Calculus

Problem 16101

Find the derivative of the function $f(x)=e^{-x^{8}}$ .

See Solution

Problem 16102

Find the absolute max and min of $f(x)=5 x^{3}-60 x$ on the interval $[0,5]$ .

See Solution

Problem 16103

Find the derivative of $f(x)=\ln(7-x)$ .

See Solution

Problem 16104

Find the derivative of the function $f(x)=\ln(7x)$ .

See Solution

Problem 16105

Evaluate the integral $\int 5 \cos \sqrt{x} \, dx$ using substitution and integration by parts.

See Solution

Problem 16106

Find the vertical distance $v$ between the line $y=x+6$ and the parabola $y=x^{2}$ for $-2 \leq x \leq 3$ . What is the maximum distance?

See Solution

Problem 16107

Find the width $x$ (in ft) of a Norman window with a perimeter of 24 ft that maximizes light.

See Solution

Problem 16108

Given the cost function $C(x)=54,000+140 x+4 x^{3/2}$ , calculate:
(a) Total cost for 1,000 units: \$ (b) Average cost for 1,000 units: \$ per unit (c) Marginal cost for 1,000 units: \$ per unit (d) Production level minimizing average cost: units (e) Minimum average cost: \$ per unit

See Solution

Problem 16109

Prove that $f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$ satisfies $f''(x)+f(x)=0$ .

See Solution

Problem 16110

Identify which of these series are alternating:
1. $\sum \infty n=1 (-1)^{n+11} / n^{3}$
2. $\sum \infty n=1 \cos n / n^{2}$
3. $\sum \infty n=1 -1 / 2^{n}$
4. $\sum \infty n=1 \cos(n \pi) / (n+4)$

See Solution

Problem 16111

Identify which statements about convergence are true:
1. Absolutely convergent implies conditionally convergent.
2. Conditionally convergent implies convergent.
3. Convergent implies conditionally convergent.
4. Conditionally convergent implies absolutely convergent.
5. Absolutely convergent implies convergent.
6. Convergent implies absolutely convergent.

See Solution

Problem 16112

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin (g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

See Solution

Problem 16113

If $\sum a_n$ diverges, what can we conclude about $\sum |a_n|$ ? Options: 1. Divergent 2. Absolutely convergent 3. Convergent

See Solution

Problem 16114

Prove that the function $f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$ satisfies $f^{\prime \prime}(x)+f(x)=0$ .

See Solution

Problem 16115

Find where the function $f(x)=8 x^{3}-11 x^{2}+7$ is concave up or down and identify points of inflection.

See Solution

Problem 16116

Find critical points $c$ for local minima of the function $f(x)=(x^{2}-4)e^{-x}$ for $x>0$ .

See Solution

Problem 16117

Prove that $f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ satisfies $f^{\prime}(x)=f(x)$ and show $f(x)=e^{x}$ .

See Solution

Problem 16118

Given that $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \infty$ , what can we conclude about $\sum a_n$ ? Choose all that apply: 1. Convergent 2. Absolutely convergent 3. Diverges to $\infty$ 4. Divergent 5. Converges to 1

See Solution

Problem 16119

Given the cost function $C(x)=54,000+140 x+4 x^{3 / 2}$ , find:
(a) Total cost at 1,000 units.
(b) Average cost at 1,000 units.
(c) Marginal cost at 1,000 units.
(d) Production level that minimizes average cost.
(e) Minimum average cost.

See Solution

Problem 16120

Find the angle $\theta$ that minimizes total resistance $T=\left(\frac{a-b \cot (\theta)}{R^{4}}+\frac{b \csc (\theta)}{r^{4}}\right)$ for $r=5$ , $R=18$ .

See Solution

Problem 16121

Fish populations grow logistically. Given $d=1000, k=9, c=0.2$ , find:
(a) Initial fish count. (b) Population at 10, 20, 30 years. (c) Limit of $P(t)$ as $t \to \infty$ .

See Solution

Problem 16122

Find the derivative of $f(t) = t^{2} \ln (1+2)$ .

See Solution

Problem 16123

Determine if the limit $\lim _{x \rightarrow 1} f(x)$ exists for the piecewise function $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-1}{x-1} & \text { if } x \neq 1 \\ 4 & \text { if } x=1\end{array}\right.$

See Solution

Problem 16124

Evaluate the function $f(x)$ and determine which statements about limits and continuity at $x=1$ are true. Options: (A) I only, (B) II only.

See Solution

Problem 16125

In 2016, mailing costs were $\$ 2.54$ for 3 oz, plus $\$ 0.20$ per extra oz. What describes the range restrictions?

See Solution

Problem 16126

Find the position of a particle on the curve $y=2x$ at $t=1$ , given $x'(t)=3t^2+1$ and starting at $(2,4)$ .

See Solution

Problem 16127

A remote-controlled car's position is given by $x(t)=2t+\sin t$ and $y(t)=2\cos(t-\sin t)$ .
a. Find the tangent line at $t=3$ . b. Calculate speed at $t=15$ . c. Determine acceleration when $x=40$ . $x^{\prime}(t)=2+\cos t, \quad y^{\prime}(t)=-2\sin(t-\sin t)(1-\cos t)$

See Solution

Problem 16128

Find the first time $t$ when the velocity of the particle at $x(t)=12 e^{-t} \sin t$ is zero. (A) $\frac{\pi}{4}$ (B) $\frac{\pi}{2}$

See Solution

Problem 16129

Find the values of $t \geq 0$ for which the particle at position $x(t)=t^{3}-3t^{2}-9t+1$ is at rest.

See Solution

Problem 16130

Find the Taylor series for $f(x)=x^{4}-3x^{2}+1$ at $a=1$ .

See Solution

Problem 16131

Evaluate the integral: $\int e^{8 \theta} \sin (9 \theta) d \theta$ (use constant $C$ ).

See Solution

Problem 16132

Find the tangent line equation at $t=\frac{\pi}{3}$ for $x=\cos t, y=\sec t$ and $\frac{d^{2} y}{d x^{2}}$ .
Tangent line: $y=\square x+\square$

See Solution

Problem 16133

Find the Taylor series for $f(x)=x-x^{3}$ centered at $a=-2$ . No need to show that $R_n(x)$ is 0.

See Solution

Problem 16134

Calculate the curve length for $x=2 \sin t-2 t \cos t$ , $y=2 \cos t+2 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ . Length = $\square$ units.

See Solution

Problem 16135

Find the tangent line equation at $t=\frac{\pi}{4}$ for $x=\sec t$ , $y=\tan t$ . Also, determine $\frac{d^{2} y}{d x^{2}}$ at this point.
Tangent line: $y=\sqrt{2} x+-1$ .
Value of $\frac{d^{2} y}{d x^{2}}=\square$ .

See Solution

Problem 16136

Find the curve length for $x=2 \sin t-2 t \cos t$ , $y=2 \cos t+2 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ .

See Solution

Problem 16137

Find the curve length for $x=\frac{t^{2}}{2}, y=\frac{(2 t+1)^{3 / 2}}{3}, 0 \leq t \leq 10$ . Length is $\square$ .

See Solution

Problem 16138

Find the integral: $\int \frac{1}{1+\sin x} d x$ and show it equals $\frac{1}{2} \tan \left(\frac{x}{2}+a\right)+C$ .

See Solution

Problem 16139

Calculate the indefinite integral of (8x - 3)^{-8}. What is $\int(8 x-3)^{-8} d x$ ?

See Solution

Problem 16140

Calculate the curve length for $x=\frac{t^{2}}{2}$ and $y=\frac{(2t+1)^{3/2}}{3}$ from $t=0$ to $t=20$ . Length = $\square$ .

See Solution

Problem 16141

Calculate the curve length for $x=3 \sin t-3 t \cos t$ , $y=3 \cos t+3 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ . Length = $\square$ units.

See Solution

Problem 16142

Calculate the surface area of the curve $x=t+\sqrt{6}, y=\frac{t}{2}+\sqrt{6} t+2$ from $-\sqrt{6}$ to $\sqrt{6}$ when revolved around the $y$ -axis.

See Solution

Problem 16143

Calculate the area under one arch of the cycloid given by $x=3 a(t-\sin t)$ and $y=2 a(1-\cos t)$ . The area is $\square$ . Use $\pi$ as needed.

See Solution

Problem 16144

Find the population of a city 16 years after incorporation if the growth rate is $\frac{d N}{d t}=900+300 \sqrt{t}$ and initial population is 7,000.

See Solution

Problem 16145

Find the tangent line equation at the point where $x=\frac{1}{t+1}$ , $y=\frac{t}{t-1}$ for $t=2$ . Also, find $\frac{d^{2} y}{d x^{2}}$ .
Write the tangent line as $y=\square x-\square$ .

See Solution

Problem 16146

Find the tangent line equation at $t=\frac{\pi}{2}$ for $x=-\cos t$ , $y=4+\sin t$ . Also, find $\frac{d^{2} y}{d x^{2}}$ at this point.

See Solution

Problem 16147

Find the tangent line equation at $t=4$ for $x=\frac{1}{t+3}, y=\frac{t}{t-3}$ and $\frac{d^{2} y}{d x^{2}}$ .
$y=\square x-\square$

See Solution

Problem 16148

Find the marginal cost $C^{\prime}(x)$ for producing $x=22$ slide rules, where $C(x)=10+\sqrt{4x+16}$ .

See Solution

Problem 16149

Find the slope of the curve defined by $x^{3}+2 t^{2}=33$ and $2 y^{3}-2 t^{2}=22$ at $t=4$ . The slope is $\square$ .

See Solution

Problem 16150

Find a point of discontinuity for the function $f(x)=\frac{1}{\sin \frac{\pi x}{2}}$ : A. 0 B. 1 C. 2 D. 4

See Solution

Problem 16151

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{-(x+\Delta x)^{3}+x^{3}}{\Delta x}$ . Choose a) $-x^{3}$ , b) $x^{3}$ , c) $-3 x^{2}$ , d) $3 x^{2}$ .

See Solution

Problem 16152

Bestimmen Sie die Extremstellen der Funktion $f_{k}(x)=\frac{1}{3} k x^{3}-k x+9$ in Abhängigkeit von $k$ .

See Solution

Problem 16153

Evaluate these integrals using FTC and Walli's Formula:
1. $\int_{0}^{1} \frac{x d x}{(x^{2}+1)^{3}}$
2. $\int_{0}^{3} \frac{d x}{\sqrt{9-x^{2}}}$
3. $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos x}{\sin^{2} x} d x$
4. $\int_{0}^{\pi} \sin^{6} \frac{x}{2} d x$
5. $\int_{0}^{\frac{\pi}{4}} \cos^{4} 2 x d x$

See Solution

Problem 16154

Find the limit of $s(n)$ as $n$ approaches infinity, where $s(n)=\frac{1}{n^{2}}\left[\frac{n(n+1)}{5}\right]$ .

See Solution

Problem 16155

1) Find the tangent line equation for $f(x)=-\sqrt{x+1}$ at $x=3$ . 2) Calculate $\lim _{\Delta x \rightarrow 0} \frac{\frac{-4}{x+\Delta x}+\frac{4}{x}}{\Delta x}$ . 3) For $f(x)=2 x^{2}+x-1$ , find $\lim _{\Delta x \rightarrow 0} \frac{f(3+\Delta x)-f(3)}{\Delta x}$ . 4) What is the derivative of $f(x)=\frac{-3}{x}$ ?

See Solution

Problem 16156

Find the sum formula for $n$ terms and calculate the limit as $n \to \infty$ : $\lim _{n \to \infty} \sum_{i=1}^{n}\left(1+\frac{6 i}{n}\right)^{3}\left(\frac{2}{n}\right)$

See Solution

Problem 16157

Analyze the convergence of the series $\sum_{n=1}^{\infty} a_{n} \frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}}$ with the limit comparison test.

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Problem 16158

Bestimme, wann die Wassermenge im Staubecken zunimmt, wann die Änderungsrate minimal ist und deren Bedeutung für $g(t)$ . Funktion: $f(t)=\frac{1}{4} t^{3}-\frac{11}{4} t^{2}-4 t+44$ .

See Solution

Problem 16159

Die Funktion $f(t)=\frac{1}{4} t^{3}-\frac{11}{4} t^{2}-4 t+44$ beschreibt die Wassermenge im Staubecken.
a) Bestimmen Sie die Zeiträume, in denen die Wassermenge zunimmt. b) Finden Sie den Zeitpunkt, an dem die Änderungsrate am geringsten ist. c) Erklären Sie die Bedeutung dieses Zeitpunkts für die Funktion $g(t)$ .

See Solution

Problem 16160

An 8 m ladder leans against a wall. If pulled away at $2 \mathrm{~cm/s}$ , find $\frac{d \theta}{d t}$ when $\theta = \frac{\pi}{3}$ .

See Solution

Problem 16161

A conical tank (radius $3 \mathrm{~cm}$ , height $15 \mathrm{~cm}$ ) fills at $6 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ .
a) Prove $V=\frac{\pi h^{3}}{75}$ . b) Find the height change rate when $h=5 \mathrm{~cm}$ .

See Solution

Problem 16162

Evaluate the integral $\int_{-2}^{2} x^{3} d x$ using the limit definition by dividing the interval into $n$ parts.

See Solution

Problem 16163

Analyze the function $f(x)=3 x^{3}+2 x^{2}-5 x+6$ : find extrema, intervals, real and imaginary zeros, and factor using synthetic division.

See Solution

Problem 16164

Untersuchen Sie die Funktion $f(x)=x^{4}-6,25 x^{2}+9$ auf Nullstellen, Symmetrie, Extrema, Wendepunkte und Monotonie. Zeichnen Sie den Graphen für $-2,5<x<2,5$ . Bestimmen Sie Wendetangenten und deren Eckpunkte sowie Innenwinkel. Finden Sie die Parabel $g$ mit Scheitel $S(0|9)$ und Nullstellen $x_1=-1,5$ , $x_2=1,5$ . Zeigen Sie, dass $f(x) \leq g(x)$ für $]-1,5;1,5[$ und vergleichen Sie die Steigungen $f'$ und $g'$ in diesem Intervall.

See Solution

Problem 16165

10) Where is $f(x)=|2x+3|$ not differentiable? A) $R-\{\frac{3}{2}\}$ B) $x=-\frac{3}{2}$ C) $x=\frac{3}{2}$ D) $R-\{-\frac{3}{2}\}$
11) Where is $f(x)=|2x+3|$ differentiable? A) $R-\{-\frac{3}{2}\}$ B) $R-\{\frac{3}{2}\}$ C) $x=\frac{3}{2}$ D) $-\frac{3}{2}$
12) Find $f^{\prime}(-1)$ for $f(x)=-2x^{3}$ . A) $\lim _{\Delta x \rightarrow 0} \frac{-2(1+\Delta x)^{3}+2}{\Delta x}$ B) $\lim _{\Delta x \rightarrow 0} \frac{-2(-1+\Delta x)^{3}+2}{\Delta x}$ C) $\lim _{\Delta x \rightarrow 0} \frac{2(-1+\Delta x)^{3}-2}{\Delta x}$ D) $\lim _{\Delta x \rightarrow 0} \frac{-2(-1+\Delta x)^{3}-2}{\Delta x}$
13) Which function is differentiable at $x=2$ ? A) $f(x)=|x-2|$ B) $f(x)=\sqrt{x-2}$ C) $f(x)=\frac{3}{x-2}$ D) $f(x)=(x-2)^{3}$
14) If the tangent line to $g$ at $(2,5)$ passes through $(a,-7)$ and $g^{\prime}(2)=4$ , find $a$ . A) -3 B) -1 C) 1 D) 3

See Solution

Problem 16166

Untersuchen Sie die Monotonie und Beschränktheit der Folgen: (a) $a_{n}=\frac{n+3}{2 n}$ , (b) $a_{n}=n \cos (\pi(n+1))$ , (c) $a_{n}=1+\left(-\frac{3}{4}\right)^{n}$ , (d) $a_{n}=\cos \left(\frac{\pi}{6}\right) \sin \left(\frac{\pi n}{2}\right)$ .

See Solution

Problem 16167

Untersuchen Sie die Funktion $f(x)=x^{4}-6,25 x^{2}+9$ auf Nullstellen, Symmetrie, Extrema, Wendepunkte und zeichnen Sie den Graphen. Bestimmen Sie die Wendetangenten und das Dreieck mit der $x$ -Achse. Finden Sie die Parabel $g$ mit $S(0|9)$ und den Nullstellen $x_{1}=-1,5$ , $x_{2}=1,5$ . Vergleichen Sie $f$ und $g$ im Intervall $]-1,5;1,5[$ .

See Solution

Problem 16168

Find the flux of the vector field $\vec{F}=4 x \vec{i}+2 y \vec{j}$ through a cylinder with $a=1$ and $b=6$ .

See Solution

Problem 16169

Find a vector equation for the ellipse on the plane $4x - 2y + z = -5$ inside the cylinder $x^2 + y^2 = 25$ . Also, compute the surface area using a double integral.

See Solution

Problem 16170

Find the area between the curves $f(x)=4x-x^{2}+2$ and $g(x)=x+2$ .

See Solution

Problem 16171

Find the $y$ -intercept of the graph of $f$ with slope $\frac{d f}{d x}=-\frac{15 x}{\sqrt{9-5 x^{2}}}$ and passing through $(1,5)$ .

See Solution

Problem 16172

Find the area under $g(x)=2x^{2}-x-1$ over the interval $[3,5]$ using 4 rectangles.

See Solution

Problem 16173

Find the upper and lower sums for $y = \sqrt{1-x^{2}}$ with given integral limits and number of rectangles $n$ .

See Solution

Problem 16174

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

See Solution

Problem 16175

Find the limit of $s(n)$ as $n$ approaches infinity, where $s(n)=\frac{1}{n^{2}}\left[\frac{n(n+1)}{7}\right]$ .

See Solution

Problem 16176

Find the sum formula for $n$ terms and calculate the limit as $n \rightarrow \infty$ : $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5+\frac{i}{n}\right)\left(\frac{9}{n}\right)$

See Solution

Problem 16177

Trouvez la dérivée de $y$ par rapport à $x$ pour l'équation $y^{7} \ln (y)-x^{4} \ln (x)=5$ . Réponse: $y^{\prime}=$

See Solution

Problem 16178

Bestimme die Ableitung der Funktion $f(x) = \frac{3}{x} - \frac{5}{x^{3}}$ .

See Solution

Problem 16179

Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{3}+1}$ is absolutely or conditionally convergent.

See Solution

Problem 16180

Calculate the indefinite integral and include the constant of integration C: $\int x^{6} d x$ .

See Solution

Problem 16181

Determine if the series $\sum_{n=1}^{\infty}(-1)^{n-1} \cdot n$ is absolutely or conditionally convergent.

See Solution

Problem 16182

Evaluate the integral $\int_{-2}^{2} x^{3} d x$ using the limit definition.

See Solution

Problem 16183

Is the series $\sum_{n=1}^{\infty}(-1)^{n-1}$ absolutely or conditionally convergent?

See Solution

Problem 16184

Evaluate the integral $\int_{-2}^{2} x^{3} d x$ using the limit definition.

See Solution

Problem 16185

Bestimmen Sie die Ableitung der Funktion $g(x) = \frac{(x^{3}+x) \cdot \sin(x)}{\ln(x)}$ für $x > 0$ .

See Solution

Problem 16186

Bestimme die Ableitung der Funktion $f(x) = \frac{x^{3}+5 x^{2}}{(x+2)^{-2 \alpha}}$ für $0<\alpha<1$ .

See Solution

Problem 16187

Find the mass of a $2.23 \mathrm{~kg}$ radioactive substance after 5 days with a decay rate of $2\%$ per day. Round to 2 decimal places.

See Solution

Problem 16188

Evaluate the integral $\int_{-2}^{2}(t^{2}-2) dt$ and verify with a graphing tool.

See Solution

Problem 16189

Bestimme den Grenzwert von $f(x)=3x^{3}-4x^{5}-x^{2}$ für $x \to \infty$ .

See Solution

Problem 16190

Berechne Nullstellen, Extrempunkte und Wendepunkte der Funktion $f_{k}(x)=\frac{1}{2} x^{3}-k x^{2}+\frac{1}{2} k^{2} x$ .

See Solution

Problem 16191

Find the values of $x$ where the second derivative of the function $f(x)$ , which has a max at $(0,4)$ and zeros at $2$ and $-2$ , is positive.

See Solution

Problem 16192

Find the derivative $f^{\prime}(1)$ for the function $f(x)=\sqrt{x}+\frac{1}{x}+2$ .

See Solution

Problem 16193

Calculate the integral from 0 to 1 of $3 \cos\left(\frac{\pi t}{2}\right) dt$ .

See Solution

Problem 16194

Find the derivative values at the extrema of $f(x)=\cos(\frac{\pi x}{2})$ : $f'(0)=$ and $f'(2)=$ .

See Solution

Problem 16195

Find the absolute extrema of $g(x)=4x^{2}-24x$ on the interval $[0,7]$ . Minimum $(x, y)=(\square \times)$ .

See Solution

Problem 16196

Find the critical numbers of the function $g(t)=t \sqrt{4-t}$ for $t<3$ . Enter answers as a comma-separated list.

See Solution

Problem 16197

Find the derivative $f^{\prime}(1)$ for the function $f(x)=\sqrt[n]{x}$ .

See Solution

Problem 16198

Find the absolute extrema of $f(x)=x^{2}-4x$ on the intervals: (a) $[-1,4]$ , (b) $(2,5]$ , (c) $(0,4)$ , (d) $[2,6)$ .

See Solution

Problem 16199

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

See Solution

Problem 16200

Encuentra $y^{\prime}=\frac{d y}{d x}$ de $e^{3 x}+e^{2 y}=x^{5}+y^{7}$ usando derivación implícita.

See Solution

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Calculus

Problem 16101

Find the derivative of the function f(x)=e−x8f(x)=e^{-x^{8}}f(x)=e−x8.

Problem 16102

Find the absolute max and min of f(x)=5x3−60xf(x)=5 x^{3}-60 xf(x)=5x3−60x on the interval [0,5][0,5][0,5].

Problem 16103

Find the derivative of f(x)=ln⁡(7−x)f(x)=\ln(7-x)f(x)=ln(7−x).

Problem 16104

Find the derivative of the function f(x)=ln⁡(7x)f(x)=\ln(7x)f(x)=ln(7x).

Problem 16105

Evaluate the integral ∫5cos⁡x dx\int 5 \cos \sqrt{x} \, dx∫5cosx​dx using substitution and integration by parts.

Problem 16106

Find the vertical distance vvv between the line y=x+6y=x+6y=x+6 and the parabola y=x2y=x^{2}y=x2 for −2≤x≤3-2 \leq x \leq 3−2≤x≤3. What is the maximum distance?

Problem 16107

Find the width xxx (in ft) of a Norman window with a perimeter of 24 ft that maximizes light.

Problem 16108

Problem 16109

Prove that f(x)=∑n=0∞(−1)nx2n(2n)!f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}f(x)=∑n=0∞​(2n)!(−1)nx2n​ satisfies f′′(x)+f(x)=0f''(x)+f(x)=0f′′(x)+f(x)=0.

Problem 16110

Problem 16111

Problem 16112

Find g′(52)g^{\prime}\left(\frac{5}{2}\right)g′(25​) given g(x)−9xsin⁡(g(x))=6x2−x−35g(x)-9 x \sin (g(x))=6 x^{2}-x-35g(x)−9xsin(g(x))=6x2−x−35 and g(52)=0g\left(\frac{5}{2}\right)=0g(25​)=0.

Problem 16113

If ∑an\sum a_n∑an​ diverges, what can we conclude about ∑∣an∣\sum |a_n|∑∣an​∣? Options: 1. Divergent 2. Absolutely convergent 3. Convergent

Problem 16114

Prove that the function f(x)=∑n=0∞(−1)nx2n(2n)!f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}f(x)=∑n=0∞​(2n)!(−1)nx2n​ satisfies f′′(x)+f(x)=0f^{\prime \prime}(x)+f(x)=0f′′(x)+f(x)=0.

Problem 16115

Find where the function f(x)=8x3−11x2+7f(x)=8 x^{3}-11 x^{2}+7f(x)=8x3−11x2+7 is concave up or down and identify points of inflection.

Problem 16116

Find critical points ccc for local minima of the function f(x)=(x2−4)e−xf(x)=(x^{2}-4)e^{-x}f(x)=(x2−4)e−x for x>0x>0x>0.

Problem 16117

Prove that f(x)=∑n=0∞xnn!f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}f(x)=∑n=0∞​n!xn​ satisfies f′(x)=f(x)f^{\prime}(x)=f(x)f′(x)=f(x) and show f(x)=exf(x)=e^{x}f(x)=ex.

Problem 16118

Problem 16119

Given the cost function C(x)=54,000+140x+4x3/2C(x)=54,000+140 x+4 x^{3 / 2}C(x)=54,000+140x+4x3/2, find:(a) Total cost at 1,000 units.(b) Average cost at 1,000 units.(c) Marginal cost at 1,000 units.(d) Production level that minimizes average cost.(e) Minimum average cost.

Problem 16120

Find the angle θ\thetaθ that minimizes total resistance T=(a−bcot⁡(θ)R4+bcsc⁡(θ)r4)T=\left(\frac{a-b \cot (\theta)}{R^{4}}+\frac{b \csc (\theta)}{r^{4}}\right)T=(R4a−bcot(θ)​+r4bcsc(θ)​) for r=5r=5r=5, R=18R=18R=18.

Problem 16121

Fish populations grow logistically. Given d=1000,k=9,c=0.2d=1000, k=9, c=0.2d=1000,k=9,c=0.2, find: (a) Initial fish count. (b) Population at 10, 20, 30 years. (c) Limit of P(t)P(t)P(t) as t→∞t \to \inftyt→∞.

Problem 16122

Find the derivative of f(t)=t2ln⁡(1+2)f(t) = t^{2} \ln (1+2)f(t)=t2ln(1+2).

Problem 16123

Problem 16124

Evaluate the function f(x)f(x)f(x) and determine which statements about limits and continuity at x=1x=1x=1 are true. Options: (A) I only, (B) II only.

Problem 16125

In 2016, mailing costs were $2.54\$ 2.54$2.54 for 3 oz, plus $0.20\$ 0.20$0.20 per extra oz. What describes the range restrictions?

Problem 16126

Find the position of a particle on the curve y=2xy=2xy=2x at t=1t=1t=1, given x′(t)=3t2+1x'(t)=3t^2+1x′(t)=3t2+1 and starting at (2,4)(2,4)(2,4).

Problem 16127

Problem 16128

Find the first time ttt when the velocity of the particle at x(t)=12e−tsin⁡tx(t)=12 e^{-t} \sin tx(t)=12e−tsint is zero. (A) π4\frac{\pi}{4}4π​ (B) π2\frac{\pi}{2}2π​

Problem 16129

Find the values of t≥0t \geq 0t≥0 for which the particle at position x(t)=t3−3t2−9t+1x(t)=t^{3}-3t^{2}-9t+1x(t)=t3−3t2−9t+1 is at rest.

Problem 16130

Find the Taylor series for f(x)=x4−3x2+1f(x)=x^{4}-3x^{2}+1f(x)=x4−3x2+1 at a=1a=1a=1.

Problem 16131

Evaluate the integral: ∫e8θsin⁡(9θ)dθ\int e^{8 \theta} \sin (9 \theta) d \theta∫e8θsin(9θ)dθ (use constant CCC).

Problem 16132

Find the tangent line equation at t=π3t=\frac{\pi}{3}t=3π​ for x=cos⁡t,y=sec⁡tx=\cos t, y=\sec tx=cost,y=sect and d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​. Tangent line: y=□x+□y=\square x+\squarey=□x+□

Problem 16133

Find the Taylor series for f(x)=x−x3f(x)=x-x^{3}f(x)=x−x3 centered at a=−2a=-2a=−2. No need to show that Rn(x)R_n(x)Rn​(x) is 0.

Problem 16134

Calculate the curve length for x=2sin⁡t−2tcos⁡tx=2 \sin t-2 t \cos tx=2sint−2tcost, y=2cos⁡t+2tsin⁡ty=2 \cos t+2 t \sin ty=2cost+2tsint, 0≤t≤π20 \leq t \leq \frac{\pi}{2}0≤t≤2π​. Length = □\square□ units.

Problem 16135

Problem 16136

Find the curve length for x=2sin⁡t−2tcos⁡tx=2 \sin t-2 t \cos tx=2sint−2tcost, y=2cos⁡t+2tsin⁡ty=2 \cos t+2 t \sin ty=2cost+2tsint, 0≤t≤π20 \leq t \leq \frac{\pi}{2}0≤t≤2π​.

Problem 16137

Find the curve length for x=t22,y=(2t+1)3/23,0≤t≤10x=\frac{t^{2}}{2}, y=\frac{(2 t+1)^{3 / 2}}{3}, 0 \leq t \leq 10x=2t2​,y=3(2t+1)3/2​,0≤t≤10. Length is □\square□.

Problem 16138

Find the integral: ∫11+sin⁡xdx\int \frac{1}{1+\sin x} d x∫1+sinx1​dx and show it equals 12tan⁡(x2+a)+C\frac{1}{2} \tan \left(\frac{x}{2}+a\right)+C21​tan(2x​+a)+C.

Problem 16139

Calculate the indefinite integral of (8x - 3)^{-8}. What is ∫(8x−3)−8dx\int(8 x-3)^{-8} d x∫(8x−3)−8dx?

Problem 16140

Calculate the curve length for x=t22x=\frac{t^{2}}{2}x=2t2​ and y=(2t+1)3/23y=\frac{(2t+1)^{3/2}}{3}y=3(2t+1)3/2​ from t=0t=0t=0 to t=20t=20t=20. Length = □\square□.

Problem 16141

Calculate the curve length for x=3sin⁡t−3tcos⁡tx=3 \sin t-3 t \cos tx=3sint−3tcost, y=3cos⁡t+3tsin⁡ty=3 \cos t+3 t \sin ty=3cost+3tsint, 0≤t≤π20 \leq t \leq \frac{\pi}{2}0≤t≤2π​. Length = □\square□ units.

Problem 16142

Calculate the surface area of the curve x=t+6,y=t2+6t+2x=t+\sqrt{6}, y=\frac{t}{2}+\sqrt{6} t+2x=t+6​,y=2t​+6​t+2 from −6-\sqrt{6}−6​ to 6\sqrt{6}6​ when revolved around the yyy-axis.

Problem 16143

Calculate the area under one arch of the cycloid given by x=3a(t−sin⁡t)x=3 a(t-\sin t)x=3a(t−sint) and y=2a(1−cos⁡t)y=2 a(1-\cos t)y=2a(1−cost). The area is □\square□. Use π\piπ as needed.

Find the derivative of the function $f(x)=e^{-x^{8}}$ .

Find the absolute max and min of $f(x)=5 x^{3}-60 x$ on the interval $[0,5]$ .

Find the derivative of $f(x)=\ln(7-x)$ .

Find the derivative of the function $f(x)=\ln(7x)$ .

Evaluate the integral $\int 5 \cos \sqrt{x} \, dx$ using substitution and integration by parts.

Find the vertical distance $v$ between the line $y=x+6$ and the parabola $y=x^{2}$ for $-2 \leq x \leq 3$ . What is the maximum distance?

Find the width $x$ (in ft) of a Norman window with a perimeter of 24 ft that maximizes light.

Prove that $f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$ satisfies $f''(x)+f(x)=0$ .

Find $g^{\prime}\left(\frac{5}{2}\right)$ given $g(x)-9 x \sin (g(x))=6 x^{2}-x-35$ and $g\left(\frac{5}{2}\right)=0$ .

If $\sum a_n$ diverges, what can we conclude about $\sum |a_n|$ ? Options: 1. Divergent 2. Absolutely convergent 3. Convergent

Prove that the function $f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}$ satisfies $f^{\prime \prime}(x)+f(x)=0$ .

Find where the function $f(x)=8 x^{3}-11 x^{2}+7$ is concave up or down and identify points of inflection.

Find critical points $c$ for local minima of the function $f(x)=(x^{2}-4)e^{-x}$ for $x>0$ .

Prove that $f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$ satisfies $f^{\prime}(x)=f(x)$ and show $f(x)=e^{x}$ .

Given the cost function $C(x)=54,000+140 x+4 x^{3 / 2}$ , find:
(a) Total cost at 1,000 units.
(b) Average cost at 1,000 units.
(c) Marginal cost at 1,000 units.
(d) Production level that minimizes average cost.
(e) Minimum average cost.

Find the angle $\theta$ that minimizes total resistance $T=\left(\frac{a-b \cot (\theta)}{R^{4}}+\frac{b \csc (\theta)}{r^{4}}\right)$ for $r=5$ , $R=18$ .

Fish populations grow logistically. Given $d=1000, k=9, c=0.2$ , find:
(a) Initial fish count. (b) Population at 10, 20, 30 years. (c) Limit of $P(t)$ as $t \to \infty$ .

Find the derivative of $f(t) = t^{2} \ln (1+2)$ .

Evaluate the function $f(x)$ and determine which statements about limits and continuity at $x=1$ are true. Options: (A) I only, (B) II only.

In 2016, mailing costs were $\$ 2.54$ for 3 oz, plus $\$ 0.20$ per extra oz. What describes the range restrictions?

Find the position of a particle on the curve $y=2x$ at $t=1$ , given $x'(t)=3t^2+1$ and starting at $(2,4)$ .

Find the first time $t$ when the velocity of the particle at $x(t)=12 e^{-t} \sin t$ is zero. (A) $\frac{\pi}{4}$ (B) $\frac{\pi}{2}$

Find the values of $t \geq 0$ for which the particle at position $x(t)=t^{3}-3t^{2}-9t+1$ is at rest.

Find the Taylor series for $f(x)=x^{4}-3x^{2}+1$ at $a=1$ .

Evaluate the integral: $\int e^{8 \theta} \sin (9 \theta) d \theta$ (use constant $C$ ).

Find the tangent line equation at $t=\frac{\pi}{3}$ for $x=\cos t, y=\sec t$ and $\frac{d^{2} y}{d x^{2}}$ .
Tangent line: $y=\square x+\square$

Find the Taylor series for $f(x)=x-x^{3}$ centered at $a=-2$ . No need to show that $R_n(x)$ is 0.

Calculate the curve length for $x=2 \sin t-2 t \cos t$ , $y=2 \cos t+2 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ . Length = $\square$ units.

Find the curve length for $x=2 \sin t-2 t \cos t$ , $y=2 \cos t+2 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ .

Find the curve length for $x=\frac{t^{2}}{2}, y=\frac{(2 t+1)^{3 / 2}}{3}, 0 \leq t \leq 10$ . Length is $\square$ .

Find the integral: $\int \frac{1}{1+\sin x} d x$ and show it equals $\frac{1}{2} \tan \left(\frac{x}{2}+a\right)+C$ .

Calculate the indefinite integral of (8x - 3)^{-8}. What is $\int(8 x-3)^{-8} d x$ ?

Calculate the curve length for $x=\frac{t^{2}}{2}$ and $y=\frac{(2t+1)^{3/2}}{3}$ from $t=0$ to $t=20$ . Length = $\square$ .

Calculate the curve length for $x=3 \sin t-3 t \cos t$ , $y=3 \cos t+3 t \sin t$ , $0 \leq t \leq \frac{\pi}{2}$ . Length = $\square$ units.

Calculate the surface area of the curve $x=t+\sqrt{6}, y=\frac{t}{2}+\sqrt{6} t+2$ from $-\sqrt{6}$ to $\sqrt{6}$ when revolved around the $y$ -axis.

Calculate the area under one arch of the cycloid given by $x=3 a(t-\sin t)$ and $y=2 a(1-\cos t)$ . The area is $\square$ . Use $\pi$ as needed.

Find the population of a city 16 years after incorporation if the growth rate is $\frac{d N}{d t}=900+300 \sqrt{t}$ and initial population is 7,000.

Find the tangent line equation at the point where $x=\frac{1}{t+1}$ , $y=\frac{t}{t-1}$ for $t=2$ . Also, find $\frac{d^{2} y}{d x^{2}}$ .
Write the tangent line as $y=\square x-\square$ .

Find the tangent line equation at $t=\frac{\pi}{2}$ for $x=-\cos t$ , $y=4+\sin t$ . Also, find $\frac{d^{2} y}{d x^{2}}$ at this point.

Find the tangent line equation at $t=4$ for $x=\frac{1}{t+3}, y=\frac{t}{t-3}$ and $\frac{d^{2} y}{d x^{2}}$ .
$y=\square x-\square$

Find the marginal cost $C^{\prime}(x)$ for producing $x=22$ slide rules, where $C(x)=10+\sqrt{4x+16}$ .

Find the slope of the curve defined by $x^{3}+2 t^{2}=33$ and $2 y^{3}-2 t^{2}=22$ at $t=4$ . The slope is $\square$ .

Find a point of discontinuity for the function $f(x)=\frac{1}{\sin \frac{\pi x}{2}}$ : A. 0 B. 1 C. 2 D. 4

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{-(x+\Delta x)^{3}+x^{3}}{\Delta x}$ . Choose a) $-x^{3}$ , b) $x^{3}$ , c) $-3 x^{2}$ , d) $3 x^{2}$ .

Bestimmen Sie die Extremstellen der Funktion $f_{k}(x)=\frac{1}{3} k x^{3}-k x+9$ in Abhängigkeit von $k$ .

Find the limit of $s(n)$ as $n$ approaches infinity, where $s(n)=\frac{1}{n^{2}}\left[\frac{n(n+1)}{5}\right]$ .

Find the sum formula for $n$ terms and calculate the limit as $n \to \infty$ : $\lim _{n \to \infty} \sum_{i=1}^{n}\left(1+\frac{6 i}{n}\right)^{3}\left(\frac{2}{n}\right)$

Analyze the convergence of the series $\sum_{n=1}^{\infty} a_{n} \frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}}$ with the limit comparison test.

Bestimme, wann die Wassermenge im Staubecken zunimmt, wann die Änderungsrate minimal ist und deren Bedeutung für $g(t)$ . Funktion: $f(t)=\frac{1}{4} t^{3}-\frac{11}{4} t^{2}-4 t+44$ .

An 8 m ladder leans against a wall. If pulled away at $2 \mathrm{~cm/s}$ , find $\frac{d \theta}{d t}$ when $\theta = \frac{\pi}{3}$ .

A conical tank (radius $3 \mathrm{~cm}$ , height $15 \mathrm{~cm}$ ) fills at $6 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ .
a) Prove $V=\frac{\pi h^{3}}{75}$ . b) Find the height change rate when $h=5 \mathrm{~cm}$ .

Evaluate the integral $\int_{-2}^{2} x^{3} d x$ using the limit definition by dividing the interval into $n$ parts.

Analyze the function $f(x)=3 x^{3}+2 x^{2}-5 x+6$ : find extrema, intervals, real and imaginary zeros, and factor using synthetic division.

Find the flux of the vector field $\vec{F}=4 x \vec{i}+2 y \vec{j}$ through a cylinder with $a=1$ and $b=6$ .

Find a vector equation for the ellipse on the plane $4x - 2y + z = -5$ inside the cylinder $x^2 + y^2 = 25$ . Also, compute the surface area using a double integral.

Find the area between the curves $f(x)=4x-x^{2}+2$ and $g(x)=x+2$ .

Find the $y$ -intercept of the graph of $f$ with slope $\frac{d f}{d x}=-\frac{15 x}{\sqrt{9-5 x^{2}}}$ and passing through $(1,5)$ .

Find the area under $g(x)=2x^{2}-x-1$ over the interval $[3,5]$ using 4 rectangles.

Find the upper and lower sums for $y = \sqrt{1-x^{2}}$ with given integral limits and number of rectangles $n$ .

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