Calculus

Problem 20101

Find the tangent line equation to $5x^{2}+4y^{4}=-9xy$ at the point $(-1,1)$ .

See Solution

Problem 20102

Determine if the series $\sum_{k=1}^{\infty} \frac{5^{k}+6^{k}}{11^{k}}$ converges or diverges.

See Solution

Problem 20103

Find the area between $f(x)=x^{3}+8$ and the $x$ -axis over the interval $[-3,-1]$ . The area is $\square$ .

See Solution

Problem 20104

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}-2$ such that $F(1)=0$ . What is $F(x)=\square$ ?

See Solution

Problem 20105

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{1}{2}} \frac{6 d x}{\sqrt{1-x^{2}}}=\square$

See Solution

Problem 20106

Analyze a trapezoidal velocity-time graph with these details: starts at rest, constant 2 m/s from 1-5s, decelerates at 4s, max velocity -1 m/s at 12s. Find: (i) max distance from start, (ii) displacement after 12s, (iii) max acceleration magnitude.

See Solution

Problem 20107

Find the derivatives of these functions:
(a) $f(x) = \frac{x^{3}}{\sec (x)}$
(b) $g(x) = \log_{5}(\sin(5x^{2}-3))$
(c) $h(x) = \tan(x) \cdot \arccos(x)$
(d) $i(x) = \ln(\ln(\ln(x)))$
(e) $j(x) = \cos^{7}(3^{x}+4^{5})$

See Solution

Problem 20108

Evaluate the integral $H(x)=\int_{0}^{x} \sqrt{4-t^{2}} d t$ for $-2 \leq x \leq 2$ :
a. Find $H(0)$ , b. $H'(1)$ , c. $H'(2)$ , d. Use geometry for $H(2)$ , e. Solve $H(x)=s H(-x)$ .

See Solution

Problem 20109

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{3} e^{-x^{4}} d x$ . Is it finite?

See Solution

Problem 20110

Determine if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges using the integral test. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ .

See Solution

Problem 20111

求函数 $f(x)=\left|x^{2}-a x-b\right|$ 在区间 $[-1,1]$ 上的导数最大值 $M(a, b)$ 的最小值。

See Solution

Problem 20112

Use the integral test to check if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ . The integral is finite, so the series converges.

See Solution

Problem 20113

Find the gradient of curve $C$ at point $P(4,-1)$ and the normal's equation $ax+by+c=0$ . Also, determine $f(x)$ .

See Solution

Problem 20114

Find the gradient of the curve $C$ at point $P(4,-1)$ , then determine the normal's equation in the form $a x + b y + c = 0$ .

See Solution

Problem 20115

Determine the values of $x$ for which the series $\sum_{n=0}^{\infty} e^{n x}$ converges and find its sum.

See Solution

Problem 20116

Determine convergence or divergence of the series $\sum_{n=1}^{\infty} n^{-4}$ using the Integral Test. Evaluate $\int_{1}^{\infty} x^{-4} dx$ .

See Solution

Problem 20117

Determine if the series $\sum_{n=1}^{\infty} n^{3} e^{-n^{4}}$ converges by evaluating the integral $\int_{1}^{\infty} x^{3} e^{-x^{4}} d x$ .

See Solution

Problem 20118

Bestimme die 4. Grades Funktion für die Punkte A(0,5), B(5,2), C(7,1.4), D(10,3), E(13,1.8). Berechne Extrem- und Wendepunkte, sowie Nullstellen der Funktion $f(x)=-0,004 x^{4}+0,1 x^{3}-0,72 x^{2}+x+5$ . Zeichne $K$ für $-2 \leq x \leq 14$ und finde Punkte, wo der Wind senkrecht auf die Küste trifft.

See Solution

Problem 20119

Evaluate the integral from 1 to 7 of $(x^{3}+1)/x^{2}$ with respect to $x$ .

See Solution

Problem 20120

9 radioaktive Stoffe haben Halbwertszeiten. a) Wie viel U235 bleibt nach 2,112 Mrd. Jahren von 1 kg? b) Was denkt Erik falsch über I131 nach 16 Tagen? Wie viel % sind zerfallen? Q10 a) Wie lange bis 3 g Rn220 bleiben? b) Wie lange bis 90 % Pu239 zerfallen? 11 a) Wie viel Cs137 bleibt am 26.4.2020? b) Wann bleibt 1 % Cs137 übrig?

See Solution

Problem 20121

Berechne die Ableitungen $f^{\prime}(x)$ für die folgenden Funktionen: a) $x^{3}+x^{2}$ , b) $x^{4}+x^{2}$ , c) $x^{5}+x^{4}$ , d) $x^{4}+1$ , e) $x^{7}+x^{4}+x^{2}$ , f) $x^{8}+x^{3}+5$ , g) $x^{5}+x+3$ , h) $x^{11}+x^{7}+x^{3}$ , i) $x^{5}+x^{4}+x^{3}+x^{2}$ .

See Solution

Problem 20122

Evaluate $\int_{1}^{0}(9-3 x^{2}) dx$ using $\int_{0}^{1} x^{2} dx=\frac{1}{3}$ .

See Solution

Problem 20123

Evaluate the integrals: (a) $\int_{5}^{3} f(x) dx$ , (b) $\int_{5}^{7} \frac{4 h(x)-g(x)}{2} dx$ , (c) $\int_{7}^{3} r(x) dx$ .

See Solution

Problem 20124

Solve the Boltzmann equation for $f_{1}$ in an electric field, find $\sigma_{\alpha \beta}(\omega)$ at $T=0$ .

See Solution

Problem 20125

Find $f^{\prime}\left(\frac{\pi}{2}\right)$ for the function $f(x)=x^{\cos x}$ .

See Solution

Problem 20126

Bestimme, wo die Tangente an $f$ in Punkt B die $x$ -Achse schneidet: a) $f(x)=0,5 x^{2}, B(2 \mid f(2))$ b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}, B(-1 \mid f(-1))$ c) $f(x)=\frac{3}{x^{\prime}}, B(-1 \mid f(-1))$

See Solution

Problem 20127

Find the approximation of $f(1.1)$ using linearization, given the normal line at $x=1$ is $y=-x+7$ .

See Solution

Problem 20128

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=x+e^{2 x}$ . Choices: a. $\frac{1}{2}$ b. Else c. $\frac{1}{6}$ d. $\frac{1}{5}$ e. $\frac{1}{4}$ f. $\frac{1}{3}$ .

See Solution

Problem 20129

Bestimme den Schnittpunkt der Tangente an $f$ bei $B$ mit der $x$ -Achse für die Funktionen: a) $f(x)=0,5 x^{2}, B(2 \mid f(2))$ b) $f(x)=\frac{1}{3} x^{3}+2 x^{2}, B(-1 \mid f(-1))$ c) $f(x)=\frac{3}{x}, B(-1 \mid f(-1))$

See Solution

Problem 20130

Find $f^{(30)}(0)$ for the function $f(x)=x^{30}+\cos x$ . Choices: a. $30 !-1$ , b. $311+1$ , c. $30 !+1$ , d. $311-1$ , e. 0, f. 1, g. Else.

See Solution

Problem 20131

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 2 x \sin x}{x^{2} \cos 8 x}=$ a. Else b. 4 c. 5 d. 3 e. Does not exist f. 2 g. 0

See Solution

Problem 20132

Find the derivative $\frac{d y}{d x}$ if $y=2 \cos (3 x)$ . Options: A. $-\sin 3 x$ , B. $-3 \sin 3 x$ , C. $6 \sin 3 x$ , D. $-6 \sin 3 x$ .

See Solution

Problem 20133

Një vijë ka ekuacionin $y=f(x)$ . Jepet $\frac{d y}{d x}=10 x^{4}-12 x^{2}+1$ dhe $f(2)=9$ . Gjeni $f(x)$ dhe tregoni që $f(-1)=-24$ .

See Solution

Problem 20134

Find $f^{\prime}(0)+f^{\prime}(-1)$ for $f(x)=\frac{2 x}{x+2}$ . Options: A. 0 B. 4 C. 5 D. 1

See Solution

Problem 20135

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (8 x) \sin (3 x)}{6 x^{2} \cos (10 x)}$ . What is the result?

See Solution

Problem 20136

Determine which functions are differentiable at $x=1$ : A. $f(x)=x^{2}(x-1)^{1/2}$ , B. $f(x)=2-(x-1)^{2/3}$ , C. $f(x)=(x-1)^{2/3}(x-2)^{1/3}$ , D. $f(x)=-1+(x-1)^{3/2}$ . Options: a. Only D, b. C&D only, c. A, B&D, d. A, B&C, e. B, C&D.

See Solution

Problem 20137

Calculate the area under $f(x) = x^2$ from $x = 0$ to $x = 2$ using a definite integral and sketch the curve.

See Solution

Problem 20138

Find the value of $x$ where the tangent to $y=2 \sqrt{2 x+20}$ is perpendicular to $6 x+2 y=1$ . Options: a. 3, b. 7, c. 5, d. 6, e. 8, f. 4, g. Else.

See Solution

Problem 20139

Find the area under the curve for $f(x)=\frac{1}{\sqrt{x}}$ from $x=16$ to $x=25$ using a definite integral.

See Solution

Problem 20140

Find the area under the curve of $f(x)=6 x^{2}+6 x-3$ from $x=1$ to $x=2$ using a definite integral.

See Solution

Problem 20141

Find the tangent line equation to the curve defined by $x=t^{2}-4 t+1$ and $y=t^{3}$ at the point $(-3,8)$ .

See Solution

Problem 20142

Find the area under the curve $f(x)=12-4 \sqrt[3]{x}$ from $x=0$ to $x=8$ using a definite integral.

See Solution

Problem 20143

Gegeben ist die Funktion $f(x)=\frac{2}{\sqrt{x}}$ für $x>0$ . Berechne die Fläche im Intervall $[1; 2]$ und die Linien, die diese halbieren.

See Solution

Problem 20144

Find the area under the curve $f(x)=12 x^{3}$ from $x=1$ to $x=3$ using a definite integral.

See Solution

Problem 20145

Ein Sandhaufen hat eine elliptische Grundfläche mit $a=5 \mathrm{~m}$ und $b=3 \mathrm{~m}$ .
a) Finde das Volumen als Integral. b) Berechne das Volumen. c) Bestimme die Höhe des Sandhaufens.

See Solution

Problem 20146

Find the speed and orbital period of the ISS, which orbits at $350 \mathrm{~km}$ above Earth. Calculate both values.

See Solution

Problem 20147

Calculate the orbital speeds of Venus and Earth around the Sun, given the Sun's mass $1.99 \times 10^{30} \mathrm{~kg}$ , Venus' radius $1.08 \times 10^{11} \mathrm{~m}$ , and Earth's radius $1.49 \times 10^{11} \mathrm{~m}$ .

See Solution

Problem 20148

Check if $f(x)=\sin^{-1} x$ on $[-1,1]$ meets the Mean Value Theorem conditions.

See Solution

Problem 20149

Check if $f(x)=\ln(x-1)$ on $[2,4]$ meets the Mean Value Theorem conditions.

See Solution

Problem 20150

Lösen Sie eine der Aufgaben:
1. Gewicht eines Tieres: $g(t)=-\frac{11}{8} t^{3}+23 t^{2}+18 t+48$ .
2. Geschwindigkeit einer Achterbahn: $v(t)=-\frac{1}{1000}(t^{4}-54 t^{3}+956 t^{2}-5754 t-3229)$ .

See Solution

Problem 20151

Bestimme die Fläche $A(t)$ , die der Graph von $f_{t}(x)=\frac{t}{x^{2}}$ über $[1; 2]$ einschließt, und finde $t$ für $A(t)=8$ .

See Solution

Problem 20152

Find the average value of $f(x)=\sqrt[4]{x}$ over $[1,16]$ and graph the function with its average value.

See Solution

Problem 20153

Calculate the area between the curves $y=-e^{-x}$ and $y=0$ from $x=-2$ to $x=2$ .

See Solution

Problem 20154

Find the volume $V$ of the solid formed by rotating the area between $y=\sqrt{x-1}$ , $y=0$ , and $x=6$ around the $x$ -axis. Answer: $V=12.5 \pi$ . Sketch the region.

See Solution

Problem 20155

Find the volume $V$ of the solid formed by rotating the area between $x=2 \sqrt{7 y}$ , $x=0$ , $y=3$ around the $y$ -axis. $V=$ Sketch the region.

See Solution

Problem 20156

Calculate the integral from 45 to 120 of the function $(172 - x - 28 + 0.2x)$ .

See Solution

Problem 20157

Bestimmen Sie die Ableitung von $f$ an $x_{0}=2$ mit dem Differenzenquotienten für $h \rightarrow 0$ für die Funktionen: a) $f(x)=x^{2}$ b) $f(x)=\frac{2}{x}$ c) $f(x)=2 x^{2}-3$ d) $f(x)=x^{4}$ e) $f(x)=x^{3}$ f) $f(x)=4 x-x^{2}$ g) $f(x)=-\frac{1}{x}$ h) $f(x)=5$

See Solution

Problem 20158

A 500 g isotope decays as $A(t)=500 e^{-0.039 t}$ . Find (a) amount left after 20 years, (b) time to half decay.

See Solution

Problem 20159

Find intervals of concavity and inflection points for $f(x)=(x^{2}-4)e^{x}$ .

See Solution

Problem 20160

Zeigen Sie, dass die Karaffe mindestens $500 \mathrm{ml}$ fasst, indem Sie die Integration durchführen: $\int_{a}^{b}[f(x)-g(x)] d x \geq 500$ .

See Solution

Problem 20161

For the function $f(x)=2x$ from $a=3$ to $b=5$ , find the Riemann sum with 5 rectangles and the exact area using integrals.

See Solution

Problem 20162

Find the max and min of $f(x)=3x-2$ on the intervals $[0,9]$ and $[-3,2]$ . Indicate the $x$ -values.

See Solution

Problem 20163

Bestimme die Fläche $A(t) = \int_{1}^{2} f_{t}(x) \, dx$ für $t > 0$ . Wann ist $A(t) = 8$ FE?

See Solution

Problem 20164

Bestimme die Fläche $A(t) = \int_{1}^{2} f_{t}(x) \, dx$ mit $f_{t}(x) = \frac{t}{x^{2}}$ . Für welches $t$ ist $A(t) = 8$ ?

See Solution

Problem 20165

Find the marginal cost function for $C(x)=4500+5x+0.01x^{2}+0.0002x^{3}$ . Then, calculate it at $x=100$ and find $C(100)$ .

See Solution

Problem 20166

Find the average rate of change of $f(x)=x^{2}-3x-5$ over the interval $-2 \leq x \leq 3$ .

See Solution

Problem 20167

How much to deposit now to have \ $2050 in 60 years at a continuous compounding rate of$ 5.5\%$?

See Solution

Problem 20168

What is the present value needed to reach \$1700 in 60 years at a continuous compounding rate of 7.5%?

See Solution

Problem 20169

Décomposez $f(x)=\mathrm{e}^{\frac{x}{x^{2}+1}}$ en $v \circ u$ et trouvez les dérivées de $u$ , $v$ et $f$ .

See Solution

Problem 20170

Bestimmen Sie die Ableitungsfunktion $f^{\prime}$ und berechnen Sie $f(2)$ für: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

See Solution

Problem 20171

Bestimmen Sie die zweite Ableitung von $r^{\prime}(x)=-2 t x^{2} e^{-x^{2}}$ .

See Solution

Problem 20172

Bestimme die Ableitungsfunktion $f'$ von $f$ und berechne $f(2)$ für die folgenden Funktionen: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

See Solution

Problem 20173

Find the number of devices $x$ to minimize average cost given $C(x)=841,000+160x+0.001x^2$ . What is the average cost and its difference from marginal cost?

See Solution

Problem 20174

Soit $u(x)=-3 x+4$ et $v(x)=x^{3}$ . Trouvez $(v \circ u)(x)$ et calculez $(v \circ u)^{\prime}(x)$ de deux manières.

See Solution

Problem 20175

Ein Stein fällt aus $35 \mathrm{~m}$ Höhe. Bestimme die Höhe nach 1,0 s und 2,0 s sowie Zeit und Geschwindigkeit beim Aufprall.

See Solution

Problem 20176

Find the number of devices $x$ to minimize average cost given $C(x)=484,000+160x+0.001x^{2}$ . What is the average cost? \$\square. How does it differ from marginal cost? \$\square.

See Solution

Problem 20177

Find the second derivatives of $g(x)$ and $h(x)$ if $g^{\prime}(x)=t e^{-x^{2}}$ and $h^{\prime}(x)=-2 t x^{2} e^{-x^{2}}$ .

See Solution

Problem 20178

Find the maximum increase of the dependency ratio $R(t)=0.00731 t^{4}-0.174 t^{3}+1.528 t^{2}+0.48 t+19.3$ for $0 \leq t \leq 6$ , and its value in 2052.

See Solution

Problem 20179

A turkey cools from $185^{\circ}$ F in a $75^{\circ}$ F room.
(a) Find its temp after 45 min if it's $141^{\circ}$ F after 30 min.
(b) When will it reach $100^{\circ}$ F? Use Newton's cooling law:
$T(t) = T_a + (T_o - T_a) \cdot e^{-kt}$

See Solution

Problem 20180

Find the reduction level $q$ for lowest average cost from $C(q)=1,000+194 q^{2}$ . Round $q$ to two decimals. What is the average cost? Round to the nearest dollar.

See Solution

Problem 20181

Find the emission reduction level $q$ that minimizes average cost $C(q)=1,900+193q^{2}$ . Round $q$ to two decimals and find average cost to nearest dollar.

See Solution

Problem 20182

Berechne die Fläche zwischen den Graphen von $f$ und $g$ für die gegebenen Funktionen und Grenzen.

See Solution

Problem 20183

Find the emf induced in a 30-turn, 2.70 mm radius coil with magnetic field $B=50.0+3.20 \sin 1046 \pi t$ mT.

See Solution

Problem 20184

Find the revenue $R$ and profit $P$ functions for the demand $p=600-0.05x$ , and cost $C(x)=0.000002x^3-0.03x^2+400x+80,000$ . Then, compute $C'(2000)$ , $R'(2000)$ , and $P'(2000)$ .

See Solution

Problem 20185

Find the minimum average cost for producing $x$ units, where $C(x)=600+100x-100\ln(x)$ , $x \geq 1$ .

See Solution

Problem 20186

Find the derivative of $f(x) = \frac{\sqrt{x-1}-1}{x-2}$ using the quotient and chain rules.

See Solution

Problem 20187

Evaluate the limit: $\lim _{x \rightarrow 16}\left(\frac{1}{\sqrt{x}-4}-\frac{8}{x-16}\right)=$

See Solution

Problem 20188

La fonction $f(x)=(1-x) \mathrm{e}^{3 x}$ est-elle concave sur $[0 ; 1]$ et où se trouve son point d'inflexion ?

See Solution

Problem 20189

Find the rate of change of consumption with respect to income for $C(x)=0.712 x+95.05$ where $C(x)$ is in \$ billions.

See Solution

Problem 20190

Find $\frac{d r}{d t}$ for the curve $r=3 \cos \theta$ when $\frac{d \theta}{d t}=2$ at $\theta=\frac{\pi}{3}$ .

See Solution

Problem 20191

Find $N(t)=\int 2000(1+0.2 t)^{-3 / 2} d t$ and evaluate $N(5)$ .

See Solution

Problem 20192

Find the tangent line equation to the curve $y=x^{2}(x+1)^{4}$ at point P(1,16).

See Solution

Problem 20193

A 1930s demand function for corn is given by $p=\frac{6,650,000}{q^{1.3}}$ .
(a) Find the price per bushel to maximize revenue (round to nearest cent).
(b) Calculate how much corn can be sold at that price.
(c) Determine the annual revenue (round to nearest cent).

See Solution

Problem 20194

Find the normal line equation to the curve $y=\frac{2 x+3}{6-x}$ at the point $(5,13)$ .

See Solution

Problem 20195

Find the stationary point of the curve $y=(x-4) \mathrm{e}^{-x}$ and determine its nature.

See Solution

Problem 20196

Find the first and second derivatives of $f(x)=4 x^{2}-2 x+1$ .

See Solution

Problem 20197

A particle's position is given by $\langle x(t), y(t)\rangle=\langle\sin(2t), t^{2}-t\rangle$ . Find speed at $t=3$ , speed trend at $t=5$ , total distance from $t=0$ to $t=6$ , and position at $t=11$ after $t=8$ .

See Solution

Problem 20198

Berechne die Steigung von $f$ an $x_{0}$ mit der $h$ -Methode für: a) $f(x)=x^{2}, x_{0}=1$ b) $f(x)=2 x^{2}, x_{0}=-1$ c) $f(x)=x^{3}, x_{0}=2$ d) $f(x)=2 x, x_{0}=1$

See Solution

Problem 20199

A particle's position is given by $\langle x(t), y(t)\rangle = \langle \sin(2t), t^2 - t \rangle$ for $0 \leq t \leq 8$ .
a. Find speed at $t=3$ seconds. b. Is speed increasing or decreasing at $t=5$ seconds? c. Total distance traveled from $t=0$ to $t=6$ seconds? d. Position at $t=11$ seconds if it moves straight from $t=8$ with the same velocity.

See Solution

Problem 20200

If the tangent to the polar curve $r=f(\theta)$ at $\theta=\frac{\pi}{6}$ is vertical, which statement is true?

See Solution

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Calculus

Problem 20101

Find the tangent line equation to 5x2+4y4=−9xy5x^{2}+4y^{4}=-9xy5x2+4y4=−9xy at the point (−1,1)(-1,1)(−1,1).

Problem 20102

Determine if the series ∑k=1∞5k+6k11k\sum_{k=1}^{\infty} \frac{5^{k}+6^{k}}{11^{k}}∑k=1∞​11k5k+6k​ converges or diverges.

Problem 20103

Find the area between f(x)=x3+8f(x)=x^{3}+8f(x)=x3+8 and the xxx-axis over the interval [−3,−1][-3,-1][−3,−1]. The area is □\square□.

Problem 20104

Find the antiderivative FFF of f(x)=x3−3x−4−2f(x)=x^{3}-3 x^{-4}-2f(x)=x3−3x−4−2 such that F(1)=0F(1)=0F(1)=0. What is F(x)=□F(x)=\squareF(x)=□?

Problem 20105

Evaluate the integral using the Fundamental Theorem of Calculus: ∫0126dx1−x2=□\int_{0}^{\frac{1}{2}} \frac{6 d x}{\sqrt{1-x^{2}}}=\square∫021​​1−x2​6dx​=□

Problem 20106

Analyze a trapezoidal velocity-time graph with these details: starts at rest, constant 2 m/s from 1-5s, decelerates at 4s, max velocity -1 m/s at 12s. Find: (i) max distance from start, (ii) displacement after 12s, (iii) max acceleration magnitude.

Problem 20107

Problem 20108

Evaluate the integral H(x)=∫0x4−t2dtH(x)=\int_{0}^{x} \sqrt{4-t^{2}} d tH(x)=∫0x​4−t2​dt for −2≤x≤2-2 \leq x \leq 2−2≤x≤2: a. Find H(0)H(0)H(0), b. H′(1)H'(1)H′(1), c. H′(2)H'(2)H′(2), d. Use geometry for H(2)H(2)H(2), e. Solve H(x)=sH(−x)H(x)=s H(-x)H(x)=sH(−x).

Problem 20109

Evaluate the integral from 1 to infinity: ∫1∞x3e−x4dx\int_{1}^{\infty} x^{3} e^{-x^{4}} d x∫1∞​x3e−x4dx. Is it finite?

Problem 20110

Determine if the series ∑n=2∞n2n3+1\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}∑n=2∞​n3+1n2​ converges or diverges using the integral test. Evaluate ∫2∞x2x3+1dx\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx∫2∞​x3+1x2​dx.

Problem 20111

求函数 f(x)=∣x2−ax−b∣f(x)=\left|x^{2}-a x-b\right|f(x)=∣∣​x2−ax−b∣∣​ 在区间 [−1,1][-1,1][−1,1] 上的导数最大值 M(a,b)M(a, b)M(a,b) 的最小值。

Problem 20112

Use the integral test to check if the series ∑n=2∞n2n3+1\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}∑n=2∞​n3+1n2​ converges or diverges. Evaluate ∫2∞x2x3+1dx\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx∫2∞​x3+1x2​dx. The integral is finite, so the series converges.

Problem 20113

Find the gradient of curve CCC at point P(4,−1)P(4,-1)P(4,−1) and the normal's equation ax+by+c=0ax+by+c=0ax+by+c=0. Also, determine f(x)f(x)f(x).

Problem 20114

Find the gradient of the curve CCC at point P(4,−1)P(4,-1)P(4,−1), then determine the normal's equation in the form ax+by+c=0a x + b y + c = 0ax+by+c=0.

Problem 20115

Determine the values of xxx for which the series ∑n=0∞enx\sum_{n=0}^{\infty} e^{n x}∑n=0∞​enx converges and find its sum.

Problem 20116

Determine convergence or divergence of the series ∑n=1∞n−4\sum_{n=1}^{\infty} n^{-4}∑n=1∞​n−4 using the Integral Test. Evaluate ∫1∞x−4dx\int_{1}^{\infty} x^{-4} dx∫1∞​x−4dx.

Problem 20117

Determine if the series ∑n=1∞n3e−n4\sum_{n=1}^{\infty} n^{3} e^{-n^{4}}∑n=1∞​n3e−n4 converges by evaluating the integral ∫1∞x3e−x4dx\int_{1}^{\infty} x^{3} e^{-x^{4}} d x∫1∞​x3e−x4dx.

Problem 20118

Problem 20119

Evaluate the integral from 1 to 7 of (x3+1)/x2(x^{3}+1)/x^{2}(x3+1)/x2 with respect to xxx.

Problem 20120

Problem 20121

Problem 20122

Evaluate ∫10(9−3x2)dx\int_{1}^{0}(9-3 x^{2}) dx∫10​(9−3x2)dx using ∫01x2dx=13\int_{0}^{1} x^{2} dx=\frac{1}{3}∫01​x2dx=31​.

Problem 20123

Evaluate the integrals: (a) ∫53f(x)dx\int_{5}^{3} f(x) dx∫53​f(x)dx, (b) ∫574h(x)−g(x)2dx\int_{5}^{7} \frac{4 h(x)-g(x)}{2} dx∫57​24h(x)−g(x)​dx, (c) ∫73r(x)dx\int_{7}^{3} r(x) dx∫73​r(x)dx.

Problem 20124

Solve the Boltzmann equation for f1f_{1}f1​ in an electric field, find σαβ(ω)\sigma_{\alpha \beta}(\omega)σαβ​(ω) at T=0T=0T=0.

Problem 20125

Find f′(π2)f^{\prime}\left(\frac{\pi}{2}\right)f′(2π​) for the function f(x)=xcos⁡xf(x)=x^{\cos x}f(x)=xcosx.

Problem 20126

Problem 20127

Find the approximation of f(1.1)f(1.1)f(1.1) using linearization, given the normal line at x=1x=1x=1 is y=−x+7y=-x+7y=−x+7.

Problem 20128

Find (f−1)′(1)\left(f^{-1}\right)^{\prime}(1)(f−1)′(1) for f(x)=x+e2xf(x)=x+e^{2 x}f(x)=x+e2x. Choices: a. 12\frac{1}{2}21​ b. Else c. 16\frac{1}{6}61​ d. 15\frac{1}{5}51​ e. 14\frac{1}{4}41​ f. 13\frac{1}{3}31​.

Problem 20129

Problem 20130

Find f(30)(0)f^{(30)}(0)f(30)(0) for the function f(x)=x30+cos⁡xf(x)=x^{30}+\cos xf(x)=x30+cosx. Choices: a. 30!−130 !-130!−1, b. 311+1311+1311+1, c. 30!+130 !+130!+1, d. 311−1311-1311−1, e. 0, f. 1, g. Else.

Problem 20131

Find the limit: lim⁡x→0sin⁡2xsin⁡xx2cos⁡8x=\lim _{x \rightarrow 0} \frac{\sin 2 x \sin x}{x^{2} \cos 8 x}=limx→0​x2cos8xsin2xsinx​= a. Else b. 4 c. 5 d. 3 e. Does not exist f. 2 g. 0

Problem 20132

Find the derivative dydx\frac{d y}{d x}dxdy​ if y=2cos⁡(3x)y=2 \cos (3 x)y=2cos(3x). Options: A. −sin⁡3x-\sin 3 x−sin3x, B. −3sin⁡3x-3 \sin 3 x−3sin3x, C. 6sin⁡3x6 \sin 3 x6sin3x, D. −6sin⁡3x-6 \sin 3 x−6sin3x.

Problem 20133

Një vijë ka ekuacionin y=f(x)y=f(x)y=f(x). Jepet dydx=10x4−12x2+1\frac{d y}{d x}=10 x^{4}-12 x^{2}+1dxdy​=10x4−12x2+1 dhe f(2)=9f(2)=9f(2)=9. Gjeni f(x)f(x)f(x) dhe tregoni që f(−1)=−24f(-1)=-24f(−1)=−24.

Problem 20134

Find f′(0)+f′(−1)f^{\prime}(0)+f^{\prime}(-1)f′(0)+f′(−1) for f(x)=2xx+2f(x)=\frac{2 x}{x+2}f(x)=x+22x​. Options: A. 0 B. 4 C. 5 D. 1

Problem 20135

Calculate the limit: lim⁡x→0sin⁡(8x)sin⁡(3x)6x2cos⁡(10x)\lim _{x \rightarrow 0} \frac{\sin (8 x) \sin (3 x)}{6 x^{2} \cos (10 x)}limx→0​6x2cos(10x)sin(8x)sin(3x)​. What is the result?

Problem 20136

Problem 20137

Calculate the area under f(x)=x2f(x) = x^2f(x)=x2 from x=0x = 0x=0 to x=2x = 2x=2 using a definite integral and sketch the curve.

Problem 20138

Find the value of xxx where the tangent to y=22x+20y=2 \sqrt{2 x+20}y=22x+20​ is perpendicular to 6x+2y=16 x+2 y=16x+2y=1. Options: a. 3, b. 7, c. 5, d. 6, e. 8, f. 4, g. Else.

Problem 20139

Find the area under the curve for f(x)=1xf(x)=\frac{1}{\sqrt{x}}f(x)=x​1​ from x=16x=16x=16 to x=25x=25x=25 using a definite integral.

Problem 20140

Find the area under the curve of f(x)=6x2+6x−3f(x)=6 x^{2}+6 x-3f(x)=6x2+6x−3 from x=1x=1x=1 to x=2x=2x=2 using a definite integral.

Problem 20141

Find the tangent line equation to the curve defined by x=t2−4t+1x=t^{2}-4 t+1x=t2−4t+1 and y=t3y=t^{3}y=t3 at the point (−3,8)(-3,8)(−3,8).

Problem 20142

Find the area under the curve f(x)=12−4x3f(x)=12-4 \sqrt[3]{x}f(x)=12−43x​ from x=0x=0x=0 to x=8x=8x=8 using a definite integral.

Problem 20143

Gegeben ist die Funktion f(x)=2xf(x)=\frac{2}{\sqrt{x}}f(x)=x​2​ für x>0x>0x>0. Berechne die Fläche im Intervall [1;2][1; 2][1;2] und die Linien, die diese halbieren.

Find the tangent line equation to $5x^{2}+4y^{4}=-9xy$ at the point $(-1,1)$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{5^{k}+6^{k}}{11^{k}}$ converges or diverges.

Find the area between $f(x)=x^{3}+8$ and the $x$ -axis over the interval $[-3,-1]$ . The area is $\square$ .

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}-2$ such that $F(1)=0$ . What is $F(x)=\square$ ?

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{1}{2}} \frac{6 d x}{\sqrt{1-x^{2}}}=\square$

Evaluate the integral $H(x)=\int_{0}^{x} \sqrt{4-t^{2}} d t$ for $-2 \leq x \leq 2$ :
a. Find $H(0)$ , b. $H'(1)$ , c. $H'(2)$ , d. Use geometry for $H(2)$ , e. Solve $H(x)=s H(-x)$ .

Evaluate the integral from 1 to infinity: $\int_{1}^{\infty} x^{3} e^{-x^{4}} d x$ . Is it finite?

Determine if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges using the integral test. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ .

求函数 $f(x)=\left|x^{2}-a x-b\right|$ 在区间 $[-1,1]$ 上的导数最大值 $M(a, b)$ 的最小值。

Use the integral test to check if the series $\sum_{n=2}^{\infty} \frac{n^{2}}{n^{3}+1}$ converges or diverges. Evaluate $\int_{2}^{\infty} \frac{x^{2}}{x^{3}+1} dx$ . The integral is finite, so the series converges.

Find the gradient of curve $C$ at point $P(4,-1)$ and the normal's equation $ax+by+c=0$ . Also, determine $f(x)$ .

Find the gradient of the curve $C$ at point $P(4,-1)$ , then determine the normal's equation in the form $a x + b y + c = 0$ .

Determine the values of $x$ for which the series $\sum_{n=0}^{\infty} e^{n x}$ converges and find its sum.

Determine convergence or divergence of the series $\sum_{n=1}^{\infty} n^{-4}$ using the Integral Test. Evaluate $\int_{1}^{\infty} x^{-4} dx$ .

Determine if the series $\sum_{n=1}^{\infty} n^{3} e^{-n^{4}}$ converges by evaluating the integral $\int_{1}^{\infty} x^{3} e^{-x^{4}} d x$ .

Evaluate the integral from 1 to 7 of $(x^{3}+1)/x^{2}$ with respect to $x$ .

Evaluate $\int_{1}^{0}(9-3 x^{2}) dx$ using $\int_{0}^{1} x^{2} dx=\frac{1}{3}$ .

Evaluate the integrals: (a) $\int_{5}^{3} f(x) dx$ , (b) $\int_{5}^{7} \frac{4 h(x)-g(x)}{2} dx$ , (c) $\int_{7}^{3} r(x) dx$ .

Solve the Boltzmann equation for $f_{1}$ in an electric field, find $\sigma_{\alpha \beta}(\omega)$ at $T=0$ .

Find $f^{\prime}\left(\frac{\pi}{2}\right)$ for the function $f(x)=x^{\cos x}$ .

Find the approximation of $f(1.1)$ using linearization, given the normal line at $x=1$ is $y=-x+7$ .

Find $\left(f^{-1}\right)^{\prime}(1)$ for $f(x)=x+e^{2 x}$ . Choices: a. $\frac{1}{2}$ b. Else c. $\frac{1}{6}$ d. $\frac{1}{5}$ e. $\frac{1}{4}$ f. $\frac{1}{3}$ .

Find $f^{(30)}(0)$ for the function $f(x)=x^{30}+\cos x$ . Choices: a. $30 !-1$ , b. $311+1$ , c. $30 !+1$ , d. $311-1$ , e. 0, f. 1, g. Else.

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 2 x \sin x}{x^{2} \cos 8 x}=$ a. Else b. 4 c. 5 d. 3 e. Does not exist f. 2 g. 0

Find the derivative $\frac{d y}{d x}$ if $y=2 \cos (3 x)$ . Options: A. $-\sin 3 x$ , B. $-3 \sin 3 x$ , C. $6 \sin 3 x$ , D. $-6 \sin 3 x$ .

Një vijë ka ekuacionin $y=f(x)$ . Jepet $\frac{d y}{d x}=10 x^{4}-12 x^{2}+1$ dhe $f(2)=9$ . Gjeni $f(x)$ dhe tregoni që $f(-1)=-24$ .

Find $f^{\prime}(0)+f^{\prime}(-1)$ for $f(x)=\frac{2 x}{x+2}$ . Options: A. 0 B. 4 C. 5 D. 1

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (8 x) \sin (3 x)}{6 x^{2} \cos (10 x)}$ . What is the result?

Calculate the area under $f(x) = x^2$ from $x = 0$ to $x = 2$ using a definite integral and sketch the curve.

Find the value of $x$ where the tangent to $y=2 \sqrt{2 x+20}$ is perpendicular to $6 x+2 y=1$ . Options: a. 3, b. 7, c. 5, d. 6, e. 8, f. 4, g. Else.

Find the area under the curve for $f(x)=\frac{1}{\sqrt{x}}$ from $x=16$ to $x=25$ using a definite integral.

Find the area under the curve of $f(x)=6 x^{2}+6 x-3$ from $x=1$ to $x=2$ using a definite integral.

Find the tangent line equation to the curve defined by $x=t^{2}-4 t+1$ and $y=t^{3}$ at the point $(-3,8)$ .

Find the area under the curve $f(x)=12-4 \sqrt[3]{x}$ from $x=0$ to $x=8$ using a definite integral.

Gegeben ist die Funktion $f(x)=\frac{2}{\sqrt{x}}$ für $x>0$ . Berechne die Fläche im Intervall $[1; 2]$ und die Linien, die diese halbieren.

Find the area under the curve $f(x)=12 x^{3}$ from $x=1$ to $x=3$ using a definite integral.

Ein Sandhaufen hat eine elliptische Grundfläche mit $a=5 \mathrm{~m}$ und $b=3 \mathrm{~m}$ .
a) Finde das Volumen als Integral. b) Berechne das Volumen. c) Bestimme die Höhe des Sandhaufens.

Find the speed and orbital period of the ISS, which orbits at $350 \mathrm{~km}$ above Earth. Calculate both values.

Calculate the orbital speeds of Venus and Earth around the Sun, given the Sun's mass $1.99 \times 10^{30} \mathrm{~kg}$ , Venus' radius $1.08 \times 10^{11} \mathrm{~m}$ , and Earth's radius $1.49 \times 10^{11} \mathrm{~m}$ .

Check if $f(x)=\sin^{-1} x$ on $[-1,1]$ meets the Mean Value Theorem conditions.

Check if $f(x)=\ln(x-1)$ on $[2,4]$ meets the Mean Value Theorem conditions.

Bestimme die Fläche $A(t)$ , die der Graph von $f_{t}(x)=\frac{t}{x^{2}}$ über $[1; 2]$ einschließt, und finde $t$ für $A(t)=8$ .

Find the average value of $f(x)=\sqrt[4]{x}$ over $[1,16]$ and graph the function with its average value.

Calculate the area between the curves $y=-e^{-x}$ and $y=0$ from $x=-2$ to $x=2$ .

Find the volume $V$ of the solid formed by rotating the area between $y=\sqrt{x-1}$ , $y=0$ , and $x=6$ around the $x$ -axis. Answer: $V=12.5 \pi$ . Sketch the region.

Find the volume $V$ of the solid formed by rotating the area between $x=2 \sqrt{7 y}$ , $x=0$ , $y=3$ around the $y$ -axis. $V=$ Sketch the region.

Calculate the integral from 45 to 120 of the function $(172 - x - 28 + 0.2x)$ .

A 500 g isotope decays as $A(t)=500 e^{-0.039 t}$ . Find (a) amount left after 20 years, (b) time to half decay.

Find intervals of concavity and inflection points for $f(x)=(x^{2}-4)e^{x}$ .

Zeigen Sie, dass die Karaffe mindestens $500 \mathrm{ml}$ fasst, indem Sie die Integration durchführen: $\int_{a}^{b}[f(x)-g(x)] d x \geq 500$ .

For the function $f(x)=2x$ from $a=3$ to $b=5$ , find the Riemann sum with 5 rectangles and the exact area using integrals.

Find the max and min of $f(x)=3x-2$ on the intervals $[0,9]$ and $[-3,2]$ . Indicate the $x$ -values.

Bestimme die Fläche $A(t) = \int_{1}^{2} f_{t}(x) \, dx$ für $t > 0$ . Wann ist $A(t) = 8$ FE?

Bestimme die Fläche $A(t) = \int_{1}^{2} f_{t}(x) \, dx$ mit $f_{t}(x) = \frac{t}{x^{2}}$ . Für welches $t$ ist $A(t) = 8$ ?

Find the marginal cost function for $C(x)=4500+5x+0.01x^{2}+0.0002x^{3}$ . Then, calculate it at $x=100$ and find $C(100)$ .

Find the average rate of change of $f(x)=x^{2}-3x-5$ over the interval $-2 \leq x \leq 3$ .

How much to deposit now to have \ $2050 in 60 years at a continuous compounding rate of$ 5.5\%$?

Décomposez $f(x)=\mathrm{e}^{\frac{x}{x^{2}+1}}$ en $v \circ u$ et trouvez les dérivées de $u$ , $v$ et $f$ .

Bestimmen Sie die Ableitungsfunktion $f^{\prime}$ und berechnen Sie $f(2)$ für: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

Bestimmen Sie die zweite Ableitung von $r^{\prime}(x)=-2 t x^{2} e^{-x^{2}}$ .

Bestimme die Ableitungsfunktion $f'$ von $f$ und berechne $f(2)$ für die folgenden Funktionen: a) $f(x)=2 x^{2}$ , b) $f(x)=4 x^{2}$ , c) $f(x)=-3 x^{2}$ , d) $f(x)=\frac{1}{2} x^{2}$ .

Find the number of devices $x$ to minimize average cost given $C(x)=841,000+160x+0.001x^2$ . What is the average cost and its difference from marginal cost?

Soit $u(x)=-3 x+4$ et $v(x)=x^{3}$ . Trouvez $(v \circ u)(x)$ et calculez $(v \circ u)^{\prime}(x)$ de deux manières.