Calculus

Problem 18101

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{x}$ .

See Solution

Problem 18102

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt[3]{1+7 x} d x$

See Solution

Problem 18103

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

See Solution

Problem 18104

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

See Solution

Problem 18105

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

See Solution

Problem 18106

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

See Solution

Problem 18107

Differentiate the function $f(x)=5 e^{8\left(x^{2}+2\right)^{2}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 18108

Find the relative maxima of $f(x)=x^{4}+x^{3}-6x^{2}$ using its derivative and the second derivative test.

See Solution

Problem 18109

Find the derivative of $y=2^{8 x+5}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 18110

Differentiate the function $f(x)=\ln(4-x^{2})$ . What is $f'(x)=\square$ ?

See Solution

Problem 18111

Find the derivative of $y=e^{\sin x}$ , i.e., calculate $\frac{d y}{d x}$ .

See Solution

Problem 18112

Find the limit: $\lim _{+\infty} \frac{\frac{\pi}{2}-\operatorname{arctg} x}{\ln \frac{x+1}{x-1}}$ .

See Solution

Problem 18113

If $\$ 1400$ is invested at $1.75 \%$ interest, how many years until it doubles? Round to the nearest hundredth.

See Solution

Problem 18114

Wyznacz granice: a) $\lim _{0} \frac{x-\operatorname{tg} x}{x^{3}}$ ; b) $\lim _{+\infty} \frac{\frac{\pi}{2}-\operatorname{arctg} x}{\ln(x+1)}$ ; c) $\lim _{+\infty} \frac{e^{x}+\sin 4 x}{e^{x}-\cos x}$ ; d) $\lim _{0+} \frac{\operatorname{ctg} x}{\ln(e^{x}-1)}$ ; e) $\lim _{0}\left(\frac{1}{x^{2}}-\operatorname{ctg}^{2} x\right)$ ; f) $\lim _{1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)$ ; g) $\lim _{+\infty}\left(x^{2} e^{\frac{1}{x}}-x^{2}-x\right)$ ; h) $\lim _{+\infty} x^{n} e^{-2 x}, n \in \mathbb{N}$ ; i) $\lim _{1} \operatorname{ctg} \frac{\pi x}{2} \ln(1-x)$ ; j) $\lim _{+\infty}\left(\frac{2}{\pi} \operatorname{arctg} x\right)^{x^{2}}$ ; k) $\lim (\operatorname{tg} x)^{\operatorname{ctg} x}; \lim (e^{3 x}+2 x)^{\frac{1}{x}}; \lim (\arcsin x)^{\operatorname{tg} x}; \lim (3^{x}-1)^{\ln \frac{1}{x}}$ .

See Solution

Problem 18115

Find the half-life of a substance with a decay rate of $9.3\%$ per day using the continuous exponential decay model.

See Solution

Problem 18116

Ein Wasserbecken hat Zufluss ( $300 \frac{1}{\min}$ ) und Abfluss ( $100 \frac{l}{\min}$ ). Berechne Liter nach 5, 10, 15, 20, 25, 30, 35 min.

See Solution

Problem 18117

Estimate the area under $f(x)=16-x^{2}$ from $x=0$ to $x=4$ using 4 rectangles: (A) right endpoints, (B) left endpoints.

See Solution

Problem 18118

Evaluate $\int_{3}^{5} \frac{4x^{2}-15x+11}{x-2} \, dx$ and simplify your answer, condensing logs if needed.

See Solution

Problem 18119

How long to double a bacteria population growing at $2\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 18120

Find the function $f(x)$ and the interval $[A, B]$ for the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}$ without evaluating it.

See Solution

Problem 18121

Find the half-life of a radioactive substance with a decay rate of $5\%$ per day. Round to the nearest hundredth.

See Solution

Problem 18122

Aufgabe: Flächenberechnung mit GTR a) Bestimme die Fläche unter $f(x)=\frac{1}{4} x^{3}+\frac{1}{2} x^{2}-2 x$ . b) Berechne die Fläche zwischen $f(x)=x^{2}+1$ und $g(x)=0,5 x+3$ . c) Zeige, dass $g(x)=x^{4}-3,75 x^{2}-1$ und $h(x)=x^{4}-3 x^{2}-4$ sind. Berechne die Fläche für das Logo.

See Solution

Problem 18123

Find the function $f(x)$ , and the interval $[A, B]$ for the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}$ . Use $A=0$ , $B=1$ .

See Solution

Problem 18124

A turkey cools from $185^{\circ}$ F in a $75^{\circ}$ F room.
(a) Find $T(45)$ if $T(30) = 144^{\circ}$ F.
(b) When is $T(t) = 100^{\circ}$ F?
Round answers to 2 decimal places.

See Solution

Problem 18125

What is the future value of \ $100 invested for 1 year at a$ 100\% $continuous compounding rate in terms of$ e$?

See Solution

Problem 18126

A bacteria culture had 300 after 20 min and 1400 after 40 min. Find initial size, doubling time, population after 90 min, and time to reach 15000.

See Solution

Problem 18127

1. Derive the labor supply function from the utility $U(C, L)=C-(1-L)^{2}$ with $h=1-L$ and no non-labor income. (5 marks)
2. Show the slope of an indifference curve as $\frac{\partial w}{\partial h}=\left(1-\frac{\bar{u}}{h^{2}}\right)$ using the budget constraint. (5 marks)
3. Prove that the slope of the indifference curve is zero at the optimal labor supply from part 2. (5 marks)
4. Write the profit function for the firm with production $f(E)=2E-E^{2}$ and derive the optimal labor demand. (5 marks)
5. Find the equilibrium wage, worker utility, and firm profit using the labor supply and demand functions. (5 marks)
6. In the monopoly union model, derive wage, employment, utility, and profit under the wage constraint on the labor demand function. (10 marks)
7. Explain why the monopoly union model outcome is inefficient. (5 marks)
8. Use the Lagrangean method to find optimal labor supply and wage under profit constraint $\left(\bar{\pi}=2h-h^{2}-wh\right)$ and compare outcomes. (10 marks)

See Solution

Problem 18128

Thema: Integralrechnung Flug eines Fesselballons
Eine Rettungsboje wird durch die Funktion $f$ konstruiert: $f: f(x)=\left\{\begin{array}{l}g(x) \text{ für } x \in[0 ; 8] \\ h(x) \text{ für } x \in(8 ; 12]\end{array}\right.$ mit $g(x)=\sqrt{c x}$ und $h(x)=a \sqrt{b-x}$ .
a) Bestimmen Sie $g(x)$ und $h(x)$ . b) Berechnen Sie die Fläche des Profilschnittes. c) Ermitteln Sie das Volumen. d) Bestimmen Sie die Oberfläche. e) Finden Sie den Schnittwinkel der Graphen von $g$ und $h$ .

See Solution

Problem 18129

Differentiate the function $\pi^{e}+\frac{1}{2 t}$ with respect to $t$ .

See Solution

Problem 18130

Calculate the integral $\int\left(t^{3}+\frac{1}{e}\right) d t$ .

See Solution

Problem 18131

Find the integral of the function $e^{\frac{x}{2}} + x$ .

See Solution

Problem 18132

Differentiate the function $\pi^{e}+\frac{1}{5+2 t}$ with respect to $t$ .

See Solution

Problem 18133

Find the integral of the function $2^{\frac{t}{2}} + \frac{1}{t}$ with respect to $t$ : $\int\left(2^{\frac{t}{2}}+\frac{1}{t}\right) d t$ .

See Solution

Problem 18134

Biologists stocked a lake with 500 fish, carrying capacity 8800. After 1 year, there were 650 fish.
a) Find the logistic model $P(t)$ . $P(t)=$ b) How long until the population reaches 4400?
It will take years.

See Solution

Problem 18135

Find the limit: $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}$ where $f(x)=3x+2e^{-3x}$ and $g'(x)=4+\frac{1}{x}$ .

See Solution

Problem 18136

Calculate the integral: $\int \frac{10}{\sqrt{1-16 x^{2}}} d x$

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Problem 18137

Evaluate the integral $\int \frac{5}{1+x^{2}} dx$ .

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Problem 18138

Determine the true statement about a differentiable function $f$ on $(0,5)$ where $f'(x)$ is negative on $(0,1)$ and $(2,3)$ .

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Problem 18139

Calculate the integral: $\int \frac{-8}{\sqrt{9-36 x^{2}}} d x$

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Problem 18140

Find the value of $x$ in the interval $[-2,2]$ where the function $g(x)=3 x^{4}-8 x^{3}$ has an absolute maximum.

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Problem 18141

Evaluate the integral: $\int \frac{7}{\sqrt{25-36 x^{2}}} d x$

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Problem 18142

Find the intervals where the graph of $h$ is concave down, given $h^{\prime}(x)=x^{4}-x^{3}+x$ .

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Problem 18143

Evaluate the integral: $\int \frac{-1}{\sqrt{16-16 x^{2}}} d x$

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Problem 18144

Gegeben ist die Funktion $f(x)=\frac{1}{4} e^{2 x}-x+2$ . a) In welchem Quadranten liegt der Extrempunkt? b) Untersuchen Sie das Krümmungsverhalten. c) Zeichnen Sie $K$ und seine Asymptote.

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Problem 18145

Evaluate the integral: $\int \frac{-4}{\sqrt{4-36 x^{2}}} d x$

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Problem 18146

Determine the intervals where the function $f$ with derivative $f^{\prime}(x)=\sin x+x \cos x$ is increasing for $0 \leq x \leq \pi$ .

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Problem 18147

Differentiate the integral $\int_{-2}^{5 x} \sin(t^{2}+1) \, dt$ with respect to $x$ .

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Problem 18148

Find $f^{\prime}(x)$ for the function $f(x)=\int_{-2}^{\sqrt{2 x}} \cos (5 t-2) d t$ .

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Problem 18149

Find $f^{\prime}(x)$ for the function $f(x)=\int_{4}^{5 x} \cos(t^{2}+4) dt$ .

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Problem 18150

A manufacturer has $R^{\prime}(q)=54 q^{-1 / 2}$ and marginal cost $0.4 q$ . Profit is \$440 at 25 units. Find profit at 36 units.

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Problem 18151

Rewrite the integral without absolute values and evaluate: $I=\int_{0}^{9}\left|x^{2}-1\right| dx$ .

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Problem 18152

How long for \$8400 to grow to \$14,600 at 9.4% interest compounded continuously? Round to the nearest hundredth.

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Problem 18153

Rewrite the integral without absolute values and evaluate: $\int_{\pi / 9}^{\pi}|\cos (x)| d x \approx$

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Problem 18154

Calculate the integral $\int_{-2}^{3} f(x) d x$ for the piecewise function $f(x)=\{2-x^{3}, x \leq 1; x^{2}, x>1\}$ .

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Problem 18155

Find the area under the curve of $f(x)=5 x^{2}$ from $x=0$ to $x=4$ using the Fundamental Theorem of Calculus. $A=$

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Problem 18156

Calculate the indefinite integral: $\int\left(-6 e^{t}-4 t\right) d t$

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Problem 18157

Calculate the integral $\int_{0}^{2 \pi} f(x) d x$ where $f(x) = \sin(x)$ for $x \leq \pi$ and $f(x) = -8 \sin(x)$ for $x > \pi$ .

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Problem 18158

Calculate the integral: $\int\left(-3 x^{-2}+3\right) d x$ .

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Problem 18159

Find the area under $f(x)=14 \cos (x)$ from $x=0$ to $x=\frac{\pi}{6}$ using the Fundamental Theorem of Calculus. $A=$

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Problem 18160

Calculez l'intégrale $\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} \mathrm{~d} x$ en trouvant $A, B, C$ et en utilisant les fractions partielles.

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Problem 18161

Find $f(4)$ and $f^{\prime}(4)$ given the tangent line to $y=f(x)$ at $(4,3)$ passes through $(0,2)$ .

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Problem 18162

Calculate the indefinite integral: $\int(-3 \sqrt[3]{x}+5 \sqrt{x}) \, dx$

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Problem 18163

Multiply and integrate: $\int(8 x+1)^{2} d x$ .

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Problem 18164

Find the antiderivative $F(x)$ where $\frac{d F}{d x}=4 x+4 e^{x}$ and $F(0)=17$ .

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Problem 18165

Find the revenue function for selling $x$ microwaves, given that marginal revenue is $159 - 0.06x$ and $R(0) = 0$ .

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Problem 18166

Calculate the indefinite integral: $\int\left(-8 x^{7}+\frac{9}{x}-\frac{5}{x^{5}}\right) d x$

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Problem 18167

Multiply and integrate: $\int x^{3}(3 x-4) d x$

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Problem 18168

Find the population function $P(t)$ after $t$ years, given $\frac{d P}{d t}=196-14 t^{\frac{1}{3}}$ and $P(0)=9200$ .

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Problem 18169

$f(x)$ fonksiyonunun tüm reel sayılar için limiti varsa, $a$ nın alabileceği değerler toplamını bulun.

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Problem 18170

Given functions $f$ and $g$ , prove the $(n+1)^{\text{th}}$ derivative of $f(x)g(x)$ using Pascal's triangle coefficients.

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Problem 18171

Calculate the integral of $(1+\sin x)^{2}$ with respect to $x$ : $\int(1+\sin x)^{2} d x$ .

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Problem 18172

Find the tangent line equation to $y=\ln x^{5}$ at the point $(1,0)$ . $y=\square$

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Problem 18173

1. Find the labor supply function from $U(C, L)=C-(1-L)^{2}$ with $h=1-L$ and no non-labor income.
2. Show the slope of an indifference curve is $\frac{\partial w}{\partial h}=\left(1-\frac{\bar{u}}{h^{2}}\right)$ .
3. Prove that at optimal labor supply, the slope of the indifference curve is zero.
4. Write the firm's profit function from $f(E)=2E-E^{2}$ and derive the optimal labor demand function.
5. Solve for equilibrium wage, worker utility, and firm profit using labor supply and demand functions.
6. Derive wage, employment, utility, and profit in the monopoly union model using previous functions.
7. Explain why the monopoly union model outcome is inefficient.
8. Use Lagrangean to find optimal labor supply and wage under profit constraint $\left(\bar{\pi}=2h-h^{2}-wh\right)$ and compare outcomes.

See Solution

Problem 18174

Find $d y / d x$ using logarithmic differentiation for $y=x \sqrt{x^{2}+9}$ , where $x>0$ .

See Solution

Problem 18175

Find the function $f(x)$ given $f^{\prime \prime}(x)=2 x^{2}+6 \cos (x)$ , $f^{\prime}(0)=7$ , $f(0)=6$ .

See Solution

Problem 18176

Find the tangent line equation at (1,0) for $x+y-1=\ln(x^{10}+y^{3})$ . What is $y$ ?

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Problem 18177

Find $d y / d x$ using logarithmic differentiation for $y=\sqrt{\frac{x^{2}-4}{x^{2}+4}}$ , where $x>2$ .

See Solution

Problem 18178

Find the function $f(x)$ given the slope $f^{\prime}(x)=e^{-x}+x^{4}$ and the point $(0,-3)$ .

See Solution

Problem 18179

Find the function $f(x)$ given the slope $f^{\prime}(x)=e^{-x}+x^{6}$ and the point $(0,-8)$ .

See Solution

Problem 18180

Find $d y / d x$ using logarithmic differentiation for $y=\frac{(x+1)(x-4)}{(x-1)(x+4)}$ , where $x>4$ .

See Solution

Problem 18181

Find the derivative of the functions: $f(x)=(12+3x)(2-x)$ and $g(x)=\frac{12+3x}{2-x}$ .

See Solution

Problem 18182

Find the function $f(x)$ given its slope $f^{\prime}(x)=e^{-x}+x^{7}$ and the point $(0,-9)$ .

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Problem 18183

Determine if the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt[3]{n-1}}$ converges or diverges using the Direct Comparison Test.

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Problem 18184

Calculate the indefinite integral: $\int \frac{1}{x^{4/5}(8+x^{1/5})} \, dx$

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Problem 18185

Find $F^{\prime}(x)$ if $F(x)=\int_{0}^{x} 3 \tan(t) dt$ . What is $F^{\prime}(x)$ ?

See Solution

Problem 18186

Find the derivative $F^{\prime}(x)$ for $F(x)=\int_{1}^{x^{4}} \frac{1}{t} dt$ .

See Solution

Problem 18187

Calculate the indefinite integral: $\int \frac{1}{x^{4 / 5}(8+x^{1 / 5})} d x$

See Solution

Problem 18188

Find the derivative of the inverse function at 0: $\left(f^{-1}\right)^{\prime}(0)$ where $f(x)=\int_{1}^{x} \sqrt{6+t^{2}} d t$ .

See Solution

Problem 18189

Calculate the limit as $x$ approaches +∞ for $\frac{e^{x}+\sin 4 x}{e^{x}-\cos x}$ .

See Solution

Problem 18190

A scientist has 110 mg of a radioactive substance. After 16 hours, 55 mg remains. How much is left after 31 hours? $\mathrm{mg}$

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Problem 18191

Find the limit as $x$ approaches $0^{+}$ for $\frac{\operatorname{ctg} x}{\ln(e^{x}-1)}$ .

See Solution

Problem 18192

Calculate $\lim _{0+} \frac{\operatorname{ctg} x}{\ln(e^{x}-1)}$ .

See Solution

Problem 18193

Find if the limit exists: $\lim _{x \rightarrow \infty} \frac{x}{4}\left(e^{7 / x}-1\right)$ and determine its value.

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Problem 18194

Find the production level $x$ at plant $A$ to minimize costs when producing 50 cars, given the cost function $C=x^{2}+2 y^{2}+2 x y+80 x+40 y+40$ .

See Solution

Problem 18195

A circle's radius decreases from $50 \mathrm{ft}$ at $3 \mathrm{ft/min}$ . Find the area change rate when radius is $22 \mathrm{ft}$ .

See Solution

Problem 18196

A balloon inflates at $48 \pi \frac{\mathrm{ft}^{3}}{\mathrm{~min}}$ . Find the radius increase rate when $r = 2 \mathrm{ft}$ .

See Solution

Problem 18197

Find if the limit exists: $\lim _{x \rightarrow 7}\left(\frac{7}{\ln (x-6)}-\frac{7}{x-7}\right)$ and its value.

See Solution

Problem 18198

Henry has 360m of fencing for a rectangular garden.
(a) Find the area function $A(x)$ in terms of $x$ .
(b) What side length $x$ maximizes the area, and what is this maximum area?

See Solution

Problem 18199

A balloon inflates at $400 \pi \frac{\mathrm{ft}^{3}}{\mathrm{~min}}$ . Find the radius increase rate when radius is $5 \mathrm{ft}$ .

See Solution

Problem 18200

Find the value of $x$ for point $P(x, y)$ on the line $2x + y = 1$ that is closest to the origin.

See Solution

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Calculus

Problem 18101

Bestimmen Sie die Ableitung der Funktion f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Problem 18102

Evaluate the integral from 0 to 1: ∫011+7x3dx\int_{0}^{1} \sqrt[3]{1+7 x} d x∫01​31+7x​dx

Problem 18103

Find the derivative of the function f(x)=1xf(x)=\frac{1}{x}f(x)=x1​.

Problem 18104

Find the derivative of f(x)=x−2+2xf(x)=x^{-2}+2xf(x)=x−2+2x.

Problem 18105

Find the derivative of f(x)=x12f(x)=x^{\frac{1}{2}}f(x)=x21​.

Problem 18106

Find the derivative of the function f(x)=x−12f(x)=x^{-\frac{1}{2}}f(x)=x−21​.

Problem 18107

Differentiate the function f(x)=5e8(x2+2)2f(x)=5 e^{8\left(x^{2}+2\right)^{2}}f(x)=5e8(x2+2)2. What is f′(x)f^{\prime}(x)f′(x)?

Problem 18108

Find the relative maxima of f(x)=x4+x3−6x2f(x)=x^{4}+x^{3}-6x^{2}f(x)=x4+x3−6x2 using its derivative and the second derivative test.

Problem 18109

Find the derivative of y=28x+5y=2^{8 x+5}y=28x+5. What is dydx=□\frac{d y}{d x}=\squaredxdy​=□?

Problem 18110

Differentiate the function f(x)=ln⁡(4−x2)f(x)=\ln(4-x^{2})f(x)=ln(4−x2). What is f′(x)=□f'(x)=\squaref′(x)=□?

Problem 18111

Find the derivative of y=esin⁡xy=e^{\sin x}y=esinx, i.e., calculate dydx\frac{d y}{d x}dxdy​.

Problem 18112

Find the limit: lim⁡+∞π2−arctg⁡xln⁡x+1x−1\lim _{+\infty} \frac{\frac{\pi}{2}-\operatorname{arctg} x}{\ln \frac{x+1}{x-1}}lim+∞​lnx−1x+1​2π​−arctgx​.

Problem 18113

If $1400\$ 1400$1400 is invested at 1.75%1.75 \%1.75% interest, how many years until it doubles? Round to the nearest hundredth.

Problem 18114

Problem 18115

Find the half-life of a substance with a decay rate of 9.3%9.3\%9.3% per day using the continuous exponential decay model.

Problem 18116

Ein Wasserbecken hat Zufluss (3001min⁡300 \frac{1}{\min}300min1​) und Abfluss (100lmin⁡100 \frac{l}{\min}100minl​). Berechne Liter nach 5, 10, 15, 20, 25, 30, 35 min.

Problem 18117

Estimate the area under f(x)=16−x2f(x)=16-x^{2}f(x)=16−x2 from x=0x=0x=0 to x=4x=4x=4 using 4 rectangles: (A) right endpoints, (B) left endpoints.

Problem 18118

Evaluate ∫354x2−15x+11x−2 dx\int_{3}^{5} \frac{4x^{2}-15x+11}{x-2} \, dx∫35​x−24x2−15x+11​dx and simplify your answer, condensing logs if needed.

Problem 18119

How long to double a bacteria population growing at 2%2\%2% per hour? Round to the nearest hundredth.

Problem 18120

Find the function f(x)f(x)f(x) and the interval [A,B][A, B][A,B] for the limit lim⁡n→∞∑i=1n4n7+4in\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}n→∞lim​i=1∑n​n4​7+n4i​​ without evaluating it.

Problem 18121

Find the half-life of a radioactive substance with a decay rate of 5%5\%5% per day. Round to the nearest hundredth.

Problem 18122

Problem 18123

Find the function f(x)f(x)f(x), and the interval [A,B][A, B][A,B] for the limit lim⁡n→∞∑i=1n4n7+4in\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}n→∞lim​i=1∑n​n4​7+n4i​​. Use A=0A=0A=0, B=1B=1B=1.

Problem 18124

A turkey cools from 185∘185^{\circ}185∘F in a 75∘75^{\circ}75∘F room. (a) Find T(45)T(45)T(45) if T(30)=144∘T(30) = 144^{\circ}T(30)=144∘F. (b) When is T(t)=100∘T(t) = 100^{\circ}T(t)=100∘F? Round answers to 2 decimal places.

Problem 18125

What is the future value of \100investedfor1yearata100 invested for 1 year at a 100investedfor1yearata100\%continuouscompoundingrateintermsof continuous compounding rate in terms of continuouscompoundingrateintermsofe$?

Problem 18126

A bacteria culture had 300 after 20 min and 1400 after 40 min. Find initial size, doubling time, population after 90 min, and time to reach 15000.

Problem 18127

Problem 18128

Problem 18129

Differentiate the function πe+12t\pi^{e}+\frac{1}{2 t}πe+2t1​ with respect to ttt.

Problem 18130

Calculate the integral ∫(t3+1e)dt\int\left(t^{3}+\frac{1}{e}\right) d t∫(t3+e1​)dt.

Problem 18131

Find the integral of the function ex2+xe^{\frac{x}{2}} + xe2x​+x.

Problem 18132

Differentiate the function πe+15+2t\pi^{e}+\frac{1}{5+2 t}πe+5+2t1​ with respect to ttt.

Problem 18133

Find the integral of the function 2t2+1t2^{\frac{t}{2}} + \frac{1}{t}22t​+t1​ with respect to ttt: ∫(2t2+1t)dt\int\left(2^{\frac{t}{2}}+\frac{1}{t}\right) d t∫(22t​+t1​)dt.

Problem 18134

Biologists stocked a lake with 500 fish, carrying capacity 8800. After 1 year, there were 650 fish. a) Find the logistic model P(t)P(t)P(t). P(t)= P(t)= P(t)= b) How long until the population reaches 4400? It will take years.

Problem 18135

Find the limit: lim⁡x→∞f(x)g(x)\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}limx→∞​g(x)f(x)​ where f(x)=3x+2e−3xf(x)=3x+2e^{-3x}f(x)=3x+2e−3x and g′(x)=4+1xg'(x)=4+\frac{1}{x}g′(x)=4+x1​.

Problem 18136

Calculate the integral: ∫101−16x2dx\int \frac{10}{\sqrt{1-16 x^{2}}} d x∫1−16x2​10​dx

Problem 18137

Evaluate the integral ∫51+x2dx\int \frac{5}{1+x^{2}} dx∫1+x25​dx.

Problem 18138

Determine the true statement about a differentiable function fff on (0,5)(0,5)(0,5) where f′(x)f'(x)f′(x) is negative on (0,1)(0,1)(0,1) and (2,3)(2,3)(2,3).

Problem 18139

Calculate the integral: ∫−89−36x2dx\int \frac{-8}{\sqrt{9-36 x^{2}}} d x∫9−36x2​−8​dx

Problem 18140

Find the value of xxx in the interval [−2,2][-2,2][−2,2] where the function g(x)=3x4−8x3g(x)=3 x^{4}-8 x^{3}g(x)=3x4−8x3 has an absolute maximum.

Problem 18141

Evaluate the integral: ∫725−36x2dx\int \frac{7}{\sqrt{25-36 x^{2}}} d x∫25−36x2​7​dx

Problem 18142

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{x}$ .

Evaluate the integral from 0 to 1: $\int_{0}^{1} \sqrt[3]{1+7 x} d x$

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

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

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

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

Differentiate the function $f(x)=5 e^{8\left(x^{2}+2\right)^{2}}$ . What is $f^{\prime}(x)$ ?

Find the relative maxima of $f(x)=x^{4}+x^{3}-6x^{2}$ using its derivative and the second derivative test.

Find the derivative of $y=2^{8 x+5}$ . What is $\frac{d y}{d x}=\square$ ?

Differentiate the function $f(x)=\ln(4-x^{2})$ . What is $f'(x)=\square$ ?

Find the derivative of $y=e^{\sin x}$ , i.e., calculate $\frac{d y}{d x}$ .

Find the limit: $\lim _{+\infty} \frac{\frac{\pi}{2}-\operatorname{arctg} x}{\ln \frac{x+1}{x-1}}$ .

If $\$ 1400$ is invested at $1.75 \%$ interest, how many years until it doubles? Round to the nearest hundredth.

Find the half-life of a substance with a decay rate of $9.3\%$ per day using the continuous exponential decay model.

Ein Wasserbecken hat Zufluss ( $300 \frac{1}{\min}$ ) und Abfluss ( $100 \frac{l}{\min}$ ). Berechne Liter nach 5, 10, 15, 20, 25, 30, 35 min.

Estimate the area under $f(x)=16-x^{2}$ from $x=0$ to $x=4$ using 4 rectangles: (A) right endpoints, (B) left endpoints.

Evaluate $\int_{3}^{5} \frac{4x^{2}-15x+11}{x-2} \, dx$ and simplify your answer, condensing logs if needed.

How long to double a bacteria population growing at $2\%$ per hour? Round to the nearest hundredth.

Find the function $f(x)$ and the interval $[A, B]$ for the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}$ without evaluating it.

Find the half-life of a radioactive substance with a decay rate of $5\%$ per day. Round to the nearest hundredth.

Find the function $f(x)$ , and the interval $[A, B]$ for the limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \sqrt{7+\frac{4 i}{n}}$ . Use $A=0$ , $B=1$ .

A turkey cools from $185^{\circ}$ F in a $75^{\circ}$ F room.
(a) Find $T(45)$ if $T(30) = 144^{\circ}$ F.
(b) When is $T(t) = 100^{\circ}$ F?
Round answers to 2 decimal places.

What is the future value of \ $100 invested for 1 year at a$ 100\% $continuous compounding rate in terms of$ e$?

Differentiate the function $\pi^{e}+\frac{1}{2 t}$ with respect to $t$ .

Calculate the integral $\int\left(t^{3}+\frac{1}{e}\right) d t$ .

Find the integral of the function $e^{\frac{x}{2}} + x$ .

Differentiate the function $\pi^{e}+\frac{1}{5+2 t}$ with respect to $t$ .

Find the integral of the function $2^{\frac{t}{2}} + \frac{1}{t}$ with respect to $t$ : $\int\left(2^{\frac{t}{2}}+\frac{1}{t}\right) d t$ .

Biologists stocked a lake with 500 fish, carrying capacity 8800. After 1 year, there were 650 fish.
a) Find the logistic model $P(t)$ . $P(t)=$ b) How long until the population reaches 4400?
It will take years.

Find the limit: $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}$ where $f(x)=3x+2e^{-3x}$ and $g'(x)=4+\frac{1}{x}$ .

Calculate the integral: $\int \frac{10}{\sqrt{1-16 x^{2}}} d x$

Evaluate the integral $\int \frac{5}{1+x^{2}} dx$ .

Determine the true statement about a differentiable function $f$ on $(0,5)$ where $f'(x)$ is negative on $(0,1)$ and $(2,3)$ .

Calculate the integral: $\int \frac{-8}{\sqrt{9-36 x^{2}}} d x$

Find the value of $x$ in the interval $[-2,2]$ where the function $g(x)=3 x^{4}-8 x^{3}$ has an absolute maximum.

Evaluate the integral: $\int \frac{7}{\sqrt{25-36 x^{2}}} d x$

Find the intervals where the graph of $h$ is concave down, given $h^{\prime}(x)=x^{4}-x^{3}+x$ .

Evaluate the integral: $\int \frac{-1}{\sqrt{16-16 x^{2}}} d x$

Gegeben ist die Funktion $f(x)=\frac{1}{4} e^{2 x}-x+2$ . a) In welchem Quadranten liegt der Extrempunkt? b) Untersuchen Sie das Krümmungsverhalten. c) Zeichnen Sie $K$ und seine Asymptote.

Evaluate the integral: $\int \frac{-4}{\sqrt{4-36 x^{2}}} d x$

Determine the intervals where the function $f$ with derivative $f^{\prime}(x)=\sin x+x \cos x$ is increasing for $0 \leq x \leq \pi$ .

Differentiate the integral $\int_{-2}^{5 x} \sin(t^{2}+1) \, dt$ with respect to $x$ .

Find $f^{\prime}(x)$ for the function $f(x)=\int_{-2}^{\sqrt{2 x}} \cos (5 t-2) d t$ .

Find $f^{\prime}(x)$ for the function $f(x)=\int_{4}^{5 x} \cos(t^{2}+4) dt$ .

A manufacturer has $R^{\prime}(q)=54 q^{-1 / 2}$ and marginal cost $0.4 q$ . Profit is \$440 at 25 units. Find profit at 36 units.

Rewrite the integral without absolute values and evaluate: $I=\int_{0}^{9}\left|x^{2}-1\right| dx$ .

Rewrite the integral without absolute values and evaluate: $\int_{\pi / 9}^{\pi}|\cos (x)| d x \approx$

Calculate the integral $\int_{-2}^{3} f(x) d x$ for the piecewise function $f(x)=\{2-x^{3}, x \leq 1; x^{2}, x>1\}$ .

Find the area under the curve of $f(x)=5 x^{2}$ from $x=0$ to $x=4$ using the Fundamental Theorem of Calculus. $A=$

Calculate the indefinite integral: $\int\left(-6 e^{t}-4 t\right) d t$

Calculate the integral $\int_{0}^{2 \pi} f(x) d x$ where $f(x) = \sin(x)$ for $x \leq \pi$ and $f(x) = -8 \sin(x)$ for $x > \pi$ .

Calculate the integral: $\int\left(-3 x^{-2}+3\right) d x$ .

Find the area under $f(x)=14 \cos (x)$ from $x=0$ to $x=\frac{\pi}{6}$ using the Fundamental Theorem of Calculus. $A=$

Calculez l'intégrale $\int_{1}^{3} \frac{2 x+2}{x^{3}+6 x} \mathrm{~d} x$ en trouvant $A, B, C$ et en utilisant les fractions partielles.

Find $f(4)$ and $f^{\prime}(4)$ given the tangent line to $y=f(x)$ at $(4,3)$ passes through $(0,2)$ .

Calculate the indefinite integral: $\int(-3 \sqrt[3]{x}+5 \sqrt{x}) \, dx$

Multiply and integrate: $\int(8 x+1)^{2} d x$ .

Find the antiderivative $F(x)$ where $\frac{d F}{d x}=4 x+4 e^{x}$ and $F(0)=17$ .

Find the revenue function for selling $x$ microwaves, given that marginal revenue is $159 - 0.06x$ and $R(0) = 0$ .

Calculate the indefinite integral: $\int\left(-8 x^{7}+\frac{9}{x}-\frac{5}{x^{5}}\right) d x$

Multiply and integrate: $\int x^{3}(3 x-4) d x$

Find the population function $P(t)$ after $t$ years, given $\frac{d P}{d t}=196-14 t^{\frac{1}{3}}$ and $P(0)=9200$ .

$f(x)$ fonksiyonunun tüm reel sayılar için limiti varsa, $a$ nın alabileceği değerler toplamını bulun.

Given functions $f$ and $g$ , prove the $(n+1)^{\text{th}}$ derivative of $f(x)g(x)$ using Pascal's triangle coefficients.

Calculate the integral of $(1+\sin x)^{2}$ with respect to $x$ : $\int(1+\sin x)^{2} d x$ .

Find the tangent line equation to $y=\ln x^{5}$ at the point $(1,0)$ . $y=\square$

Find $d y / d x$ using logarithmic differentiation for $y=x \sqrt{x^{2}+9}$ , where $x>0$ .