Calculus

Problem 28101

Solve the differential equation: $y^{\prime \prime}=2 e^{t} \sec ^{2} t \tan t+e^{t} \tan t+2 e^{t} \sec ^{2} t$ .

See Solution

Problem 28102

How much work is needed to stretch a spring $1.5 \mathrm{~m}$ if a force of $66 \mathrm{~N}$ stretches it $60 \mathrm{~cm}$ ?

See Solution

Problem 28103

Solve the differential equation: $y' = e^t \tan t + e^t \sec^2 t$ .

See Solution

Problem 28104

Calculate the integral of $t \sin (4 t)$ with respect to $t$ : $\int t \sin (4 t) d t$ .

See Solution

Problem 28105

Calculate the center of mass $\bar{x}$ for the wave on $-1 \leq x \leq 8$ with density $\delta(x)=\sqrt[3]{x}$ .

See Solution

Problem 28106

Calculate the work done by the motor to lift a 250 lb elevator car and 200 ft of cable weighing 6 lb/ft from the 1st to the 10th floor.

See Solution

Problem 28107

Find the derivative of $f(x)=e^{x}(\sin x-\cos x)$ .

See Solution

Problem 28108

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

See Solution

Problem 28109

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

See Solution

Problem 28110

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=x-1$ and $y=(x-1)^{2}$ ) around the $x$ -axis using the Washer Method.

See Solution

Problem 28111

Find $y^{\prime}$ if $x^{2}-y^{2}=2y$ .

See Solution

Problem 28112

Find the second derivative of $f(x)=\frac{1}{\sin x}$ . Options: (A) $f^{\prime \prime}(x)=\frac{1-\cos ^{2} x}{\sin ^{3} x}$ , (B) $f^{\prime \prime}(x)=\frac{1+\cos ^{2} x}{\sin ^{3} x}$ , (C) $f^{\prime \prime}(x)=\frac{1+2 \cos ^{2} x}{\sin ^{3} x}$ .

See Solution

Problem 28113

Given the signal $y(t)$ defined as:
$y(t)=\left\{\begin{array}{cl} \sin t, & 0 \leq t<2 \pi \\ 0, & \text { elsewhere } \end{array}\right.$
Let $z(t)=x(t) * y(t)$ , the convolution of $x$ and $y$ .
(a) Sketch signals $x$ and $y$ . (b) What is the duration of $z(t)$ ? (c) Find $z(\pi), z(3 \pi / 2)$ , and $z(2 \pi)$ .

See Solution

Problem 28114

Find a function $y=f(x)$ such that $\frac{d y}{d x}=\ln (\sin x+5)$ and $f(2)=3$ .

See Solution

Problem 28115

Let $h(x) = \int_{1}^{x} f(t) \, dt$ . Is $h(0)$ positive or negative? Justify your answer based on $f$ .

See Solution

Problem 28116

Finde die Ableitung von $f(x) = 5 e^{7 x^{2}+4 x}$ .

See Solution

Problem 28117

Find the volume of the solid formed by revolving region $B$ (bounded by $x=y^2+3$ , $2y=x-11$ , and the $x$ -axis) around $x=3$ using the Washer Method, with limits from $-2$ to $4$ .

See Solution

Problem 28118

Differentiate $y=(x+4)(9-x)^{2}$ with respect to $x$ .

See Solution

Problem 28119

Find the volume of the solid formed by revolving region $R$ (bounded by $y=x^2$ and $y=x+2$ ) around $y=5$ using the Washer Method.

See Solution

Problem 28120

Find the value of $k$ for which $f(x)=k \sqrt{x}-\ln x$ has a critical point at $x=1$ , and analyze its nature.

See Solution

Problem 28121

Find the first and second derivatives of the function $f(x)=k \sqrt{x}-\ln x$ for $x>0$ , where $k>0$ .

See Solution

Problem 28122

Find the value of $x$ where $h(x)=\int_{1}^{x} f(t) \, dt$ is a local maximum, given the graph of $f$ .

See Solution

Problem 28123

Find $h(1)$ for $h(x) = \int_{1}^{x} f(t) \, dt$ where $f$ is defined on [0,8].

See Solution

Problem 28124

Find the value of the positive constant $k$ such that the function $f(x)=k \sqrt{x}-\ln x$ has a point of inflection on the $x$ -axis.

See Solution

Problem 28125

Find the value of $x$ where $h(x) = \int_{1}^{x} f(t) \, dt$ has a local maximum, given the graph of $f(x)$ from 0 to 8.

See Solution

Problem 28126

Bestimmen Sie die Ableitung von $f(x)=(2 x+1) \cdot \mathrm{e}^{5 x+2}$ .

See Solution

Problem 28127

Evaluate the integral from $\frac{\pi}{2}$ to $\frac{3\pi}{2}$ of $\cos x \, dx$ . What is the result?

See Solution

Problem 28128

Estimate $\int_{0}^{1} \tan ^{2} 2 x d x$ using the trapezium rule with 4 strips. Round to two decimal places.

See Solution

Problem 28129

Find the volume of the region bounded by $x=6y-y^{2}$ and $x=0$ when revolved around the $x$ axis using the shell method. Volume: $\square$ cubic units.

See Solution

Problem 28130

Find the area between the curve $y=4 x^{3}$ and the line $y=7 x$ . The answer is 6.13 (rounded to two decimal places).

See Solution

Problem 28131

Leite die Funktion $f(x)=3 \cdot e^{0,3 x^{3}-1,2 x}-3$ dreimal ab und finde die Extrempunkte sowie deren Art.

See Solution

Problem 28132

Find the area between the curve $y=9 x^{3}$ and the line $y=5 x$ . Round your answer to two decimal places.

See Solution

Problem 28133

Calculate the area between the curve $y=25-x^{2}$ and the $x$ axis from $x=-4$ to $x=6$ .

See Solution

Problem 28134

Evaluate the integral: $\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}}+1 \, dx$ .

See Solution

Problem 28135

Find the volume using the shell method for the region bounded by $x=8y-y^{2}$ and $x=0$ revolved around the $x$ axis.

See Solution

Problem 28136

Evaluate the integral $\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}+1} \, dx$ .

See Solution

Problem 28137

Calculate the area between the curve $y=9-x^{2}$ and the $x$ axis from $x=-2$ to $x=6$ .

See Solution

Problem 28138

Find the limit: $\lim_{h \rightarrow 0} \frac{1}{h} \ln\left(\frac{2+h}{2}\right)$ .

See Solution

Problem 28139

Approximate the integral $\int_{0}^{10} \sqrt{100-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ . Compare with the exact value, $25$ .

See Solution

Problem 28140

Estimate $\int_{0}^{1} \tan ^{2} 3 x \, dx$ using the trapezium rule with 4 strips. Approximation: $\approx 50.32$ .

See Solution

Problem 28141

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=12$ around the $x$ axis. The volume is $\square$ cubic units in terms of $\pi$ .

See Solution

Problem 28142

Find the limit: $\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{2 \sin ^{2} \theta}$ .

See Solution

Problem 28143

Prüfe die Aussagen über Extremstellen einer Funktion $f$ und begründe sie oder gib Gegenbeispiele an.

See Solution

Problem 28144

Calculate the area between the curve $y=1-x^{2}$ and the $x$ -axis from $x=0$ to $x=4$ .

See Solution

Problem 28145

Find the derivative $f^{\prime}(9)$ for $f(x)=\sqrt{x-5}$ using the limit definition $m_{\tan }=\lim _{x \rightarrow 9} \frac{f(x)-f(9)}{x-9}$ .

See Solution

Problem 28146

Find the slope of the tangent line for $f(x)=9-x-x^{2}$ at $a=0$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, determine the tangent line equation.

See Solution

Problem 28147

Find the volume from revolving the area between $x=8y-y^2$ and $x=0$ around the $x$ -axis using the shell method. Volume = $\square$ cubic units.

See Solution

Problem 28148

Find the area between the curve $y=6 x^{3}$ and the line $y=7 x$ . Round your answer to two decimal places.

See Solution

Problem 28149

Bestimmen Sie die Wendepunkte der Funktionen $f$ : a) $f(x)=2 x^{3}-6 x^{2}+4,5 x$ , b) $f(x)=x^{4}-24 x^{2}$ durch die dritte Ableitung.

See Solution

Problem 28150

Find $f^{\prime}(2)$ for $f(x)=x^{2}+8x$ using $m_{\tan }=\lim _{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}$ .

See Solution

Problem 28151

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=3$ around the $x$ axis. The volume is $\square$ cubic units in terms of $\pi$ .

See Solution

Problem 28152

Find $f^{\prime}(-2)$ for $f(x)=\frac{1}{x-7}$ using $m_{\tan }=\lim _{x \rightarrow -2} \frac{f(x)-f(-2)}{x+2}$ .

See Solution

Problem 28153

Calculate the area between the curve $y=25-x^{2}$ and the $x$ axis from $x=2$ to $x=6$ . The area is $\square$ .

See Solution

Problem 28154

Estimate $\int_{0}^{1} \tan ^{2} x \, dx$ using the trapezium rule with 4 strips: $\approx \square$ (round to 2 decimals).

See Solution

Problem 28155

Approximate the integral $\int_{0}^{3} \sqrt{9-x^{2}} d x$ using Simpson's rule with $n=8$ and compare to $\frac{9 \pi}{4}$ .

See Solution

Problem 28156

Find the slope of the tangent line for $f(x)=\sqrt{x+3}$ at $a=1$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, determine the equation of the tangent line at this point.

See Solution

Problem 28157

Find the slope of the tangent line for $f(x)=\frac{-9}{x-1}$ at $a=7$ and its equation.

See Solution

Problem 28158

Find the slope of the tangent line for $f(x)=9-x-x^{2}$ at $a=0$ and its equation. Slope: $m_{\tan}=-1$ .

See Solution

Problem 28159

Analyze pressure variation in a static fluid around a sphere of radius R using Euler's equation in spherical coordinates.

See Solution

Problem 28160

Bestimmen Sie die Wendepunkte der Funktionen $f$ : a) $f(x)=2 x^{3}-6 x^{2}+4,5 x$ , b) $f(x)=x^{4}-24 x^{2}$ .

See Solution

Problem 28161

Find the limits:
$\lim _{x \rightarrow 3^{-}} \frac{f(x)-3}{x-3}$
and
$\lim _{x \rightarrow 3^{+}} \frac{f(x)-3}{x-3}$
to show the derivative does not exist at $x=3$ .

See Solution

Problem 28162

A stone is tossed with an initial velocity of $19 \mathrm{~m/s}$ . Find average velocity over intervals: [1,1.05], [1,1.01], [1,1.005], [1,1.001]. Guess instantaneous velocity at $t=1 \mathrm{~s}$ .

See Solution

Problem 28163

Find the area between the curve $y=7 x^{3}$ and the line $y=9 x$ . Round your answer to two decimal places.

See Solution

Problem 28164

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=10$ around the $x$ axis. The volume is $\square$ cubic units (express in terms of $\pi$ ).

See Solution

Problem 28165

Given the function $y=f(x)=10 e^{0.7 x}$ at point $P(0,10)$ , find slopes of secant lines for given $x$ values and estimate the tangent line slope. Then derive the tangent line equation at $P$ .

See Solution

Problem 28166

Find the area between the curve $y=16-x^{2}$ and the $x$ -axis from $x=0$ to $x=6$ . What is it?

See Solution

Problem 28167

Find the area between the curve $y=5 x^{3}$ and the line $y=3 x$ . Round your answer to two decimal places.

See Solution

Problem 28168

Find the volume of the solid formed by revolving the region between $x=16y-y^{2}$ and $x=0$ around the $x$ axis. Volume = $\square$ cubic units.

See Solution

Problem 28169

Leiten Sie die Funktion $g_{k}(x)$ zweimal ab: a) $g_{k}(x)=k e^{2 x}-k$ und b) $g_{k}(x)=x e^{k x+1}$ .

See Solution

Problem 28170

Find the limit as $x$ approaches 7 for the expression $\frac{8x^2 - 49x - 49}{2x^2 - 98}$ .

See Solution

Problem 28171

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=7$ around the $x$ axis. The volume is $\square$ cubic units (exact answer in terms of $\pi$ ).

See Solution

Problem 28172

Find the area between the curve $y=2 x^{3}$ and the line $y=11 x$ . Round your answer to two decimal places: $\square$ .

See Solution

Problem 28173

Find the volume using the shell method for the region bounded by $x=6y-y^2$ and $x=0$ rotated around the $x$ axis. Volume: $\square$ cubic units.

See Solution

Problem 28174

Estimate $\int_{0}^{1} \tan ^{2} 3 x \, dx$ using the trapezium rule with 5 strips. Round to two decimal places.

See Solution

Problem 28175

Approximate the integral $\int_{0}^{2} \sqrt{4-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ and compare to $\pi$ . Round to three decimal places.

See Solution

Problem 28176

Find the volume using the shell method for the region bounded by $x=7y-y^{2}$ and $x=0$ around the $x$ axis. Volume: $\square$ cubic units.

See Solution

Problem 28177

Untersuchen Sie die Flächeninhalte der Funktionen $f(x)=\frac{1}{x^{3}}$ , $f(x)=\frac{1}{x^{2}}$ , $f(x)=\frac{1}{\sqrt{x}}$ für $x \geq 1$ .

See Solution

Problem 28178

Show that the sequence $a_{1}=1, a_{n+1}=\frac{a_{n}+5}{2}$ is bounded by 5, increasing, and find its limit.

See Solution

Problem 28179

Approximate the area in unit $^{2}$ between the $x$ -axis and the half circle graph of $f$ over $[-2,2]$ . Round to three decimal places.

See Solution

Problem 28180

Approximate the integral $\int_{0}^{1} \sqrt{1-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ . Compare with $\frac{\pi}{4}$ . Round to three decimal places.

See Solution

Problem 28181

Approximate the integral $\int_{0}^{6} \sqrt{36-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ and compare to $9 \pi$ . $\int_{0}^{6} \sqrt{36-x^{2}} d x \approx \square$

See Solution

Problem 28182

Trouver la dérivée de la fonction $f(x) = \frac{3}{7} x + \frac{5}{2}$ .

See Solution

Problem 28183

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{7 x^{3}-4 x+1}{x^{2}}$ .

See Solution

Problem 28184

Find the derivative of the function $f(x)=\frac{4-3x}{1+2x}$ . What is $f'(x)$ ?

See Solution

Problem 28185

Determine for which $p>1$ the sequence $a_{n}=\sqrt{n^{p}+n+1}-\sqrt{n^{p}-n+1}$ is bounded.

See Solution

Problem 28186

Calculate the integral $\int_{0}^{1} \tan^{2}(2x) \, dx$ and round to two decimal places.

See Solution

Problem 28187

Find the accumulation points for the sequences $a_{n}=i^{n}$ , $b_{n}=(-1)^{n}+\frac{n^{2}+1}{n^{2}}$ , and $c_{n}=\sin(n \frac{\pi}{6})$ .

See Solution

Problem 28188

Calculate the area under the curve $y=4-x^{2}$ from $x=0$ to $x=6$ . The area is $\square$ .

See Solution

Problem 28189

Calculate the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=8$ about the $x$ axis. The volume is $\square$ cubic units (in terms of $\pi$ ).

See Solution

Problem 28190

Find the volume from revolving the area between $x=11y-y^{2}$ and $x=0$ around the $x$ axis using the shell method: $\square$ cubic units.

See Solution

Problem 28191

Determine if the function $f(x)=\frac{8}{(x-2)^{4}+1}$ for $x \in[2, \infty)$ has an inverse.

See Solution

Problem 28192

Calculate the limit: $\lim _{x \rightarrow 0} \frac{x^{3}}{\sin \left(3 x^{2}\right)}$ . State if it does not exist.

See Solution

Problem 28193

Estimate $\int_{0}^{1} \tan ^{2} x \, dx$ using the trapezium rule with 5 strips. Result: $\int_{0}^{1} \tan ^{2} x \, dx \approx \square$ (Round to two decimal places.)

See Solution

Problem 28194

Estimate $\int_{0}^{1} \tan ^{2} x \, dx$ using the trapezium rule with 5 strips. Round to two decimal places: $\int_{0}^{1} \tan ^{2} x \, dx \approx \square$

See Solution

Problem 28195

Examine the continuity of the piecewise function $f$ at $x=0$ and $x=3$ .

See Solution

Problem 28196

Evaluate these limits: 1) $\lim _{x \rightarrow 1} \frac{x^{2}-7 x+6}{x-1}$ 2) $\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}$

See Solution

Problem 28197

Bestimme die dritte Ableitung von $f(x)=3 \cdot e^{0,3 x^{3}-1,2 x}-3$ und finde die Wendestelle.

See Solution

Problem 28198

Approximate the integral $\int_{0}^{5} \sqrt{25-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ . Compare to $\frac{25 \pi}{4}$ . Round answers as specified.

See Solution

Problem 28199

Find the limit as $x$ approaches $-\infty$ for $\frac{4x-3}{\sqrt{2x^2+5}}$ .

See Solution

Problem 28200

Find the rate of change of area of a circle $A(r)=\pi r^{2}$ when the radius $r$ is 5. $A'(5)=$

See Solution

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Calculus

Problem 28101

Solve the differential equation: y′′=2etsec⁡2ttan⁡t+ettan⁡t+2etsec⁡2ty^{\prime \prime}=2 e^{t} \sec ^{2} t \tan t+e^{t} \tan t+2 e^{t} \sec ^{2} ty′′=2etsec2ttant+ettant+2etsec2t.

Problem 28102

How much work is needed to stretch a spring 1.5 m1.5 \mathrm{~m}1.5 m if a force of 66 N66 \mathrm{~N}66 N stretches it 60 cm60 \mathrm{~cm}60 cm?

Problem 28103

Solve the differential equation: y′=ettan⁡t+etsec⁡2ty' = e^t \tan t + e^t \sec^2 ty′=ettant+etsec2t.

Problem 28104

Calculate the integral of tsin⁡(4t)t \sin (4 t)tsin(4t) with respect to ttt: ∫tsin⁡(4t)dt\int t \sin (4 t) d t∫tsin(4t)dt.

Problem 28105

Calculate the center of mass xˉ\bar{x}xˉ for the wave on −1≤x≤8-1 \leq x \leq 8−1≤x≤8 with density δ(x)=x3\delta(x)=\sqrt[3]{x}δ(x)=3x​.

Problem 28106

Calculate the work done by the motor to lift a 250 lb elevator car and 200 ft of cable weighing 6 lb/ft from the 1st to the 10th floor.

Problem 28107

Find the derivative of f(x)=ex(sin⁡x−cos⁡x)f(x)=e^{x}(\sin x-\cos x)f(x)=ex(sinx−cosx).

Problem 28108

Find the derivative of f(x)=cos⁡(2x)x2f(x)=\frac{\cos(2x)}{x^{2}}f(x)=x2cos(2x)​.

Problem 28109

Find the derivative of f(x)=−5(2x)3f(x)=-\frac{5}{(2 x)^{3}}f(x)=−(2x)35​.

Problem 28110

Find the volume of the solid formed by rotating the region RRR (bounded by y=x−1y=x-1y=x−1 and y=(x−1)2y=(x-1)^{2}y=(x−1)2) around the xxx-axis using the Washer Method.

Problem 28111

Find y′y^{\prime}y′ if x2−y2=2yx^{2}-y^{2}=2yx2−y2=2y.

Problem 28112

Problem 28113

Problem 28114

Find a function y=f(x)y=f(x)y=f(x) such that dydx=ln⁡(sin⁡x+5)\frac{d y}{d x}=\ln (\sin x+5)dxdy​=ln(sinx+5) and f(2)=3f(2)=3f(2)=3.

Problem 28115

Let h(x)=∫1xf(t) dth(x) = \int_{1}^{x} f(t) \, dth(x)=∫1x​f(t)dt. Is h(0)h(0)h(0) positive or negative? Justify your answer based on fff.

Problem 28116

Finde die Ableitung von f(x)=5e7x2+4x f(x) = 5 e^{7 x^{2}+4 x} f(x)=5e7x2+4x.

Problem 28117

Find the volume of the solid formed by revolving region BBB (bounded by x=y2+3x=y^2+3x=y2+3, 2y=x−112y=x-112y=x−11, and the xxx-axis) around x=3x=3x=3 using the Washer Method, with limits from −2-2−2 to 444.

Problem 28118

Differentiate y=(x+4)(9−x)2y=(x+4)(9-x)^{2}y=(x+4)(9−x)2 with respect to xxx.

Problem 28119

Find the volume of the solid formed by revolving region RRR (bounded by y=x2y=x^2y=x2 and y=x+2y=x+2y=x+2) around y=5y=5y=5 using the Washer Method.

Problem 28120

Find the value of kkk for which f(x)=kx−ln⁡xf(x)=k \sqrt{x}-\ln xf(x)=kx​−lnx has a critical point at x=1x=1x=1, and analyze its nature.

Problem 28121

Find the first and second derivatives of the function f(x)=kx−ln⁡xf(x)=k \sqrt{x}-\ln xf(x)=kx​−lnx for x>0x>0x>0, where k>0k>0k>0.

Problem 28122

Find the value of xxx where h(x)=∫1xf(t) dth(x)=\int_{1}^{x} f(t) \, dth(x)=∫1x​f(t)dt is a local maximum, given the graph of fff.

Problem 28123

Find h(1) h(1) h(1) for h(x)=∫1xf(t) dt h(x) = \int_{1}^{x} f(t) \, dt h(x)=∫1x​f(t)dt where f f f is defined on [0,8].

Problem 28124

Find the value of the positive constant kkk such that the function f(x)=kx−ln⁡xf(x)=k \sqrt{x}-\ln xf(x)=kx​−lnx has a point of inflection on the xxx-axis.

Problem 28125

Find the value of xxx where h(x)=∫1xf(t) dth(x) = \int_{1}^{x} f(t) \, dth(x)=∫1x​f(t)dt has a local maximum, given the graph of f(x)f(x)f(x) from 0 to 8.

Problem 28126

Bestimmen Sie die Ableitung von f(x)=(2x+1)⋅e5x+2f(x)=(2 x+1) \cdot \mathrm{e}^{5 x+2}f(x)=(2x+1)⋅e5x+2.

Problem 28127

Evaluate the integral from π2\frac{\pi}{2}2π​ to 3π2\frac{3\pi}{2}23π​ of cos⁡x dx\cos x \, dxcosxdx. What is the result?

Problem 28128

Estimate ∫01tan⁡22xdx\int_{0}^{1} \tan ^{2} 2 x d x∫01​tan22xdx using the trapezium rule with 4 strips. Round to two decimal places.

Problem 28129

Find the volume of the region bounded by x=6y−y2x=6y-y^{2}x=6y−y2 and x=0x=0x=0 when revolved around the xxx axis using the shell method. Volume: □\square□ cubic units.

Problem 28130

Find the area between the curve y=4x3y=4 x^{3}y=4x3 and the line y=7xy=7 xy=7x. The answer is 6.13 (rounded to two decimal places).

Problem 28131

Leite die Funktion f(x)=3⋅e0,3x3−1,2x−3f(x)=3 \cdot e^{0,3 x^{3}-1,2 x}-3f(x)=3⋅e0,3x3−1,2x−3 dreimal ab und finde die Extrempunkte sowie deren Art.

Problem 28132

Find the area between the curve y=9x3y=9 x^{3}y=9x3 and the line y=5xy=5 xy=5x. Round your answer to two decimal places.

Problem 28133

Calculate the area between the curve y=25−x2y=25-x^{2}y=25−x2 and the xxx axis from x=−4x=-4x=−4 to x=6x=6x=6.

Problem 28134

Evaluate the integral: ∫14x416+1x4+1 dx\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}}+1 \, dx∫14​16x4​+x41​​+1dx.

Problem 28135

Find the volume using the shell method for the region bounded by x=8y−y2x=8y-y^{2}x=8y−y2 and x=0x=0x=0 revolved around the xxx axis.

Problem 28136

Evaluate the integral ∫14x416+1x4+1 dx\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}+1} \, dx∫14​16x4​+x41​+1​dx.

Problem 28137

Calculate the area between the curve y=9−x2y=9-x^{2}y=9−x2 and the xxx axis from x=−2x=-2x=−2 to x=6x=6x=6.

Problem 28138

Find the limit: lim⁡h→01hln⁡(2+h2)\lim_{h \rightarrow 0} \frac{1}{h} \ln\left(\frac{2+h}{2}\right)limh→0​h1​ln(22+h​).

Problem 28139

Approximate the integral ∫010100−x2 dx\int_{0}^{10} \sqrt{100-x^{2}} \mathrm{~d} x∫010​100−x2​ dx using Simpson's rule with n=8n=8n=8. Compare with the exact value, 252525.

Problem 28140

Estimate ∫01tan⁡23x dx\int_{0}^{1} \tan ^{2} 3 x \, dx∫01​tan23xdx using the trapezium rule with 4 strips. Approximation: ≈50.32\approx 50.32≈50.32.

Problem 28141

Solve the differential equation: $y^{\prime \prime}=2 e^{t} \sec ^{2} t \tan t+e^{t} \tan t+2 e^{t} \sec ^{2} t$ .

How much work is needed to stretch a spring $1.5 \mathrm{~m}$ if a force of $66 \mathrm{~N}$ stretches it $60 \mathrm{~cm}$ ?

Solve the differential equation: $y' = e^t \tan t + e^t \sec^2 t$ .

Calculate the integral of $t \sin (4 t)$ with respect to $t$ : $\int t \sin (4 t) d t$ .

Calculate the center of mass $\bar{x}$ for the wave on $-1 \leq x \leq 8$ with density $\delta(x)=\sqrt[3]{x}$ .

Find the derivative of $f(x)=e^{x}(\sin x-\cos x)$ .

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

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

Find the volume of the solid formed by rotating the region $R$ (bounded by $y=x-1$ and $y=(x-1)^{2}$ ) around the $x$ -axis using the Washer Method.

Find $y^{\prime}$ if $x^{2}-y^{2}=2y$ .

Find a function $y=f(x)$ such that $\frac{d y}{d x}=\ln (\sin x+5)$ and $f(2)=3$ .

Let $h(x) = \int_{1}^{x} f(t) \, dt$ . Is $h(0)$ positive or negative? Justify your answer based on $f$ .

Finde die Ableitung von $f(x) = 5 e^{7 x^{2}+4 x}$ .

Find the volume of the solid formed by revolving region $B$ (bounded by $x=y^2+3$ , $2y=x-11$ , and the $x$ -axis) around $x=3$ using the Washer Method, with limits from $-2$ to $4$ .

Differentiate $y=(x+4)(9-x)^{2}$ with respect to $x$ .

Find the volume of the solid formed by revolving region $R$ (bounded by $y=x^2$ and $y=x+2$ ) around $y=5$ using the Washer Method.

Find the value of $k$ for which $f(x)=k \sqrt{x}-\ln x$ has a critical point at $x=1$ , and analyze its nature.

Find the first and second derivatives of the function $f(x)=k \sqrt{x}-\ln x$ for $x>0$ , where $k>0$ .

Find the value of $x$ where $h(x)=\int_{1}^{x} f(t) \, dt$ is a local maximum, given the graph of $f$ .

Find $h(1)$ for $h(x) = \int_{1}^{x} f(t) \, dt$ where $f$ is defined on [0,8].

Find the value of the positive constant $k$ such that the function $f(x)=k \sqrt{x}-\ln x$ has a point of inflection on the $x$ -axis.

Find the value of $x$ where $h(x) = \int_{1}^{x} f(t) \, dt$ has a local maximum, given the graph of $f(x)$ from 0 to 8.

Bestimmen Sie die Ableitung von $f(x)=(2 x+1) \cdot \mathrm{e}^{5 x+2}$ .

Evaluate the integral from $\frac{\pi}{2}$ to $\frac{3\pi}{2}$ of $\cos x \, dx$ . What is the result?

Estimate $\int_{0}^{1} \tan ^{2} 2 x d x$ using the trapezium rule with 4 strips. Round to two decimal places.

Find the volume of the region bounded by $x=6y-y^{2}$ and $x=0$ when revolved around the $x$ axis using the shell method. Volume: $\square$ cubic units.

Find the area between the curve $y=4 x^{3}$ and the line $y=7 x$ . The answer is 6.13 (rounded to two decimal places).

Leite die Funktion $f(x)=3 \cdot e^{0,3 x^{3}-1,2 x}-3$ dreimal ab und finde die Extrempunkte sowie deren Art.

Find the area between the curve $y=9 x^{3}$ and the line $y=5 x$ . Round your answer to two decimal places.

Calculate the area between the curve $y=25-x^{2}$ and the $x$ axis from $x=-4$ to $x=6$ .

Evaluate the integral: $\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}}+1 \, dx$ .

Find the volume using the shell method for the region bounded by $x=8y-y^{2}$ and $x=0$ revolved around the $x$ axis.

Evaluate the integral $\int_{1}^{4} \sqrt{\frac{x^{4}}{16}+\frac{1}{x^{4}}+1} \, dx$ .

Calculate the area between the curve $y=9-x^{2}$ and the $x$ axis from $x=-2$ to $x=6$ .

Find the limit: $\lim_{h \rightarrow 0} \frac{1}{h} \ln\left(\frac{2+h}{2}\right)$ .

Approximate the integral $\int_{0}^{10} \sqrt{100-x^{2}} \mathrm{~d} x$ using Simpson's rule with $n=8$ . Compare with the exact value, $25$ .

Estimate $\int_{0}^{1} \tan ^{2} 3 x \, dx$ using the trapezium rule with 4 strips. Approximation: $\approx 50.32$ .

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=12$ around the $x$ axis. The volume is $\square$ cubic units in terms of $\pi$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{2 \sin ^{2} \theta}$ .

Prüfe die Aussagen über Extremstellen einer Funktion $f$ und begründe sie oder gib Gegenbeispiele an.

Calculate the area between the curve $y=1-x^{2}$ and the $x$ -axis from $x=0$ to $x=4$ .

Find the derivative $f^{\prime}(9)$ for $f(x)=\sqrt{x-5}$ using the limit definition $m_{\tan }=\lim _{x \rightarrow 9} \frac{f(x)-f(9)}{x-9}$ .

Find the slope of the tangent line for $f(x)=9-x-x^{2}$ at $a=0$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, determine the tangent line equation.

Find the volume from revolving the area between $x=8y-y^2$ and $x=0$ around the $x$ -axis using the shell method. Volume = $\square$ cubic units.

Find the area between the curve $y=6 x^{3}$ and the line $y=7 x$ . Round your answer to two decimal places.

Bestimmen Sie die Wendepunkte der Funktionen $f$ : a) $f(x)=2 x^{3}-6 x^{2}+4,5 x$ , b) $f(x)=x^{4}-24 x^{2}$ durch die dritte Ableitung.

Find $f^{\prime}(2)$ for $f(x)=x^{2}+8x$ using $m_{\tan }=\lim _{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}$ .

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=3$ around the $x$ axis. The volume is $\square$ cubic units in terms of $\pi$ .

Find $f^{\prime}(-2)$ for $f(x)=\frac{1}{x-7}$ using $m_{\tan }=\lim _{x \rightarrow -2} \frac{f(x)-f(-2)}{x+2}$ .

Calculate the area between the curve $y=25-x^{2}$ and the $x$ axis from $x=2$ to $x=6$ . The area is $\square$ .

Estimate $\int_{0}^{1} \tan ^{2} x \, dx$ using the trapezium rule with 4 strips: $\approx \square$ (round to 2 decimals).

Approximate the integral $\int_{0}^{3} \sqrt{9-x^{2}} d x$ using Simpson's rule with $n=8$ and compare to $\frac{9 \pi}{4}$ .

Find the slope of the tangent line for $f(x)=\sqrt{x+3}$ at $a=1$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ . Then, determine the equation of the tangent line at this point.

Find the slope of the tangent line for $f(x)=\frac{-9}{x-1}$ at $a=7$ and its equation.

Find the slope of the tangent line for $f(x)=9-x-x^{2}$ at $a=0$ and its equation. Slope: $m_{\tan}=-1$ .

Bestimmen Sie die Wendepunkte der Funktionen $f$ : a) $f(x)=2 x^{3}-6 x^{2}+4,5 x$ , b) $f(x)=x^{4}-24 x^{2}$ .

Find the limits:
$\lim _{x \rightarrow 3^{-}} \frac{f(x)-3}{x-3}$
and
$\lim _{x \rightarrow 3^{+}} \frac{f(x)-3}{x-3}$
to show the derivative does not exist at $x=3$ .

A stone is tossed with an initial velocity of $19 \mathrm{~m/s}$ . Find average velocity over intervals: [1,1.05], [1,1.01], [1,1.005], [1,1.001]. Guess instantaneous velocity at $t=1 \mathrm{~s}$ .

Find the area between the curve $y=7 x^{3}$ and the line $y=9 x$ . Round your answer to two decimal places.

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=10$ around the $x$ axis. The volume is $\square$ cubic units (express in terms of $\pi$ ).

Given the function $y=f(x)=10 e^{0.7 x}$ at point $P(0,10)$ , find slopes of secant lines for given $x$ values and estimate the tangent line slope. Then derive the tangent line equation at $P$ .

Find the area between the curve $y=16-x^{2}$ and the $x$ -axis from $x=0$ to $x=6$ . What is it?

Find the area between the curve $y=5 x^{3}$ and the line $y=3 x$ . Round your answer to two decimal places.

Find the volume of the solid formed by revolving the region between $x=16y-y^{2}$ and $x=0$ around the $x$ axis. Volume = $\square$ cubic units.

Leiten Sie die Funktion $g_{k}(x)$ zweimal ab: a) $g_{k}(x)=k e^{2 x}-k$ und b) $g_{k}(x)=x e^{k x+1}$ .

Find the limit as $x$ approaches 7 for the expression $\frac{8x^2 - 49x - 49}{2x^2 - 98}$ .

Find the volume of the solid formed by rotating the area under $y=x^{2}$ from $x=1$ to $x=7$ around the $x$ axis. The volume is $\square$ cubic units (exact answer in terms of $\pi$ ).

Find the area between the curve $y=2 x^{3}$ and the line $y=11 x$ . Round your answer to two decimal places: $\square$ .

Find the volume using the shell method for the region bounded by $x=6y-y^2$ and $x=0$ rotated around the $x$ axis. Volume: $\square$ cubic units.

Estimate $\int_{0}^{1} \tan ^{2} 3 x \, dx$ using the trapezium rule with 5 strips. Round to two decimal places.