ .

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Problem 15176

Berechnen Sie das Integral: $\int_{1}^{3} 2 \, dx$ mit dem Hauptsatz der Differential- und Integralrechnung.

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Problem 15177

Determine the tangent plane equation for $z=e^{x}(\sin y+1)$ at point $P\left(0, \frac{\pi}{2}, 2\right)$ .

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Problem 15178

A ball is thrown down from a $58.0 \mathrm{~m}$ tower at $11.0 \mathrm{~m/s}$ . Find its speed before hitting the ground. $\mathrm{m/s}$

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Problem 15179

Differentiate $2xy + 3 = 0$ to find $\frac{dy}{dx}$ .

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Problem 15180

Differentiate $x^{2} y^{3} + x^{3} y^{4} = 11$ implicitly to find $\frac{dy}{dx}$ .

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Problem 15181

Differentiate $y^{5}=x^{3}$ implicitly to find $\frac{dy}{dx}$ .

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Problem 15182

Evaluate the integral $\int_{1}^{2} k x^{3} d x$ and express it as $\left[\frac{1}{4} k x^{4}\right]_{1}^{2}$ .

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Problem 15183

Welche Bedingung müssen die Koeffizienten a und $b$ der Funktion $f(x)=a x^{6}+b x^{4}$ erfüllen, damit $f$ zwei Wendestellen hat?

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Problem 15184

A 17-ft ladder leans against a wall. If the base moves away at $0.5 \mathrm{ft/s}$ , how fast does the top slide down after 12s?

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Problem 15185

Find the limit of $\frac{f(3+h)-f(3)}{h}$ for $f(x)=-2x^{2}$ .

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Problem 15186

Bestimme die Ableitung von $f(x) = x^{2}$ .

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Problem 15187

Find the limit of $\frac{f(2+h)-f(2)}{h}$ for $f(x)=0.5x^2$ .

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Problem 15188

Bestimme die x-Werte für waagrechte Tangenten der Funktion $f(x)=0,25 x^{4}-x^{3}+4$ und analysiere die Steigung.

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Problem 15189

Bestimme die Bedingungen für $\mathrm{a}, \mathrm{b}, \mathrm{c}$ , damit die Ableitung von $f(x)=a x^{3}+b x^{2}+c x+d$ 0, 1 oder 2 Nullstellen hat. Finde $\mathrm{a}$ , damit 6 eine lokale Extremstelle von $f(x)=x^{3}(a-x)$ ist und zeichne den Graphen. Wo ist $f(x)<0$ ?

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Problem 15190

Find the function $f(x)$ if $f^{\prime}(x)=-5 x^{-6}$ .

See Solution

Problem 15191

Skateboard in Halfpipe: Geschwindigkeit $f(x)=4 x \cdot(x-2) \cdot(x-3)$ , bestimme die Stammfunktion von 0 bis 3 s und deren Bedeutung.

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Problem 15192

Calculate the derivative of $f(x)$ at $a$ using the limit: $\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ .

See Solution

Problem 15193

Find the sum to infinity of the series $2+\frac{1}{2}+\frac{9}{8}+\frac{27}{32}+\ldots$

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Problem 15194

Eine Ampel regelt den Verkehr. Gegeben ist die Beschleunigung $f(x)=0,00002 x \cdot(x-60) \cdot(x-90)$ für $0 \leq x \leq 90$ .
a. Beschreibe die Fahrt des Autos. b. Berechne die Stammfunktion von $f(x)$ im Intervall $[0, 90]$ . c. Was bedeutet dieser Wert?

See Solution

Problem 15195

Bestimme die Ableitungen für die Funktionen: a) $f(x)=x^{3}$ , b) $f(x)=a x^{8}$ , c) $f(x)=x^{6}+3 a x^{2}-8 x$ , d) $f(x)=7 b x^{7}$ , e) $f(x)=9 x^{2}+12$ .

See Solution

Problem 15196

Eine Ampel regelt den Verkehr. Gegeben ist die Beschleunigung $f(x)=0,00002 x \cdot(x-60) \cdot(x-90)$ .
a. Beschreibe die Fahrt des Autos. b. Berechne die Stammfunktion von 0 bis 90 Sekunden. c. Was bedeutet dieser Wert?

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Problem 15197

Finde die Punkte, wo die Tangente von $f$ parallel zu $y=3x+4$ ist. a) $f(x)=x^{3}$ b) $f(x)=x^{2}$ c) $f(x)=x^{\frac{3}{2}}$ d) $f(x)=x^{6}$

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Problem 15198

Find $\frac{\partial z}{\partial x}$ for the implicit function defined by $F(x, y, z)=x^{3} e^{y+z}-y \sin (x-z)$ .

See Solution

Problem 15199

Bestimme die Ableitung von $-\sqrt{2} \cdot x^{3}$ .

See Solution

Problem 15200

Berechne die Ableitung der Funktion $f(x)=x^{5}$ .

See Solution

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Calculus

Problem 15101

Find the slope of the tangent line for f(x)=1+(x2−1)3f(x)=\sqrt{1+(x^{2}-1)^{3}}f(x)=1+(x2−1)3​ at x=2x=2x=2.

Problem 15102

Find the marginal revenue from the demand function p=15ln⁡(q+1)p=\frac{15}{\ln (q+1)}p=ln(q+1)15​. Calculate drdq=□\frac{d r}{d q}=\squaredqdr​=□.

Problem 15103

Differentiate the function y=(9x)xy=(9 x)^{x}y=(9x)x.

Problem 15104

Find the slope of the tangent line at x=−2x=-2x=−2 for h(x)=(3x−4)2xh(x)=\frac{(3x-4)^{2}}{x}h(x)=x(3x−4)2​.

Problem 15105

Find the critical numbers of the function f(x)=x24−xf(x)=x^{2} \sqrt{4-x}f(x)=x24−x​ on the interval [1,4][1,4][1,4].

Problem 15106

Find the derivative of the function h(x)=7x7h(x)=7^{x^{7}}h(x)=7x7. What is h′(x)=□h^{\prime}(x)=\squareh′(x)=□?

Problem 15107

Find the rate of change of pressure P(x)=104e−0.00012xP(x)=10^{4} e^{-0.00012 x}P(x)=104e−0.00012x at x=1020x=1020x=1020 m and x=1240x=1240x=1240 m, rounded to two decimals.

Problem 15108

Find the slope of the tangent line to h(x)=f(2x2−x)h(x)=f(2x^{2}-x)h(x)=f(2x2−x) at x=−1x=-1x=−1 for f(x)=12x+3.5f(x)=\frac{1}{2}x+3.5f(x)=21​x+3.5.

Problem 15109

Find the marginal-cost function from the average cost cˉ=3250eq650q\bar{c}=\frac{3250 e^{\frac{q}{650}}}{q}cˉ=q3250e650q​​ for q=325q=325q=325 and q=650q=650q=650.

Problem 15110

Find the number of tractors xxx that minimizes the cost given by C(x)=11000−60x+0.1x2C(x)=11000-60x+0.1x^{2}C(x)=11000−60x+0.1x2.

Problem 15111

Find the rate of change of price ppp with respect to quantity qqq for p=e−0.004qp=e^{-0.004 q}p=e−0.004q at q=400q=400q=400.

Problem 15112

A cylinder's radius increases at 7 ft/min. Volume is constant at 554 ft³. Find height change rate when radius is 888 ft. Use V=πr2hV=\pi r^{2} hV=πr2h. Round to three decimal places.

Problem 15113

Find the general antiderivative of f(x)=33x2f(x)=33 x^{2}f(x)=33x2 and verify by differentiation. What is ∫f(x)dx=\int f(x) d x =∫f(x)dx=?

Problem 15114

Find the antiderivative of f(x)=5x−3/5f(x)=5 x^{-3/5}f(x)=5x−3/5 and verify by differentiating. ∫f(x)dx=\int f(x) dx =∫f(x)dx=

Problem 15115

Find the marginal-cost function for cˉ=9500eq950q\bar{c}=\frac{9500 e^{\frac{q}{950}}}{q}cˉ=q9500e950q​​ at q=475q=475q=475 and q=950q=950q=950.

Problem 15116

Find the antiderivative of f(x)=29x+17x−2f(x)=29 x+17 x^{-2}f(x)=29x+17x−2 and verify by differentiating. Use CCC for the constant. ∫f(x)dx=\int f(x) dx = ∫f(x)dx=

Problem 15117

Find f′f^{\prime}f′ and fff given f′′(x)=cos⁡(x)f^{\prime \prime}(x)=\cos(x)f′′(x)=cos(x), f′(π2)=15f^{\prime}(\frac{\pi}{2})=15f′(2π​)=15, f(π2)=6f(\frac{\pi}{2})=6f(2π​)=6.

Problem 15118

Show that the function f(x)=2x2+3x+5f(x)=2x^2+3x+5f(x)=2x2+3x+5 takes a specific value using the Intermediate Value Theorem in a given interval.

Problem 15119

Determine where the function f(x)=x+5x2−3x−40f(x)=\frac{x+5}{x^{2}-3x-40}f(x)=x2−3x−40x+5​ is increasing or decreasing.

Problem 15120

Calculate the limit of 2x+h1+h2x2h2+xh3\frac{2x+h}{1}+\frac{h^{2}}{x^{2}h^{2}+xh^{3}}12x+h​+x2h2+xh3h2​ as hhh approaches 0.

Problem 15121

Find the marginal-cost function and its values for cˉ=6500eq650q\bar{c}=\frac{6500 e^{\frac{q}{650}}}{q}cˉ=q6500e650q​​ at q=325q=325q=325 and q=650q=650q=650.

Problem 15122

A hammer falls for 7 seconds under gravity −9.8 m/s2-9.8 \mathrm{~m/s}^{2}−9.8 m/s2. Find the distance fallen, s(7)s(7)s(7). Round to one decimal place.

Problem 15123

Problem 15124

Calculate the limit: lim⁡h→0x2+2xh+h2+1x+h−x2−1xh\lim _{h \rightarrow 0} \frac{x^{2}+2 x h+h^{2}+\frac{1}{x+h}-x^{2}-\frac{1}{x}}{h}limh→0​hx2+2xh+h2+x+h1​−x2−x1​​.

Problem 15125

Problem 15126

Evaluate f(x)=2x2+3x+5f(x)=2 x^{2}+3 x+5f(x)=2x2+3x+5 using the Intermediate Value Theorem on the interval [1,4][1,4][1,4].

Problem 15127

Find the limit: lim⁡h→0((x+h)2+1x+h)−(x2+1x)h\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)}{h}limh→0​h((x+h)2+x+h1​)−(x2+x1​)​.

Problem 15128

Differentiate y=11ln⁡(x24x+53)y=11 \ln \left(x^{2} \sqrt[3]{4 x+5}\right)y=11ln(x234x+5​).

Problem 15129

Find the limit: lim⁡h→0((x+h)2+1x+h)−(x2+1x)2h\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)^{2}}{h}limh→0​h((x+h)2+x+h1​)−(x2+x1​)2​

Problem 15130

Find constants c1c_{1}c1​ and c2c_{2}c2​ so that F(x)=c1xe−x+c2e−xF(x)=c_{1} x e^{-x}+c_{2} e^{-x}F(x)=c1​xe−x+c2​e−x is an antiderivative of f(x)=5xe−xf(x)=5 x e^{-x}f(x)=5xe−x.

Problem 15131

Find the marginal-cost function and its values for q=450q=450q=450 and q=900q=900q=900 given cˉ=9000eq900q\bar{c}=\frac{9000 e^{\frac{q}{900}}}{q}cˉ=q9000e900q​​.

Problem 15132

Evaluate the integral ∫−22∫−4−x24−x2∫x2+y22(x2+y2) dz dy dx\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{\sqrt{x^{2}+y^{2}}}^{2}(x^{2}+y^{2}) \, dz \, dy \, dx∫−22​∫−4−x2​4−x2​​∫x2+y2​2​(x2+y2)dzdydx. Use cylindrical coordinates.

Problem 15133

Find the derivative of f(x)=(3x5−2x2+1)(3−x3)f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})f(x)=(3x5−2x2+1)(3−x3) using the product rule.

Problem 15134

Find the mass of a 2138-gram radioactive sample after 2 hours with a decay rate of 11%11\%11% per hour. Round to the nearest tenth.

Problem 15135

Find the derivative of f(x)=(3x5−2x2+1)(3−x3)f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})f(x)=(3x5−2x2+1)(3−x3).

Problem 15136

Find the indefinite integral ∫x−3(3x−2)2dx\int \frac{x-3}{(3 x-2)^{2}} d x∫(3x−2)2x−3​dx using the substitution u=3x−2u=3 x-2u=3x−2.

Problem 15137

Evaluate the integral: ∫116∫14(xy+yx)dydx\int_{1}^{16} \int_{1}^{4}\left(\frac{x}{y}+\frac{y}{x}\right) d y d x∫116​∫14​(yx​+xy​)dydx

Problem 15138

Find the limit: lim⁡x→−1+[3−log⁡2(x+1)]\lim _{x \rightarrow-1^{+}}\left[3-\log _{2}(x+1)\right]limx→−1+​[3−log2​(x+1)].

Problem 15139

Calculez la dérivée de f(x)=e−x+ln⁡(x3+1)f(x)=e^{-\sqrt{x}}+\ln(x^{3}+1)f(x)=e−x​+ln(x3+1) (30 points). Justifiez votre réponse.

Problem 15140

Find the saddle point of the function where f′=6(x−2)(x+1)f' = 6(x-2)(x+1)f′=6(x−2)(x+1).

Problem 15141

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

Find the marginal revenue from the demand function $p=\frac{15}{\ln (q+1)}$ . Calculate $\frac{d r}{d q}=\square$ .

Differentiate the function $y=(9 x)^{x}$ .

Find the slope of the tangent line at $x=-2$ for $h(x)=\frac{(3x-4)^{2}}{x}$ .

Find the critical numbers of the function $f(x)=x^{2} \sqrt{4-x}$ on the interval $[1,4]$ .

Find the derivative of the function $h(x)=7^{x^{7}}$ . What is $h^{\prime}(x)=\square$ ?

Find the rate of change of pressure $P(x)=10^{4} e^{-0.00012 x}$ at $x=1020$ m and $x=1240$ m, rounded to two decimals.

Find the slope of the tangent line to $h(x)=f(2x^{2}-x)$ at $x=-1$ for $f(x)=\frac{1}{2}x+3.5$ .

Find the marginal-cost function from the average cost $\bar{c}=\frac{3250 e^{\frac{q}{650}}}{q}$ for $q=325$ and $q=650$ .

Find the number of tractors $x$ that minimizes the cost given by $C(x)=11000-60x+0.1x^{2}$ .

Find the rate of change of price $p$ with respect to quantity $q$ for $p=e^{-0.004 q}$ at $q=400$ .

A cylinder's radius increases at 7 ft/min. Volume is constant at 554 ft³. Find height change rate when radius is $8$ ft. Use $V=\pi r^{2} h$ . Round to three decimal places.

Find the general antiderivative of $f(x)=33 x^{2}$ and verify by differentiation. What is $\int f(x) d x =$ ?

Find the antiderivative of $f(x)=5 x^{-3/5}$ and verify by differentiating. $\int f(x) dx =$

Find the marginal-cost function for $\bar{c}=\frac{9500 e^{\frac{q}{950}}}{q}$ at $q=475$ and $q=950$ .

Find the antiderivative of $f(x)=29 x+17 x^{-2}$ and verify by differentiating. Use $C$ for the constant. $\int f(x) dx =$

Find $f^{\prime}$ and $f$ given $f^{\prime \prime}(x)=\cos(x)$ , $f^{\prime}(\frac{\pi}{2})=15$ , $f(\frac{\pi}{2})=6$ .

Show that the function $f(x)=2x^2+3x+5$ takes a specific value using the Intermediate Value Theorem in a given interval.

Determine where the function $f(x)=\frac{x+5}{x^{2}-3x-40}$ is increasing or decreasing.

Calculate the limit of $\frac{2x+h}{1}+\frac{h^{2}}{x^{2}h^{2}+xh^{3}}$ as $h$ approaches 0.

Find the marginal-cost function and its values for $\bar{c}=\frac{6500 e^{\frac{q}{650}}}{q}$ at $q=325$ and $q=650$ .

A hammer falls for 7 seconds under gravity $-9.8 \mathrm{~m/s}^{2}$ . Find the distance fallen, $s(7)$ . Round to one decimal place.

Calculate the limit: $\lim _{h \rightarrow 0} \frac{x^{2}+2 x h+h^{2}+\frac{1}{x+h}-x^{2}-\frac{1}{x}}{h}$ .

Evaluate $f(x)=2 x^{2}+3 x+5$ using the Intermediate Value Theorem on the interval $[1,4]$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)}{h}$ .

Differentiate $y=11 \ln \left(x^{2} \sqrt[3]{4 x+5}\right)$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\left((x+h)^{2}+\frac{1}{x+h}\right)-\left(x^{2}+\frac{1}{x}\right)^{2}}{h}$

Find constants $c_{1}$ and $c_{2}$ so that $F(x)=c_{1} x e^{-x}+c_{2} e^{-x}$ is an antiderivative of $f(x)=5 x e^{-x}$ .

Find the marginal-cost function and its values for $q=450$ and $q=900$ given $\bar{c}=\frac{9000 e^{\frac{q}{900}}}{q}$ .

Evaluate the integral $\int_{-2}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} \int_{\sqrt{x^{2}+y^{2}}}^{2}(x^{2}+y^{2}) \, dz \, dy \, dx$ . Use cylindrical coordinates.

Find the derivative of $f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})$ using the product rule.

Find the mass of a 2138-gram radioactive sample after 2 hours with a decay rate of $11\%$ per hour. Round to the nearest tenth.

Find the derivative of $f(x)=(3x^{5}-2x^{2}+1)(3-x^{3})$ .

Find the indefinite integral $\int \frac{x-3}{(3 x-2)^{2}} d x$ using the substitution $u=3 x-2$ .

Evaluate the integral: $\int_{1}^{16} \int_{1}^{4}\left(\frac{x}{y}+\frac{y}{x}\right) d y d x$

Find the limit: $\lim _{x \rightarrow-1^{+}}\left[3-\log _{2}(x+1)\right]$ .

Calculez la dérivée de $f(x)=e^{-\sqrt{x}}+\ln(x^{3}+1)$ (30 points). Justifiez votre réponse.

Find the saddle point of the function where $f' = 6(x-2)(x+1)$ .

Find the derivative of $5 x^{2}-e^{4 y^{2}}=-6$ with respect to $x$ .

How long does it take a marble to fall 1420 feet using $h(t)=-16 t^{2}+1420$ ? Ignore air resistance.

Find the derivative of $\ln x e^{3 y} = 2 y^{2}$ with respect to $x$ .

Find the partial derivative of $x \sin 2y = y \cos 2x$ with respect to $x$ at $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ .

Find the consumer surplus when the demand function is $p(x)=11000-0.048 x^{2}$ and price is set at 8000.

Find the linearization $L(x)$ of $f(x)=\ln(x^2)$ at $x=e$ , estimate $f(3)$ , and explain the approximation's validity.

Find the consumers' surplus when the price is set at 400 for the demand function $p(x)=1200-\left(\frac{1}{10000}\right) x^{3}$ .

Find the marginal-cost function and its values for $\bar{c}=\frac{8000 e^{\frac{q}{800}}}{q}$ at $q=400$ and $q=800$ .

Determine the restrictions of $f(x)=\frac{x-4}{(x-1)^{2}(x+1)^{3}}$ for asymptotes and similar functions near them.

Use the Ratio Test on the series $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$ to determine convergence or divergence. Show your work.

Find the average cost per ton for mining 1000 tons, and the marginal cost at this production level using $C(x)=4200+5.40 x-0.001 x^{2}+0.000002 x^{3}$ .

Find the sum of the series: $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$ .

Determine if the series converge or diverge using the Ratio Test or another method. Show all work.
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$
2. $\sum_{n=1}^{\infty} \frac{(-3)^{n}}{n^{3}}$

Evaluate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5+n}$ .

Calculate the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n-1} n}{5+n}$ .

Calculate the sum: $\sum_{n=1}^{\infty} e^{-n}(n+2)!$

Determine the long term behavior of $f(x) = \frac{5x}{x + 9}$ and identify any horizontal asymptote.

Given the demand equation $q=\frac{55}{p}+\ln \left(9-p^{3}\right)$ , find:
(a) Elasticity at $p=2$ and classify it. (b) Estimate quantity change with a 2% price drop from \$2.00 to \$1.96. (c) Will revenue increase or decrease? Explain.

The cost function for mining $x$ tons is $C(x) = 4200 + 5.40x - 0.001x^2 + 0.000002x^3$ .
(a) Find average cost for 1000 tons. (b) Find marginal cost for 1000 tons. (c) Explain average cost change if production exceeds 1000 tons.

Find the long-term behavior of $f(x)=\frac{2 x^{2}+7 x-2}{6 x^{3}+5}$ . Does it have a horizontal asymptote? What is its equation?

Find the long-term behavior of $f(x)=\frac{5 x^{4}+3 x^{2}-8 x}{x^{3}+2}$ . Does it have a horizontal asymptote?

Analyze the function $f(x)=\frac{5 x^{4}+3 x^{2}-8 x}{x^{3}+2}$ for long term behavior and check for horizontal asymptotes.

Find the long-term behavior of $f(x)=\frac{(2 x-3)(x-7)}{x(x+2)(x-2)^{2}}$ and state if there's a horizontal asymptote.

Expand $f(z)=\frac{1}{z(1-z)^{2}}$ in a Laurent series for $|z|>1$ with terms $a_{k}(z-z_{0})^{k}$ for $-5 \leq k \leq 5$ .

Find $f'(x)$ if $f(x)=\int_{-2}^{x^{3}} t^{4} dt$ .

Approximate the area under the curve $y=x^{3}$ from $x=0$ to $x=2$ using 4 Right Endpoint subdivisions.

Find the displacement and total distance of an object moving with velocity $v(t)=t^{2}+2t-3$ from $t=0$ to $t=7$ .

Evaluate the definite integrals of the function: $\int_{0}^{2} f(x) dx = -3$ and $\int_{2}^{7} f(x) dx = 8$ .

Analyze the end behavior of $f(x)=\left(\frac{2}{3}\right)^{x}-2$ . What happens as $x \to \pm \infty$ ?

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