Calculus

Problem 29601

Find constants $a$ , $b$ , and $c$ in the derivative equation: $\frac{d(12 e^{3 x}+2 x^{2}-7)}{d x}=a e^{3 x}+b x+c$ .

See Solution

Problem 29602

Find the derivative of the function $e^{\frac{2}{x-1}}$ with respect to $x$ .

See Solution

Problem 29603

Find the derivative of $e^{\cos (x)}$ and show it equals $a e^{\cos (x)} \sin (x)$ .

See Solution

Problem 29604

Find the derivative $\frac{d y}{d x}$ for the function $y=e^{\sqrt{1+x}}$ . Choose the correct option.

See Solution

Problem 29605

Find the potential energy at $x=2 \mathrm{~m}$ given $F(x)=3 x^{2}-1$ and $U(0)=+13 \mathrm{~J}$ . Options: a. $11 \mathrm{~J}$ b. $7 \mathrm{~J}$ c. $16 \mathrm{~J}$ d. $9 \mathrm{~J}$ e. $13 \mathrm{~J}$

See Solution

Problem 29606

Find the volumes of solids formed by revolving region $R$ (bounded by $y=x^{3}+1$ and $y=9$ ) around $y=9$ (Disk) and $y=1$ (Shell).

See Solution

Problem 29607

احسب النهاية $\lim _{x \rightarrow-1} \frac{x^{2}+x}{x^{2}-x-2}$ واختر من الخيارات: 3، $\frac{1}{3}$ ، $-\frac{1}{3}$ ، 0.

See Solution

Problem 29608

Find the output level that maximizes profit for the demand curve $q=90-2 P$ and total cost $TC=2 q^{2}+20 q+60$ . Choices: a. 5 b. 4 c. 20 d. 4.4

See Solution

Problem 29609

A particle's position is $s(t)=2 t^{3}-21 t^{2}+72 t$ . Find velocity at $t=0$ , rest times, position at $t=14$ , and total distance traveled.

See Solution

Problem 29610

Calculate the integral from 0 to 1 of the function $e^{x} + 2x$ .

See Solution

Problem 29611

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (x)}{x}$ . Choices: $2 / \pi$ , 0, $\pi / 2$ , 1.

See Solution

Problem 29612

Find the limit as $x$ approaches 0 for $\frac{x^{2}-x+\sin x}{2 x}$ . Options: 0, 0.5, -0.5, Does not exist.

See Solution

Problem 29613

Find $t$ such that the derivative of $\sin(e^{5x})$ with respect to $x$ equals: $\frac{d\left(\sin \left(e^{5 x}\right)\right)}{d x}$

See Solution

Problem 29614

Q1. (5 pts) If $\lim _{x \rightarrow 1} f(x)=3$ , are these true or false? (1) $\lim _{x \rightarrow 1^{+}} f(x)=3$ ; (2) $f(1)=3$ . Justify.

See Solution

Problem 29615

Find the limit as $x$ approaches 0 for $\frac{2x + 3\sin x}{5x}$ . Options: 0, $2/5$ , $3/5$ , 1.

See Solution

Problem 29616

Find the volume of region $R$ bounded by $x=8-y^{2}$ , $x+y=6$ , and the $x$ -axis when rotated about $x=8$ (Washer) and $y=-1$ (Shell).

See Solution

Problem 29617

A particle's position is $s(t)=2 t^{3}-27 t^{2}+84 t$ . Find velocity at $t=0$ , rest times, position at $t=18$ , and total distance from $t=0$ to $t=18$ .

See Solution

Problem 29618

Find the correct first derivative of $f(x) = 3x^2 \cdot (x-3)^2$ step by step, correcting $f'(x) = 6x \cdot 2(x-3)$ .

See Solution

Problem 29619

Calculate the integral $\int_{0}^{1}\left(e^{x}+2 x\right) d x$ .

See Solution

Problem 29620

Evaluate $f_{x x}$ for $z=x^{\alpha} y^{\beta}$ . What is $f_{x x}$ ?

See Solution

Problem 29621

Bestimmen Sie die Ableitung von $f_{a}(x)=-x^{2}+a x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

See Solution

Problem 29622

Find the derivative of $f(x)=e^{\log x}$ . Choose the correct expression for $f'(x)$ .

See Solution

Problem 29623

Find the integral function $F(x)$ of $f(x)=\frac{1}{3 x^{2}}$ .

See Solution

Problem 29624

Leiten Sie die Funktionen nach $x, y$ und ggf. $z$ ab: d) $f(x, y)=\frac{x^{2}}{y}$ , e) $f(x, y, z)=x z+\mathrm{e}^{x y z}$ , f) $f(x, y)=a \cdot \sin (x)+y \cdot \sin (a)-x y$ , g) $f(x, y)=x^{2} \sqrt{4-(x y)^{2}}$ .

See Solution

Problem 29625

Leiten Sie die Funktionen nach $x, y$ und ggf. $z$ ab: a) $f(x, y)=6 x^{2}+3 x y+y$ b) $f(x, y)=x^{3}+x^{2} y^{2}+y^{4}-6 x y$ c) $f(x, y)=x^{0.3} y^{0.7}$ d) $f(x, y)=\frac{x^{2}}{y}$ e) $f(x, y, z)=x z+\mathrm{e}^{x y z}$ f) $f(x, y)=a \cdot \sin (x)+y \cdot \sin (a)-x y$ g) $f(x, y)=x^{2} \sqrt{4-(x y)^{2}}$

See Solution

Problem 29626

Gegeben ist die Funktion $f(x, y) = y^{3} + 3x^{2}y - 12x - 15y$ . Finde die kritischen Stellen und bestimme die Art der Extremstellen.

See Solution

Problem 29627

Bestimmen Sie den Gradienten und die Hesse-Matrix von $f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2}+e^{x_{3}+x_{4}}-x_{5}(x_{1}^{2}-2 x_{3})$ .

See Solution

Problem 29628

Gegeben ist die Funktion $f(x, y) = x^{2} - 6xy - y^{3}$ . Finde die kritischen Punkte und bestimme die Art der Extremstellen.

See Solution

Problem 29629

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x$ .
a) Zeigen Sie die Punktsymmetrie zum Ursprung. b) Beweisen Sie, dass $f_{a}$ durch $P(-2,0)$ und $Q(2,0)$ verläuft. c) Zeigen Sie, dass es je einen Hoch- und Tiefpunkt gibt. d) Bestimmen Sie die Wendetangente und für welche $a$ die Steigung $m=8$ ist.

See Solution

Problem 29630

Find the derivative of $\sqrt{a-2 x}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 29631

Find the derivative of $f(x)=\frac{x-1}{x+2}$ . Which option is correct? (1) $f'(x)=\frac{1}{(x+2)^{2}}$ , (2) $f'(x)=\frac{3}{(x+2)^{2}}$ , (3) $f'(x)=\frac{-3}{(x+2)^{2}}$ ?

See Solution

Problem 29632

Find the derivative of $(8 - \cos(4x))^2$ with respect to $x$ .

See Solution

Problem 29633

Find the derivative of $\frac{p x-1}{4 x+3}$ and set it equal to $\frac{7}{(4 x+3)^{2}}$ .

See Solution

Problem 29634

Calculate the limit: $\lim _{x \rightarrow+\infty} x\left[\left(1+\frac{1}{x}\right)^{x}-e\right]$ .

See Solution

Problem 29635

Find $a$ and $b$ such that $\frac{d(4(x^{2}+y^{2}))}{dx} = a x + b y \frac{dy}{dx}$ .

See Solution

Problem 29636

Find the derivative $\frac{d y}{d x}$ for the function $y=\left(3 x^{2}-4\right) \ln (5 x+2)$ .

See Solution

Problem 29637

Gegeben ist die Funktion $f(x)=x^{3}-3 x^{2}+3$ . Finde die Tangentengleichung im Wendepunkt und den Flächeninhalt mit den Achsen.

See Solution

Problem 29638

Find the derivative of the function $f(x)=\frac{\cos (x)}{1-\sin (x)}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 29639

Find the limit as $x$ approaches 2 from the left for $\frac{\sin(x^{2}-4)}{x-2}$ .

See Solution

Problem 29640

أوجد $\lim _{x \rightarrow \infty} \frac{2 x+4 x^{4}}{2 x^{4}+3}$ . الخيارات: $\frac{2}{3}$ ، $\frac{3}{2}$ ، 2، 0.

See Solution

Problem 29641

Gegeben ist $f_{b}(x)=x \cdot e^{x+b}$ . Berechnen Sie $f_{3}(-3)$ , die Steigung im Ursprung, den Tiefpunkt und das Verhalten für $x \rightarrow-\infty$ . Erklären Sie den Einfluss von $b$ auf den Graphen.

See Solution

Problem 29642

Die Temperatur eines Kaffees wird durch $T(t)=20+70 \cdot e^{-k t}$ beschrieben.
a) Was ist die Anfangstemperatur? b) Was ist die Raumtemperatur? c) Finde $k$ , wenn $T(30) = 35,6^{\circ} \mathrm{C}$ . d) Wann ist die Änderungsrate $-1 \frac{{ }^{\circ} \mathrm{C}}{\min }$ für $k=0,05$ ?

See Solution

Problem 29643

Find the average acceleration of a rocket that rises 1.60 x 10^5 m in 4 min, reaching 7.60 km/s.

See Solution

Problem 29644

Calculate the following limits using L'Hospital's rule if needed: a) $\lim_{x \rightarrow 0} \frac{e^{x}-1}{\sin 2x}$ , b) $\lim_{x \rightarrow 1} \frac{1-x+\ln x}{1+\cos(\pi x)}$ , c) $\lim_{x \rightarrow 0} \frac{\sin x}{x+x^{2}}$ , d) $\lim_{x \rightarrow+\infty} \frac{e^{x}}{x+x^{2}}$ , e) $\lim_{x \rightarrow+\infty} \frac{\ln x}{\sqrt{x}}$ , f) $\lim_{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$ , g) $\lim_{x \rightarrow 0}\left(\frac{1}{x}-\operatorname{ctg} x\right)$ , h) $\lim_{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{1}{\ln(x-1)}\right)$ , i) $\lim_{x \rightarrow+\infty}\left(\sqrt{x^{2}+3x}-x\right)$ , j) $\lim_{x \rightarrow 0^{+}} x \ln x$ , k) $\lim_{x \rightarrow 0}(\cos x)^{\frac{1}{x^{2}}}$ , l) $\lim_{x \rightarrow+\infty} x\left[\left(1+\frac{1}{x}\right)^{x}-e\right]$ .

See Solution

Problem 29645

Calculate the Taylor series and radius of convergence for these functions at given points: a) $f(x)=\frac{1}{1+x}$ , b) $f(x)=\frac{1}{2+x^{2}}$ , c) $f(x)=\frac{1+x}{2+3 x^{2}}$ , d) $f(x)=\frac{1}{(x-2)(4-x)}$ (at $x_{0}=3$ ), e) $f(x)=\frac{1}{(x-2)(4-x)}$ (at $x_{0}=\frac{5}{2}$ ), f) $f(x)=\frac{1}{(x-4)^{2}}$ (at $x_{0}=5$ ), g) $f(x)=(1+x)^{2}$ (at $x_{0}=2$ ), h) $f(x)=(1+x)^{\frac{1}{2}}$ , i) $f(x)=(1-x)^{\frac{1}{3}}$ , j) $f(x)=(1-x)^{-\frac{1}{3}}$ , k) $f(x)=\sin(x^{2})$ , l) $f(x)=\sin^{2}(x)$ .

See Solution

Problem 29646

Choose the true statement about the derivative of $\ln(x^2 + x)$ from the options below:
1. $\frac{d}{d x}\left(\ln \left(x^{2}+x\right)\right)=\frac{d\left(\ln \left(x^{2}\right)\right)}{d x}+\frac{d(\ln (x))}{d x}$
2. $\frac{d}{d x}\left(\ln \left(x^{2}+x\right)\right)=\frac{d(\ln (x))}{d x} \cdot \frac{d(\ln (x+1))}{d x}$
3. $\frac{d}{d x}\left(\ln \left(x^{2}+x\right)\right)=\frac{d(\ln (x))}{d x}+\frac{d(\ln (x+1))}{d x}$
4. $\frac{d}{d x}\left(\ln \left(x^{2}+x\right)\right)=\ln \left(\frac{d}{d x}\left(x^{2}+x\right)\right)$

See Solution

Problem 29647

Find $\lim _{x \rightarrow 5^{+}} \frac{\sqrt{4(\ln (x-5))^{2}+1}}{\ln (x-5)}$ .

See Solution

Problem 29648

Differentiate: $y=x^{4}+5x$

See Solution

Problem 29649

Find the integral: $\int e^{-4 x} d x$

See Solution

Problem 29650

If $f$ is continuous on $[a, b]$ and $f'(x)<0$ for $x \in (a, b)$ , what can we say about $f$ ?

See Solution

Problem 29651

Find the average rate of change of $f(x)=\frac{1}{3x+1}$ over $[2,5]$ and approximate the instantaneous rate at $x=2$ .

See Solution

Problem 29652

Find the critical points of $f^{\prime}(x)=2 x-3$ for $x \in[0,4]$ . Options: a) None b) $x=\frac{3}{2}$ c) $0, \frac{3}{2}$ d) $-\frac{3}{2}$ .

See Solution

Problem 29653

Find critical points, increasing, and decreasing intervals for $f(x)=8 \ln x-x^{2}$ .

See Solution

Problem 29654

Find the interval where $f(x)=(2+3x)^{3}$ is decreasing. Options: a) $x \leq \frac{-2}{3}$ b) $0 \geq \frac{2}{3}$ c) $\frac{-2}{3} \leq x \leq \frac{2}{3}$ d) Never Decreasing.

See Solution

Problem 29655

Find the limit: $\lim _{x \rightarrow 1^{-}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

See Solution

Problem 29656

Evaluate the integral of $e^{-\frac{\varepsilon}{k_{B} T}}$ with respect to $\varepsilon$ .

See Solution

Problem 29657

Calculate the limit: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x-3}{|x-1|}$ or state if it doesn't exist.

See Solution

Problem 29658

Find the average rate of change of $f(x)=2 x^{2}-9$ from $x=4$ to $x=b$ . Simplify your answer in terms of $b$ .

See Solution

Problem 29659

Find the limit: $\lim _{x \rightarrow-1} \frac{2|x+1|}{x+1}$ .

See Solution

Problem 29660

Find the value of $q$ that maximizes total revenue, satisfying $TR'(q) = 0$ and $TR''(q) < 0$ .

See Solution

Problem 29661

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-2$ .

See Solution

Problem 29662

Is the function $f(x)=\begin{cases}3 \ln (1+x) & x \geq 0 \\ x^{5}+x^{5} & x<0\end{cases}$ continuous on R? (YES / NO)

See Solution

Problem 29663

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{2}$ .

See Solution

Problem 29664

Find the limit: $\lim _{x \rightarrow \infty}-\frac{x^{3}}{3 x^{2}-3}$ .

See Solution

Problem 29665

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on R? (YES / NO)

See Solution

Problem 29666

Find the instantaneous rate of change for $y=-\frac{1}{x-2}$ at $x=-3$ .

See Solution

Problem 29667

Find critical points of $\sin x + \cos y = 2y$ and classify them using first and second derivative tests.

See Solution

Problem 29668

Find critical points of $\sin x+\cos y=2 y$ and classify them using first and second derivative tests.

See Solution

Problem 29669

Differentiate the function $y=-\frac{14 x^{3}}{5-6 x}$ . Identify the appropriate differentiation rule to use.

See Solution

Problem 29670

Find the numerator of the derivative of $y=-\frac{14 x^{3}}{5-6 x}$ using the quotient rule.

See Solution

Problem 29671

Analyze the function $f(x)=\frac{4}{3} x-\tan x$ for concavity in the intervals given.

See Solution

Problem 29672

Find the surface area of the curve $x=2 \sqrt{\cos (4 y)}$ from $y=-\frac{\pi}{16}$ to $y=\frac{\pi}{16}$ around the $y$ -axis.

See Solution

Problem 29673

Find the limits: $\lim _{x \rightarrow 3^{-}} f(x)$ and $\lim _{x \rightarrow 3^{+}} f(x)$ using the given data.

See Solution

Problem 29674

Untersuche den Abbau eines Kontrastmittels:
a) Finde die Funktionsgleichung. b) Wie viel bleibt nach 60 Minuten? c) Wann sind 95 % abgebaut? d) Ist das Kontrastmittel nach 24 Stunden fast weg?

See Solution

Problem 29675

Find the limit as $x$ approaches 2 from the left: $\lim _{x \rightarrow 2^{-}}(x^{2}-4) \ln (2-x)$ .

See Solution

Problem 29676

Find the tangent points for the function $y = -\frac{2}{x^2 - 9}$ .

See Solution

Problem 29677

Find the limit: $\lim _{n \rightarrow \infty} 6\left(n+\frac{1}{n}\right)^{\frac{1}{n}}+|x|$ .

See Solution

Problem 29678

Show using the Mean Value Theorem that: a) $\frac{x}{x+1}<\ln (1+x)<x$ for $x>0$ ; b) $|\arctan x-\arctan y| \leq|x-y|$ for $x, y \in \mathbb{R}$ .

See Solution

Problem 29679

Calculate the limit: $\lim _{x \rightarrow \frac{\pi}{2}}\left(\operatorname{tg} x-\frac{1}{\frac{\pi}{2}-x}\right)$ .

See Solution

Problem 29680

Calculate the limit: $\lim _{x \rightarrow \frac{\pi}{2}}\left(\operatorname{tg}(x)-\frac{1}{\frac{\pi}{2}-x}\right)$ .

See Solution

Problem 29681

Find the vertical asymptote at $x=3$ and determine the horizontal asymptotes for $f(x) = \frac{1}{x-3} + 10$ .

See Solution

Problem 29682

Calculate the limit: $\lim _{x \rightarrow 0^{+}} \left( \frac{1}{\sin 4 x}-\frac{1}{4 x} \right)$ .

See Solution

Problem 29683

Calculate the following limits:
1. $\lim _{x \rightarrow+\infty}\left(\frac{2 x^{2}+1}{3 x^{2}+1}\right)^{x-x^{2}}$
2. $\lim _{x \rightarrow 0}\left(\frac{2 x^{2}+1}{3 x^{2}+1}\right)^{-x^{2}+x^{3}}$
3. $\lim _{x \rightarrow \frac{\pi}{2}}\left(\operatorname{tg} x-\frac{1}{\frac{\pi}{2}-x}\right)$ .

See Solution

Problem 29684

Find where the following functions are increasing or decreasing: a) $f(x)=\frac{x^{3}}{x-2}$ . b) $f(x)=x \cdot \ln x, x>0$ . c) $f(x)=e^{x} \cos x$ .

See Solution

Problem 29685

Find the first derivative of the function $y=(2x^{2}-6x-14)(5x^{4}-9)$ using the product rule. What is $\frac{dy}{dx}=$ ?

See Solution

Problem 29686

Determine if the function $f(x)=\frac{1}{1-x^{2}}$ has a max, min, or is increasing (YES/NO for each).

See Solution

Problem 29687

Find the instantaneous rate of change of height $s(t)=8t-t^{2}$ at $t=3$ hours.

See Solution

Problem 29688

Tickets for the Taylor Swift concert sold out in 9 hours. Answer the following:
(a) Estimate the rate of change of $L(t)$ at $t=5.5$ . (b) Use a trapezoidal sum to estimate average $L(t)$ in the first 4 hours. (c) How many times does $L'(t)=0$ for $0 \leq t \leq 9$ ? (d) How many tickets were sold by $t=4$ using $r(t)=560 t e^{-\frac{1}{2}t}$ ?

See Solution

Problem 29689

If $f$ and $g$ are continuous with $g^{\prime}(x)=f(x)$ , find $\int_{2}^{3} f(x) dx=$ (A) $g^{\prime}(2)-g^{\prime}(3)$ (B) $g^{\prime}(3)-g^{\prime}(2)$ (C) $g(3)-g(2)$ (D) $f(3)-f(2)$ (E) $f^{\prime}(3)-f^{\prime}(2)$ .

See Solution

Problem 29690

A particle moves along the $y$ -axis with velocity $v(t)=\sin(t^2)$ . Answer the following questions about its motion.

See Solution

Problem 29691

Find critical points of $\sin x + \cos y = 2y$ using implicit differentiation. Classify them with first and second derivative tests.

See Solution

Problem 29692

If $f(2)=f'(2)=f''(2)=0$ , which statements about $f$ at $x=2$ are true? A) Local extremum B) Point of inflection C) Horizontal tangent D) No local extremum E) None

See Solution

Problem 29693

Berechne die Ableitung von $f$ an $x_0$ mit der h-Methode für: a. $f(x)=6x+1$ , $x_0=2$ ; b. $f(x)=x^2$ , $x_0=3$ ; c. $f(x)=2x^3$ , $x_0=-1$ ; d. $f(x)=-x^2+4$ , $x_0=-4$ ; e. $f(x)=-4x^4+2x$ , $x_0=1$ .

See Solution

Problem 29694

Find the future value of \ $2500 compounded continuously for 6 years at an 8% annual interest rate. Use the formula$ A = Pe^{rt}$.

See Solution

Problem 29695

Berechne das Integral von $f(x)=3 x^{2}-2$ im Intervall $[0 ; 3]$ .

See Solution

Problem 29696

Berechne den Flächeninhalt zwischen $f(x)=\frac{x^{2}}{5}+1$ für $x=2$ und $x=4$ .

See Solution

Problem 29697

Berechne das Integral von $f(x)=\frac{x^{3}}{4}$ im Intervall $[1 ; 3]$ und finde eine Stammfunktion $F$ .

See Solution

Problem 29698

Berechne den Flächeninhalt unter der Kurve von $f(x)=\frac{6}{x^{2}}+2$ zwischen $x=2$ und $x=3$ .

See Solution

Problem 29699

Which technique is best for solving the integral $\int x^{5} \sqrt{x^{2}+7} d x$ ? A-F options provided.

See Solution

Problem 29700

Calculate the integral $\int_{1}^{10}\left(\frac{1}{x}+2 x^{-2}\right)$ . Choose the correct answer: 1) -2 2) 3.91 3) 4.10 4) 42.

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 29601

Find constants aaa, bbb, and ccc in the derivative equation: d(12e3x+2x2−7)dx=ae3x+bx+c\frac{d(12 e^{3 x}+2 x^{2}-7)}{d x}=a e^{3 x}+b x+cdxd(12e3x+2x2−7)​=ae3x+bx+c.

Problem 29602

Find the derivative of the function e2x−1e^{\frac{2}{x-1}}ex−12​ with respect to xxx.

Problem 29603

Find the derivative of ecos⁡(x)e^{\cos (x)}ecos(x) and show it equals aecos⁡(x)sin⁡(x)a e^{\cos (x)} \sin (x)aecos(x)sin(x).

Problem 29604

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=e1+xy=e^{\sqrt{1+x}}y=e1+x​. Choose the correct option.

Problem 29605

Find the potential energy at x=2 mx=2 \mathrm{~m}x=2 m given F(x)=3x2−1F(x)=3 x^{2}-1F(x)=3x2−1 and U(0)=+13 JU(0)=+13 \mathrm{~J}U(0)=+13 J. Options: a. 11 J11 \mathrm{~J}11 J b. 7 J7 \mathrm{~J}7 J c. 16 J16 \mathrm{~J}16 J d. 9 J9 \mathrm{~J}9 J e. 13 J13 \mathrm{~J}13 J

Problem 29606

Find the volumes of solids formed by revolving region RRR (bounded by y=x3+1y=x^{3}+1y=x3+1 and y=9y=9y=9) around y=9y=9y=9 (Disk) and y=1y=1y=1 (Shell).

Problem 29607

احسب النهاية lim⁡x→−1x2+xx2−x−2\lim _{x \rightarrow-1} \frac{x^{2}+x}{x^{2}-x-2}limx→−1​x2−x−2x2+x​ واختر من الخيارات: 3، 13\frac{1}{3}31​، −13-\frac{1}{3}−31​، 0.

Problem 29608

Find the output level that maximizes profit for the demand curve q=90−2Pq=90-2 Pq=90−2P and total cost TC=2q2+20q+60TC=2 q^{2}+20 q+60TC=2q2+20q+60. Choices: a. 5 b. 4 c. 20 d. 4.4

Problem 29609

A particle's position is s(t)=2t3−21t2+72ts(t)=2 t^{3}-21 t^{2}+72 ts(t)=2t3−21t2+72t. Find velocity at t=0t=0t=0, rest times, position at t=14t=14t=14, and total distance traveled.

Problem 29610

Calculate the integral from 0 to 1 of the function ex+2xe^{x} + 2xex+2x.

Problem 29611

Find the limit: lim⁡x→π2sin⁡(x)x\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (x)}{x}limx→2π​​xsin(x)​. Choices: 2/π2 / \pi2/π, 0, π/2\pi / 2π/2, 1.

Problem 29612

Find the limit as xxx approaches 0 for x2−x+sin⁡x2x\frac{x^{2}-x+\sin x}{2 x}2xx2−x+sinx​. Options: 0, 0.5, -0.5, Does not exist.

Problem 29613

Find ttt such that the derivative of sin⁡(e5x)\sin(e^{5x})sin(e5x) with respect to xxx equals: d(sin⁡(e5x))dx\frac{d\left(\sin \left(e^{5 x}\right)\right)}{d x}dxd(sin(e5x))​

Problem 29614

Q1. (5 pts) If lim⁡x→1f(x)=3\lim _{x \rightarrow 1} f(x)=3limx→1​f(x)=3, are these true or false? (1) lim⁡x→1+f(x)=3\lim _{x \rightarrow 1^{+}} f(x)=3limx→1+​f(x)=3; (2) f(1)=3f(1)=3f(1)=3. Justify.

Problem 29615

Find the limit as xxx approaches 0 for 2x+3sin⁡x5x\frac{2x + 3\sin x}{5x}5x2x+3sinx​. Options: 0, 2/52/52/5, 3/53/53/5, 1.

Problem 29616

Find the volume of region RRR bounded by x=8−y2x=8-y^{2}x=8−y2, x+y=6x+y=6x+y=6, and the xxx-axis when rotated about x=8x=8x=8 (Washer) and y=−1y=-1y=−1 (Shell).

Problem 29617

A particle's position is s(t)=2t3−27t2+84ts(t)=2 t^{3}-27 t^{2}+84 ts(t)=2t3−27t2+84t. Find velocity at t=0t=0t=0, rest times, position at t=18t=18t=18, and total distance from t=0t=0t=0 to t=18t=18t=18.

Problem 29618

Find the correct first derivative of f(x)=3x2⋅(x−3)2 f(x) = 3x^2 \cdot (x-3)^2 f(x)=3x2⋅(x−3)2 step by step, correcting f′(x)=6x⋅2(x−3) f'(x) = 6x \cdot 2(x-3) f′(x)=6x⋅2(x−3).

Problem 29619

Calculate the integral ∫01(ex+2x)dx\int_{0}^{1}\left(e^{x}+2 x\right) d x∫01​(ex+2x)dx.

Problem 29620

Evaluate fxxf_{x x}fxx​ for z=xαyβz=x^{\alpha} y^{\beta}z=xαyβ. What is fxxf_{x x}fxx​?

Problem 29621

Bestimmen Sie die Ableitung von fa(x)=−x2+axf_{a}(x)=-x^{2}+a xfa​(x)=−x2+ax und die Steigung bei x=0x=0x=0. Für welches aaa ist die Steigung 1?

Problem 29622

Find the derivative of f(x)=elog⁡xf(x)=e^{\log x}f(x)=elogx. Choose the correct expression for f′(x)f'(x)f′(x).

Problem 29623

Find the integral function F(x)F(x)F(x) of f(x)=13x2f(x)=\frac{1}{3 x^{2}}f(x)=3x21​.

Problem 29624

Problem 29625

Problem 29626

Gegeben ist die Funktion f(x,y)=y3+3x2y−12x−15yf(x, y) = y^{3} + 3x^{2}y - 12x - 15yf(x,y)=y3+3x2y−12x−15y. Finde die kritischen Stellen und bestimme die Art der Extremstellen.

Problem 29627

Bestimmen Sie den Gradienten und die Hesse-Matrix von f(x1,x2,x3,x4,x5)=x1x2+ex3+x4−x5(x12−2x3)f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2}+e^{x_{3}+x_{4}}-x_{5}(x_{1}^{2}-2 x_{3})f(x1​,x2​,x3​,x4​,x5​)=x1​x2​+ex3​+x4​−x5​(x12​−2x3​).

Problem 29628

Gegeben ist die Funktion f(x,y)=x2−6xy−y3f(x, y) = x^{2} - 6xy - y^{3}f(x,y)=x2−6xy−y3. Finde die kritischen Punkte und bestimme die Art der Extremstellen.

Problem 29629

Problem 29630

Find the derivative of a−2x\sqrt{a-2 x}a−2x​ with respect to xxx, where aaa is a constant.

Problem 29631

Find the derivative of f(x)=x−1x+2f(x)=\frac{x-1}{x+2}f(x)=x+2x−1​. Which option is correct? (1) f′(x)=1(x+2)2f'(x)=\frac{1}{(x+2)^{2}}f′(x)=(x+2)21​, (2) f′(x)=3(x+2)2f'(x)=\frac{3}{(x+2)^{2}}f′(x)=(x+2)23​, (3) f′(x)=−3(x+2)2f'(x)=\frac{-3}{(x+2)^{2}}f′(x)=(x+2)2−3​?

Problem 29632

Find the derivative of (8−cos⁡(4x))2(8 - \cos(4x))^2(8−cos(4x))2 with respect to xxx.

Problem 29633

Find the derivative of px−14x+3\frac{p x-1}{4 x+3}4x+3px−1​ and set it equal to 7(4x+3)2\frac{7}{(4 x+3)^{2}}(4x+3)27​.

Problem 29634

Calculate the limit: lim⁡x→+∞x[(1+1x)x−e]\lim _{x \rightarrow+\infty} x\left[\left(1+\frac{1}{x}\right)^{x}-e\right]limx→+∞​x[(1+x1​)x−e].

Problem 29635

Find aaa and bbb such that d(4(x2+y2))dx=ax+bydydx\frac{d(4(x^{2}+y^{2}))}{dx} = a x + b y \frac{dy}{dx}dxd(4(x2+y2))​=ax+bydxdy​.

Problem 29636

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=(3x2−4)ln⁡(5x+2)y=\left(3 x^{2}-4\right) \ln (5 x+2)y=(3x2−4)ln(5x+2).

Problem 29637

Gegeben ist die Funktion f(x)=x3−3x2+3f(x)=x^{3}-3 x^{2}+3f(x)=x3−3x2+3. Finde die Tangentengleichung im Wendepunkt und den Flächeninhalt mit den Achsen.

Problem 29638

Find the derivative of the function f(x)=cos⁡(x)1−sin⁡(x)f(x)=\frac{\cos (x)}{1-\sin (x)}f(x)=1−sin(x)cos(x)​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 29639

Find the limit as xxx approaches 2 from the left for sin⁡(x2−4)x−2\frac{\sin(x^{2}-4)}{x-2}x−2sin(x2−4)​.

Problem 29640

أوجد lim⁡x→∞2x+4x42x4+3\lim _{x \rightarrow \infty} \frac{2 x+4 x^{4}}{2 x^{4}+3}limx→∞​2x4+32x+4x4​. الخيارات: 23\frac{2}{3}32​، 32\frac{3}{2}23​، 2، 0.

Problem 29641

Gegeben ist fb(x)=x⋅ex+bf_{b}(x)=x \cdot e^{x+b}fb​(x)=x⋅ex+b. Berechnen Sie f3(−3)f_{3}(-3)f3​(−3), die Steigung im Ursprung, den Tiefpunkt und das Verhalten für x→−∞x \rightarrow-\inftyx→−∞. Erklären Sie den Einfluss von bbb auf den Graphen.

Find constants $a$ , $b$ , and $c$ in the derivative equation: $\frac{d(12 e^{3 x}+2 x^{2}-7)}{d x}=a e^{3 x}+b x+c$ .

Find the derivative of the function $e^{\frac{2}{x-1}}$ with respect to $x$ .

Find the derivative of $e^{\cos (x)}$ and show it equals $a e^{\cos (x)} \sin (x)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=e^{\sqrt{1+x}}$ . Choose the correct option.

Find the potential energy at $x=2 \mathrm{~m}$ given $F(x)=3 x^{2}-1$ and $U(0)=+13 \mathrm{~J}$ . Options: a. $11 \mathrm{~J}$ b. $7 \mathrm{~J}$ c. $16 \mathrm{~J}$ d. $9 \mathrm{~J}$ e. $13 \mathrm{~J}$

Find the volumes of solids formed by revolving region $R$ (bounded by $y=x^{3}+1$ and $y=9$ ) around $y=9$ (Disk) and $y=1$ (Shell).

احسب النهاية $\lim _{x \rightarrow-1} \frac{x^{2}+x}{x^{2}-x-2}$ واختر من الخيارات: 3، $\frac{1}{3}$ ، $-\frac{1}{3}$ ، 0.

Find the output level that maximizes profit for the demand curve $q=90-2 P$ and total cost $TC=2 q^{2}+20 q+60$ . Choices: a. 5 b. 4 c. 20 d. 4.4

A particle's position is $s(t)=2 t^{3}-21 t^{2}+72 t$ . Find velocity at $t=0$ , rest times, position at $t=14$ , and total distance traveled.

Calculate the integral from 0 to 1 of the function $e^{x} + 2x$ .

Find the limit: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (x)}{x}$ . Choices: $2 / \pi$ , 0, $\pi / 2$ , 1.

Find the limit as $x$ approaches 0 for $\frac{x^{2}-x+\sin x}{2 x}$ . Options: 0, 0.5, -0.5, Does not exist.

Find $t$ such that the derivative of $\sin(e^{5x})$ with respect to $x$ equals: $\frac{d\left(\sin \left(e^{5 x}\right)\right)}{d x}$

Q1. (5 pts) If $\lim _{x \rightarrow 1} f(x)=3$ , are these true or false? (1) $\lim _{x \rightarrow 1^{+}} f(x)=3$ ; (2) $f(1)=3$ . Justify.

Find the limit as $x$ approaches 0 for $\frac{2x + 3\sin x}{5x}$ . Options: 0, $2/5$ , $3/5$ , 1.

Find the volume of region $R$ bounded by $x=8-y^{2}$ , $x+y=6$ , and the $x$ -axis when rotated about $x=8$ (Washer) and $y=-1$ (Shell).

A particle's position is $s(t)=2 t^{3}-27 t^{2}+84 t$ . Find velocity at $t=0$ , rest times, position at $t=18$ , and total distance from $t=0$ to $t=18$ .

Find the correct first derivative of $f(x) = 3x^2 \cdot (x-3)^2$ step by step, correcting $f'(x) = 6x \cdot 2(x-3)$ .

Calculate the integral $\int_{0}^{1}\left(e^{x}+2 x\right) d x$ .

Evaluate $f_{x x}$ for $z=x^{\alpha} y^{\beta}$ . What is $f_{x x}$ ?

Bestimmen Sie die Ableitung von $f_{a}(x)=-x^{2}+a x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

Find the derivative of $f(x)=e^{\log x}$ . Choose the correct expression for $f'(x)$ .

Find the integral function $F(x)$ of $f(x)=\frac{1}{3 x^{2}}$ .

Gegeben ist die Funktion $f(x, y) = y^{3} + 3x^{2}y - 12x - 15y$ . Finde die kritischen Stellen und bestimme die Art der Extremstellen.

Bestimmen Sie den Gradienten und die Hesse-Matrix von $f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2}+e^{x_{3}+x_{4}}-x_{5}(x_{1}^{2}-2 x_{3})$ .

Gegeben ist die Funktion $f(x, y) = x^{2} - 6xy - y^{3}$ . Finde die kritischen Punkte und bestimme die Art der Extremstellen.

Find the derivative of $\sqrt{a-2 x}$ with respect to $x$ , where $a$ is a constant.

Find the derivative of $f(x)=\frac{x-1}{x+2}$ . Which option is correct? (1) $f'(x)=\frac{1}{(x+2)^{2}}$ , (2) $f'(x)=\frac{3}{(x+2)^{2}}$ , (3) $f'(x)=\frac{-3}{(x+2)^{2}}$ ?

Find the derivative of $(8 - \cos(4x))^2$ with respect to $x$ .

Find the derivative of $\frac{p x-1}{4 x+3}$ and set it equal to $\frac{7}{(4 x+3)^{2}}$ .

Calculate the limit: $\lim _{x \rightarrow+\infty} x\left[\left(1+\frac{1}{x}\right)^{x}-e\right]$ .

Find $a$ and $b$ such that $\frac{d(4(x^{2}+y^{2}))}{dx} = a x + b y \frac{dy}{dx}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\left(3 x^{2}-4\right) \ln (5 x+2)$ .

Gegeben ist die Funktion $f(x)=x^{3}-3 x^{2}+3$ . Finde die Tangentengleichung im Wendepunkt und den Flächeninhalt mit den Achsen.

Find the derivative of the function $f(x)=\frac{\cos (x)}{1-\sin (x)}$ . What is $f^{\prime}(x)$ ?

Find the limit as $x$ approaches 2 from the left for $\frac{\sin(x^{2}-4)}{x-2}$ .

أوجد $\lim _{x \rightarrow \infty} \frac{2 x+4 x^{4}}{2 x^{4}+3}$ . الخيارات: $\frac{2}{3}$ ، $\frac{3}{2}$ ، 2، 0.

Gegeben ist $f_{b}(x)=x \cdot e^{x+b}$ . Berechnen Sie $f_{3}(-3)$ , die Steigung im Ursprung, den Tiefpunkt und das Verhalten für $x \rightarrow-\infty$ . Erklären Sie den Einfluss von $b$ auf den Graphen.

Find $\lim _{x \rightarrow 5^{+}} \frac{\sqrt{4(\ln (x-5))^{2}+1}}{\ln (x-5)}$ .

Differentiate: $y=x^{4}+5x$

Find the integral: $\int e^{-4 x} d x$

If $f$ is continuous on $[a, b]$ and $f'(x)<0$ for $x \in (a, b)$ , what can we say about $f$ ?

Find the average rate of change of $f(x)=\frac{1}{3x+1}$ over $[2,5]$ and approximate the instantaneous rate at $x=2$ .

Find the critical points of $f^{\prime}(x)=2 x-3$ for $x \in[0,4]$ . Options: a) None b) $x=\frac{3}{2}$ c) $0, \frac{3}{2}$ d) $-\frac{3}{2}$ .

Find critical points, increasing, and decreasing intervals for $f(x)=8 \ln x-x^{2}$ .

Find the interval where $f(x)=(2+3x)^{3}$ is decreasing. Options: a) $x \leq \frac{-2}{3}$ b) $0 \geq \frac{2}{3}$ c) $\frac{-2}{3} \leq x \leq \frac{2}{3}$ d) Never Decreasing.

Find the limit: $\lim _{x \rightarrow 1^{-}} \arctan \left(\frac{x^{2}}{1-x^{2}}\right)$ . Does it exist?

Evaluate the integral of $e^{-\frac{\varepsilon}{k_{B} T}}$ with respect to $\varepsilon$ .

Calculate the limit: $\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x-3}{|x-1|}$ or state if it doesn't exist.

Find the average rate of change of $f(x)=2 x^{2}-9$ from $x=4$ to $x=b$ . Simplify your answer in terms of $b$ .

Find the limit: $\lim _{x \rightarrow-1} \frac{2|x+1|}{x+1}$ .

Find the value of $q$ that maximizes total revenue, satisfying $TR'(q) = 0$ and $TR''(q) < 0$ .

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=0$ , $x=-\frac{\pi}{4}$ , $x=\frac{\pi}{4}$ around $y=-2$ .

Is the function $f(x)=\begin{cases}3 \ln (1+x) & x \geq 0 \\ x^{5}+x^{5} & x<0\end{cases}$ continuous on R? (YES / NO)

Find the volume of the solid formed by revolving the area between $y=\sec x$ , $y=\sqrt{2}$ , $x=0$ , and $x=\frac{\pi}{4}$ around $x=\frac{\pi}{2}$ .

Find the limit: $\lim _{x \rightarrow \infty}-\frac{x^{3}}{3 x^{2}-3}$ .

Is the function $f(x)=\begin{cases} 3 \ln (1+x) & x \geq 0 \\ x^{5}+x & x<0 \end{cases}$ continuous on R? (YES / NO)

Find the instantaneous rate of change for $y=-\frac{1}{x-2}$ at $x=-3$ .

Find critical points of $\sin x + \cos y = 2y$ and classify them using first and second derivative tests.

Find critical points of $\sin x+\cos y=2 y$ and classify them using first and second derivative tests.

Differentiate the function $y=-\frac{14 x^{3}}{5-6 x}$ . Identify the appropriate differentiation rule to use.

Find the numerator of the derivative of $y=-\frac{14 x^{3}}{5-6 x}$ using the quotient rule.

Analyze the function $f(x)=\frac{4}{3} x-\tan x$ for concavity in the intervals given.

Find the surface area of the curve $x=2 \sqrt{\cos (4 y)}$ from $y=-\frac{\pi}{16}$ to $y=\frac{\pi}{16}$ around the $y$ -axis.

Find the limits: $\lim _{x \rightarrow 3^{-}} f(x)$ and $\lim _{x \rightarrow 3^{+}} f(x)$ using the given data.

Untersuche den Abbau eines Kontrastmittels:
a) Finde die Funktionsgleichung. b) Wie viel bleibt nach 60 Minuten? c) Wann sind 95 % abgebaut? d) Ist das Kontrastmittel nach 24 Stunden fast weg?