Calculus

Problem 18601

Finde die Stammfunktion von $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

See Solution

Problem 18602

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ .
a. Verhalten von $f_{0}$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ ? b. Nachweis und Koordinaten des lokalen Tiefpunkts von $G_{0}$ . c. Tangentengleichung $t_{0}$ an $G_{0}$ im Schnitt mit der $y$ -Achse und deren Allgemeingültigkeit. d. Gibt es ein $a$ , sodass $y=2 x-54$ Tangente an $G_{a}$ bei $(6, f_{a}(6))$ ist? e. Warum ist $(0, f_{0}(0))$ kein Wendepunkt von $G_{0}$ ?

See Solution

Problem 18603

Estimate $y(0.6)$ using Multistep Euler's method with step size 0.1 for $y' = y + xy$ , $y(0) = 1$ .

See Solution

Problem 18604

A particle starts at the origin at $t=1$ with velocity $v(t)=21 t^{2}-t$ . Write the differential equation and find $s(t)$ . $\frac{d s}{d t}=$ $s(t)=$

See Solution

Problem 18605

Calculate the integral from $\pi$ to $\frac{4\pi}{3}$ of $\sec(\theta) \tan(\theta) \, d\theta$ .

See Solution

Problem 18606

Find $f^{\prime}(2)$ given $40 + 4 f(x) + x^{2}(f(x))^{3} = 0$ and $f(2) = -2$ .

See Solution

Problem 18607

Find the tangent slope $m_{\tan }$ for $f(x)=\sqrt{3x+7}$ at $P(3,4)$ and the tangent line equation $y=\square$ .

See Solution

Problem 18608

Gegeben ist die Funktion $f(x)=-\frac{1}{2} x^{2}+2 x+\frac{5}{2}$ .
a) Zeichne den Graphen von $f$ für $-2 \leq x \leq 5$ .
b) Berechne die mittlere Steigung von $f$ im Intervall $I=[-1,1]$ und die lokale Steigung bei $x_0=0$ .
c) Überprüfe, ob die mittlere und lokale Steigung übereinstimmen.

See Solution

Problem 18609

Berechnen Sie die unbestimmten Integrale mit der linearen Substitutionsregel: a) $\int(2 x+1)^{2} d x$ b) $\int\left(\frac{1}{2} x+1\right)^{2} d x$ c) $\int \frac{1}{(3 x+2)^{2}} d x$ d) $\int 2 \sqrt{2 x+1} d x$

See Solution

Problem 18610

Calculate the indefinite integral: $\int 12 x^{3} dx = \square$

See Solution

Problem 18611

Find the indefinite integral of $6 x^{\frac{1}{2}}$ : $\int 6 x^{\frac{1}{2}} d x=\square$

See Solution

Problem 18612

Evaluate the integral: $\int 2 \, dx = \square$

See Solution

Problem 18613

Calculate the indefinite integral: $\int x^{-11} d x = \square$

See Solution

Problem 18614

Evaluate the integral: $\int x^{7} dx = \square$

See Solution

Problem 18615

Evaluate $\int_{3}^{7} f(x) d x$ , $\int_{3}^{4} f(x) d x$ , and $\int_{4}^{7} g(x) d x$ .

See Solution

Problem 18616

A family has 160 feet of fencing for a garden with one side already fenced. Find dimensions for max area, then max area value.

See Solution

Problem 18617

Calculate the integral from 0 to $2\pi$ of $\sin(x)$ with respect to $x$ .

See Solution

Problem 18618

Given $\int_{2}^{4} f(x) d x=-4$ , $\int_{2}^{7} f(x) d x=-2$ , and $\int_{2}^{7} g(x) d x=8$ , find:
1. $\int_{7}^{2} g(x) d x=\square$
2. $\int_{2}^{7} 4 g(x) d x=\square$
3. $\int_{2}^{7}[g(x)-f(x)] d x=\square$
4. $\int_{2}^{7}[9 g(x)-f(x)] d x=\square$

See Solution

Problem 18619

Evaluate the integral: $\int \frac{3}{x} d x=\square$

See Solution

Problem 18620

Evaluate the limit:
$\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u - 7 \cot u}{u - \frac{\pi}{4}}.$
Use l'Hôpital's Rule if needed. What is the limit's value?

See Solution

Problem 18621

Evaluate the integral: $\int \frac{2}{x} d x=\square$

See Solution

Problem 18622

Calculate the indefinite integral and verify by differentiation: $\int 16 e^{u} d u=\square$ .

See Solution

Problem 18623

Evaluate the integral: $\int \frac{4}{\sqrt{x}} dx = \square$

See Solution

Problem 18624

Evaluate the integral $\int \frac{1}{6 x^{7}} d x=\square($ Type an exact answer.)

See Solution

Problem 18625

Bestimmen Sie die Grenzwerte $\lim _{x \rightarrow-\infty} f(x)$ und $\lim _{x \rightarrow+\infty} f(x)$ für die Funktionen: a) $f(x)=2 x^{3}+x$ , b) $f(x)=x-5 x^{3}$ , c) $f(x)=-x^{4}+2 x^{2}$ .

See Solution

Problem 18626

Find the indefinite integral: $\int \frac{1}{6 x^{6}} d x=\square($ Type an exact answer.)

See Solution

Problem 18627

Is the statement true? If $n$ is an integer, then $\frac{x^{n+1}}{n+1}$ is an antiderivative of $x^{n}$ . Choose A, B, C, or D.

See Solution

Problem 18628

Find the limit: $\lim _{x \rightarrow-1} \frac{8 x^{4}+4 x^{3}+6 x+2}{x+1}$ using l'Hôpital's Rule if applicable.

See Solution

Problem 18629

Evaluate the integral: $\int \frac{1}{9 x^{7}} d x=\square($ Type an exact answer.)

See Solution

Problem 18630

Rewrite the limit using l'Hôpital's Rule: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}=\lim _{u \rightarrow \frac{\pi}{4}}(\square)$

See Solution

Problem 18631

Evaluate the limit: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}$ using l'Hôpital's Rule.

See Solution

Problem 18632

Evaluate the integral: $\int \frac{9+u}{u} d u = \square$

See Solution

Problem 18633

Calculate the indefinite integral and verify by differentiating: $\int(3 e^{z}+4) dz = \square$

See Solution

Problem 18634

Evaluate the integral: $\int\left(5 x^{4}-\frac{4}{x^{4}}\right) d x=\square$

See Solution

Problem 18635

Find the antiderivative $C(x)$ of $C'(x)=3 x^{2}-4 x$ with $C(0)=1,000$ . What is $C(x)=\square$ ?

See Solution

Problem 18636

Find $f$ where $f^{\prime}(x)=\frac{9}{\sqrt{x}}$ and $f(9)=60$ . What is $f(x)=\square$ ?

See Solution

Problem 18637

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-16,16]$ . Conclusion holds for $\mathrm{c}=\square$ .

See Solution

Problem 18638

Find the antiderivative of $4 x^{-2}+8 x^{-1}-1$ with the condition $y(1)=5$ . What is $y(x)=\square$ ?

See Solution

Problem 18639

Find $f$ where $f'(x)=\frac{4}{\sqrt{x}}$ and $f(9)=39$ . What is $f(x)=\square$ ?

See Solution

Problem 18640

Find the slope of the tangent line to $f(x)=\frac{1}{3+2x}$ at $P=(1,\frac{1}{5})$ and its equation. Options: A, B, C, D.

See Solution

Problem 18641

Find the antiderivative of $\frac{\mathrm{dx}}{\mathrm{dt}}=4 e^{\mathrm{t}}-8$ with $\mathrm{x}(0)=1$ . What is $x(t)=\square$ ?

See Solution

Problem 18642

Find the curve equation passing through (3,4) with slope $\frac{d y}{d x}=3 x-7$ . What is $y(x)=\square$ ?

See Solution

Problem 18643

Find the absolute extreme values of $f(x)=-2x^{3}+27x^{2}-108x$ on $[2,7]$ . Where are the max/min values?

See Solution

Problem 18644

Evaluate the integral $\int_{1/2}^{1} (x^{-3} - 6) \, dx$ using the Fundamental Theorem of Calculus. Find $\int (x^{-3} - 6) \, dx = \square$ .

See Solution

Problem 18645

Find the second derivative $y^{\prime \prime}$ of the function $y=\frac{\sin x}{6 x}$ .

See Solution

Problem 18646

Find the antiderivative $F$ of $f(x)=x^{5}-3 x^{-4}-1$ such that $F(1)=1$ . What is $F(x)=\square$ ?

See Solution

Problem 18647

Find the limits: a. $\lim _{x \rightarrow 8^{+}} \frac{6}{x-8}$ , b. $\lim _{x \rightarrow 8^{-}} \frac{6}{x-8}$ , c. $\lim _{x \rightarrow 8} \frac{6}{x-8}$ .

See Solution

Problem 18648

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=4x-3$ , given $f^{\prime}(-3)=-2$ and $f(-3)=-6$ . Also, find $f(3)$ .

See Solution

Problem 18649

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten von $f_{0}$ und die Eigenschaften von $G_{0}$ .

See Solution

Problem 18650

Find the mean slope of $f(x)=3-6 x^{2}$ on $[-5,7]$ and the value of $c$ where $f'(c)$ equals this slope.

See Solution

Problem 18651

Bestimmen Sie die Punkte mit waagerechter Tangente für die Funktionen: a) $f(x)=x \cdot e^{x}$ , b) $f(x)=x^{2} \cdot e^{2 x}$ , c) $f(x)=(x-2) \cdot e^{-x}$ .

See Solution

Problem 18652

Prove the $(n+1)^{th}$ derivative of $f(x)g(x)$ using Pascal's triangle coefficients $b_k$ for row $n+1$ .

See Solution

Problem 18653

Approximate $\int_{0}^{1} \frac{2}{1+x^{2}} dx$ using the trapezium rule (5 strips) and $\int_{0}^{30} \frac{2}{1+x^{2}} dx$ using Simpson's rule (6 strips). Discuss one advantage and disadvantage of Simpson's rule.

See Solution

Problem 18654

A cell culture starts with 1200 cells and grows as $p(t)=1200+23 t^{3}$ . Find $p^{\prime}(t)$ , its units, and when it's least/greatest on $[0,3]$ .

See Solution

Problem 18655

Find $f(2)$ for the function with $f^{\prime \prime}(x)=5 x+10 \sin (x)$ , given $f(0)=3$ and $f^{\prime}(0)=3$ .

See Solution

Problem 18656

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten, lokale Tiefpunkte und Tangenten.

See Solution

Problem 18657

Graph $f(x)=x+\frac{4}{x}$ and the secant line through $(1,5)$ and $(8,8.5)$ . Find $c$ for the Mean Value Theorem on $[1,8]$ : $c=$

See Solution

Problem 18658

Bestimmen Sie die Abmessungen eines rechteckigen Gatters mit 50 m Draht, das aus zwei unterschiedlich großen Rechtecken besteht, um die Fläche zu maximieren.

See Solution

Problem 18659

Find the maxima, minima, and intervals of increase/decrease for the function $y=4 x^{3}-21 x^{2}-90 x+25$ . Then sketch the graph.

See Solution

Problem 18660

Find the position of a particle at time $t=14$ given $a(t)=24t+18$ , $s(0)=2$ , and $v(0)=10$ .

See Solution

Problem 18661

Find the average slope of $f(x)=\frac{1}{x}$ on $[3,11]$ and values of $c$ in $(3,11)$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 18662

Find $c$ using the Mean Value Theorem for $f(x)=6 x^{2}+11 x+11$ on $[9,12]$ . If not applicable, enter DNE.

See Solution

Problem 18663

Check if the Mean Value Theorem applies to $f(x)=10 \ln (x)+2$ on $[1,20]$ . If yes, find $c$ in $[1,20]$ .

See Solution

Problem 18664

Find the mean slope of $f(x)=4 \sqrt{x}+6$ on $[3,7]$ and values of $c$ where $f^{\prime}(c)$ equals this slope.

See Solution

Problem 18665

Évaluez les intégrales suivantes : a) $\int x \sqrt{1+4 x} d x$ , b) $\int \frac{1+2 x}{1+16 x^{2}} d x$ , c) $\int(4 t+2) e^{t^{2}+t} d x$ , d) $\int \frac{e^{x}}{e^{x}+e^{-x}} d x$ , e) $\int x^{2} 2^{x} d x$ , f) $\int x^{2} e^{4 x} d x$ , g) $\int 3 t^{3} \sin \left(t^{2}\right) d t$ , h) $\int x^{2}(\ln x)^{2} d x$ . Trouvez le profit maximal d'un appareil acheté à 2500 avec $R$ et $C$ données par $\frac{d R}{d t}=100(18-3 \sqrt{t})$ et $\frac{d C}{d t}=100(2+\sqrt{t})$ .

See Solution

Problem 18666

Évaluez ces intégrales : a) $\int x \sqrt{1+4 x} d x$ , b) $\int \frac{1+2 x}{1+16 x^{2}} d x$ , c) $\int(4 t+2) e^{t^{2}+t} d x$ , d) $\int \frac{e^{x}}{e^{x}+e^{-x}} d x$ , e) $\int x^{2} 2^{x} d x$ , f) $\int x^{2} e^{4 x} d x$ , g) $\int 3 t^{3} \sin \left(t^{2}\right) d t$ , h) $\int x^{2}(\ln x)^{2} d x$ .

See Solution

Problem 18667

Finde die Stammfunktionen für die folgenden Funktionen: a) $f(x)=x^{3}-2x+1$ b) $f(x)=x+\sin(x)$ c) $f(x)=3\sqrt{x}$ d) $f(x)=4x^{6}$ e) $f(x)=3a\cdot\sin(x)$ f) $f(x)=\frac{1}{\sqrt{x}}$ und für Training 2: a) $f(x)=\sqrt{x}+x$ b) $f(x)=(x-1)(x-3)$ c) $f(x)=3x^{2}-4x+5$ d) $f(x)=(x+2)^{2}$ e) $f(x)=ax\cdot(4x-2)$ f) $f(x)=(x-a)^{2}$ .

See Solution

Problem 18668

A turkey cools from $185^{\circ} \mathrm{F}$ to $77^{\circ} \mathrm{F}$ . Find its temp after 45 min and time to reach $100^{\circ} \mathrm{F}$ .

See Solution

Problem 18669

Estimate the time for the world population to double and triple given it was 7.1 billion in 2013 with a growth rate of $1.1\%$ per year.

See Solution

Problem 18670

Trouver les dimensions d'un enclos rectangulaire de 100m de clôture, divisé en deux, pour maximiser l'aire $A$ .

See Solution

Problem 18671

Check if the Mean Value Theorem applies to $f(x)=6x^2+11x+11$ on $[9,12]$ . If yes, find $c$ . If no, enter DNE.

See Solution

Problem 18672

Sketch the graph of relation $f$ with these limits: $\lim _{x \rightarrow 2^{+}} f=3$ , $\lim _{x \rightarrow 2^{-}} f=5$ , $f(2)=4$ , $\lim _{x \rightarrow 1^{+}} f=\infty$ , $\lim _{x \rightarrow 0^{-}} f=-\infty$ , $\lim _{x \rightarrow-3^{+}} f=-2$ , $\lim _{x \rightarrow-3^{-}} f=1$ , $f(-3)=-4$ , $\lim _{x \rightarrow \infty} f=2$ , $\lim _{x \rightarrow-\infty} f=2$ . Is $f$ a function? Justify your answer!

See Solution

Problem 18673

Determine which two of the following series converge:
1. $\sum_{n=1}^{\infty}\left(\frac{n^{2}}{2 n+1}\right)^{n}$
2. $\sum_{n=1}^{\infty}\left(1+\frac{1}{n^{2}}\right)^{n^{3}}$
3. $\sum_{n=1}^{\infty}\left(\frac{n-1}{n+1}\right)^{n^{2}+n}$
4. $\sum_{n=1}^{\infty} \frac{3^{n}}{n^{3}}$
5. $\sum_{n=1}^{\infty} \frac{n !}{(2 n) !}$
6. $\sum_{n=1}^{\infty} \frac{n^{3}+1}{\sqrt[3]{n^{10}+n}}$
7. $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{n !}$

See Solution

Problem 18674

Solve these differential equations and find $C$ if there's an initial value: 1a) $\frac{d y}{d x}=x^{2} y-y+x^{2}-1$ 1b) $x+3 y^{2} \sqrt{x^{2}+1} \frac{d y}{d x}=0, \quad y(0)=1$
2a) $2 x y^{\prime}-2 y=x^{2}$ 2b) $x^{2} y^{\prime}+2 x y=\ln (x), \quad y(1)=2$ .

See Solution

Problem 18675

Find the projected population in 2039 given a 3% growth rate (a) and a reduced 2% growth rate (b) from 150 million in 2011.

See Solution

Problem 18676

Berechnen Sie die folgenden Integrale mit dem Hauptsatz der Differenzial- und Integralrechnung: a) $\int_{1}^{2} 4 x^{3}$ , b) $\int_{-1}^{1}\left(9 x^{2}-1\right)$ , c) $\int_{0}^{\frac{\pi}{2}} \sin (x)$ , d) $\int_{1}^{3} \frac{1}{x^{2}}$ , e) $\int_{0}^{4} x(x-1)$ , f) $\int_{1}^{25} \frac{1}{\sqrt{x}}$ .

See Solution

Problem 18677

Find the tangent line equation for $y=5 \ln (x^{3}-7)$ at point $(2,0)$ . $y=$

See Solution

Problem 18678

Find the tangent line equation for $y=14 x e^{x}$ at $(0,0)$ in the form $y=m x+b$ . Determine $m$ and $b$ .

See Solution

Problem 18679

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

See Solution

Problem 18680

Find the point on the graph of $y=\ln(x^2+1)$ where the tangent line is perpendicular to $y=5-x$ . Report as $(x, y)$ .

See Solution

Problem 18681

Find $A(1.5)$ , $A(3)$ , $A^{\prime}(1.5)$ , and $A^{\prime}(3)$ for $A(x)=\int_{0}^{x} f(t) dt$ where $f(x)=2$ for $0 \leq x < 2$ and $f(x)=x$ for $2 \leq x \leq 4$ .

See Solution

Problem 18682

Find the differential $d y$ for $y=\ln(1+x^4)$ in terms of $x$ and $dx$ . Enter $dx$ as $\mathrm{dx}$ . $d y=$

See Solution

Problem 18683

Find the linearization of $f(x)=\sqrt{1+5 x}$ at $a=0$ : $L(x)=A+B x$ . Calculate $A$ and $B$ .

See Solution

Problem 18684

Approximate $f^{\prime}(3.9)$ given $f(3.9)=4.6$ and $f(4.2)=-2.3$ . $f^{\prime}(3.9) \approx I \quad I$

See Solution

Problem 18685

Find the general antiderivative of $f(x)=\frac{4}{\sqrt{x}}+4 \sqrt{x}$ , denoted as $F(x)=\square$ (use C for constant).

See Solution

Problem 18686

Find the general antiderivative of $f(r)=\frac{7}{r^{5}}-7 r^{5}$ . The result is $F(r)=\square$ with constant $\mathrm{C}$ .

See Solution

Problem 18687

Find the linearization $L(x)$ of $f(x)=\ln(x)$ at $a=1$ . What is $L(x)=\square$ ?

See Solution

Problem 18688

Find the general antiderivative of $f(x)=(3 x+4)^{2}$ . Let $F(x)=\square$ (use $\mathrm{C}$ as the constant).

See Solution

Problem 18689

Find the differential $dy$ for the function $y=x^{6}-4\sqrt{x}$ in terms of $x$ and $dx$ .

See Solution

Problem 18690

Approximate $\sqrt{49.4}$ using local linearization with $f(x)=\sqrt{x}$ at $x=49$ . Find $L_{49}(49.4)$ .

See Solution

Problem 18691

Find $d y$ for $y=\tan(3x+7)$ at $x=1$ with $d x=0.2$ and $d x=0.4$ .

See Solution

Problem 18692

Rewrite the integral using $u$ and $du$ , then evaluate: $\int 12 x e^{-x^{2}} dx, \quad u=-x^{2}$ . What is the result?

See Solution

Problem 18693

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+3}-2}{x^{3}-4 x+3}$ .

See Solution

Problem 18694

Rewrite the integral using $u$ and $du$ , then evaluate:
$\int x(x+1)^{11} dx, \quad u=x+1$
Find:
$\int x(x+1)^{11} dx=$

See Solution

Problem 18695

Berechnen Sie die lokale Änderungsrate von $f$ an $x_{0}$ durch Grenzwertrechnung: a) $f(x)=0,5 x^{2}, x_{0}=2$ b) $f(x)=1-x^{2}, x_{0}=2$ c) $f(x)=2 x+1; x_{0}=3$

See Solution

Problem 18696

Find the Taylor series for $f(x) = \frac{10}{x}$ centered at $a = -4$ .

See Solution

Problem 18697

Find the limit: $\lim _{x \rightarrow 1} \frac{\sqrt{x+3}-2}{x^{3}-4 x+3}$ .

See Solution

Problem 18698

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 5 x}{2 x}=$

See Solution

Problem 18699

Find the general antiderivative of $f(x)=5 x^{\frac{1}{3}}+4 x^{-\frac{1}{3}}+7$ . What is $F(x)=\square$ (use $C$ )?

See Solution

Problem 18700

Find the general antiderivative of $f(x)=5 x^{\frac{1}{3}}+4 x^{-\frac{1}{3}}+7$ . Express as $F(x)=\square$ (use C for constant).

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 18601

Finde die Stammfunktion von f(x)=x2+2xx5f(x)=\frac{x^{2}+2 x}{x^{5}}f(x)=x5x2+2x​.

Problem 18602

Problem 18603

Estimate y(0.6)y(0.6)y(0.6) using Multistep Euler's method with step size 0.1 for y′=y+xyy' = y + xyy′=y+xy, y(0)=1y(0) = 1y(0)=1.

Problem 18604

A particle starts at the origin at t=1t=1t=1 with velocity v(t)=21t2−tv(t)=21 t^{2}-tv(t)=21t2−t. Write the differential equation and find s(t)s(t)s(t). dsdt= \frac{d s}{d t}= dtds​= s(t)= s(t)= s(t)=

Problem 18605

Calculate the integral from π\piπ to 4π3\frac{4\pi}{3}34π​ of sec⁡(θ)tan⁡(θ) dθ\sec(\theta) \tan(\theta) \, d\thetasec(θ)tan(θ)dθ.

Problem 18606

Find f′(2)f^{\prime}(2)f′(2) given 40+4f(x)+x2(f(x))3=040 + 4 f(x) + x^{2}(f(x))^{3} = 040+4f(x)+x2(f(x))3=0 and f(2)=−2f(2) = -2f(2)=−2.

Problem 18607

Find the tangent slope mtan⁡m_{\tan }mtan​ for f(x)=3x+7f(x)=\sqrt{3x+7}f(x)=3x+7​ at P(3,4)P(3,4)P(3,4) and the tangent line equation y=□y=\squarey=□.

Problem 18608

Problem 18609

Problem 18610

Calculate the indefinite integral: ∫12x3dx=□\int 12 x^{3} dx = \square∫12x3dx=□

Problem 18611

Find the indefinite integral of 6x126 x^{\frac{1}{2}}6x21​: ∫6x12dx=□\int 6 x^{\frac{1}{2}} d x=\square∫6x21​dx=□

Problem 18612

Evaluate the integral: ∫2 dx=□\int 2 \, dx = \square∫2dx=□

Problem 18613

Calculate the indefinite integral: ∫x−11dx=□\int x^{-11} d x = \square∫x−11dx=□

Problem 18614

Evaluate the integral: ∫x7dx=□\int x^{7} dx = \square∫x7dx=□

Problem 18615

Evaluate ∫37f(x)dx\int_{3}^{7} f(x) d x∫37​f(x)dx, ∫34f(x)dx\int_{3}^{4} f(x) d x∫34​f(x)dx, and ∫47g(x)dx\int_{4}^{7} g(x) d x∫47​g(x)dx.

Problem 18616

A family has 160 feet of fencing for a garden with one side already fenced. Find dimensions for max area, then max area value.

Problem 18617

Calculate the integral from 0 to 2π2\pi2π of sin⁡(x)\sin(x)sin(x) with respect to xxx.

Problem 18618

Problem 18619

Evaluate the integral: ∫3xdx=□\int \frac{3}{x} d x=\square∫x3​dx=□

Problem 18620

Evaluate the limit: lim⁡u→π47tan⁡u−7cot⁡uu−π4. \lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u - 7 \cot u}{u - \frac{\pi}{4}}. u→4π​lim​u−4π​7tanu−7cotu​. Use l'Hôpital's Rule if needed. What is the limit's value?

Problem 18621

Evaluate the integral: ∫2xdx=□\int \frac{2}{x} d x=\square∫x2​dx=□

Problem 18622

Calculate the indefinite integral and verify by differentiation: ∫16eudu=□\int 16 e^{u} d u=\square∫16eudu=□.

Problem 18623

Evaluate the integral: ∫4xdx=□\int \frac{4}{\sqrt{x}} dx = \square∫x​4​dx=□

Problem 18624

Evaluate the integral ∫16x7dx=□(\int \frac{1}{6 x^{7}} d x=\square(∫6x71​dx=□( Type an exact answer.)

Problem 18625

Problem 18626

Find the indefinite integral: ∫16x6dx=□(\int \frac{1}{6 x^{6}} d x=\square(∫6x61​dx=□( Type an exact answer.)

Problem 18627

Is the statement true? If nnn is an integer, then xn+1n+1\frac{x^{n+1}}{n+1}n+1xn+1​ is an antiderivative of xnx^{n}xn. Choose A, B, C, or D.

Problem 18628

Find the limit: lim⁡x→−18x4+4x3+6x+2x+1\lim _{x \rightarrow-1} \frac{8 x^{4}+4 x^{3}+6 x+2}{x+1}x→−1lim​x+18x4+4x3+6x+2​ using l'Hôpital's Rule if applicable.

Problem 18629

Evaluate the integral: ∫19x7dx=□(\int \frac{1}{9 x^{7}} d x=\square(∫9x71​dx=□( Type an exact answer.)

Problem 18630

Rewrite the limit using l'Hôpital's Rule: lim⁡u→π47tan⁡u−7cot⁡uu−π4=lim⁡u→π4(□)\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}=\lim _{u \rightarrow \frac{\pi}{4}}(\square)u→4π​lim​u−4π​7tanu−7cotu​=u→4π​lim​(□)

Problem 18631

Evaluate the limit: lim⁡u→π47tan⁡u−7cot⁡uu−π4\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}limu→4π​​u−4π​7tanu−7cotu​ using l'Hôpital's Rule.

Problem 18632

Evaluate the integral: ∫9+uudu=□\int \frac{9+u}{u} d u = \square∫u9+u​du=□

Problem 18633

Calculate the indefinite integral and verify by differentiating: ∫(3ez+4)dz=□\int(3 e^{z}+4) dz = \square∫(3ez+4)dz=□

Problem 18634

Evaluate the integral: ∫(5x4−4x4)dx=□\int\left(5 x^{4}-\frac{4}{x^{4}}\right) d x=\square∫(5x4−x44​)dx=□

Problem 18635

Find the antiderivative C(x)C(x)C(x) of C′(x)=3x2−4xC'(x)=3 x^{2}-4 xC′(x)=3x2−4x with C(0)=1,000C(0)=1,000C(0)=1,000. What is C(x)=□C(x)=\squareC(x)=□?

Problem 18636

Find fff where f′(x)=9xf^{\prime}(x)=\frac{9}{\sqrt{x}}f′(x)=x​9​ and f(9)=60f(9)=60f(9)=60. What is f(x)=□f(x)=\squaref(x)=□?

Problem 18637

Find points ccc where the Mean Value Theorem applies for f(x)=x3f(x)=x^{3}f(x)=x3 on [−16,16][-16,16][−16,16]. Conclusion holds for c=□\mathrm{c}=\squarec=□.

Problem 18638

Find the antiderivative of 4x−2+8x−1−14 x^{-2}+8 x^{-1}-14x−2+8x−1−1 with the condition y(1)=5y(1)=5y(1)=5. What is y(x)=□y(x)=\squarey(x)=□?

Problem 18639

Find fff where f′(x)=4xf'(x)=\frac{4}{\sqrt{x}}f′(x)=x​4​ and f(9)=39f(9)=39f(9)=39. What is f(x)=□f(x)=\squaref(x)=□?

Problem 18640

Find the slope of the tangent line to f(x)=13+2xf(x)=\frac{1}{3+2x}f(x)=3+2x1​ at P=(1,15)P=(1,\frac{1}{5})P=(1,51​) and its equation. Options: A, B, C, D.

Problem 18641

Find the antiderivative of dxdt=4et−8\frac{\mathrm{dx}}{\mathrm{dt}}=4 e^{\mathrm{t}}-8dtdx​=4et−8 with x(0)=1\mathrm{x}(0)=1x(0)=1. What is x(t)=□x(t)=\squarex(t)=□?

Problem 18642

Find the curve equation passing through (3,4) with slope dydx=3x−7\frac{d y}{d x}=3 x-7dxdy​=3x−7. What is y(x)=□y(x)=\squarey(x)=□?

Finde die Stammfunktion von $f(x)=\frac{x^{2}+2 x}{x^{5}}$ .

Estimate $y(0.6)$ using Multistep Euler's method with step size 0.1 for $y' = y + xy$ , $y(0) = 1$ .

A particle starts at the origin at $t=1$ with velocity $v(t)=21 t^{2}-t$ . Write the differential equation and find $s(t)$ . $\frac{d s}{d t}=$ $s(t)=$

Calculate the integral from $\pi$ to $\frac{4\pi}{3}$ of $\sec(\theta) \tan(\theta) \, d\theta$ .

Find $f^{\prime}(2)$ given $40 + 4 f(x) + x^{2}(f(x))^{3} = 0$ and $f(2) = -2$ .

Find the tangent slope $m_{\tan }$ for $f(x)=\sqrt{3x+7}$ at $P(3,4)$ and the tangent line equation $y=\square$ .

Calculate the indefinite integral: $\int 12 x^{3} dx = \square$

Find the indefinite integral of $6 x^{\frac{1}{2}}$ : $\int 6 x^{\frac{1}{2}} d x=\square$

Evaluate the integral: $\int 2 \, dx = \square$

Calculate the indefinite integral: $\int x^{-11} d x = \square$

Evaluate the integral: $\int x^{7} dx = \square$

Evaluate $\int_{3}^{7} f(x) d x$ , $\int_{3}^{4} f(x) d x$ , and $\int_{4}^{7} g(x) d x$ .

Calculate the integral from 0 to $2\pi$ of $\sin(x)$ with respect to $x$ .

Evaluate the integral: $\int \frac{3}{x} d x=\square$

Evaluate the limit:
$\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u - 7 \cot u}{u - \frac{\pi}{4}}.$
Use l'Hôpital's Rule if needed. What is the limit's value?

Evaluate the integral: $\int \frac{2}{x} d x=\square$

Calculate the indefinite integral and verify by differentiation: $\int 16 e^{u} d u=\square$ .

Evaluate the integral: $\int \frac{4}{\sqrt{x}} dx = \square$

Evaluate the integral $\int \frac{1}{6 x^{7}} d x=\square($ Type an exact answer.)

Find the indefinite integral: $\int \frac{1}{6 x^{6}} d x=\square($ Type an exact answer.)

Is the statement true? If $n$ is an integer, then $\frac{x^{n+1}}{n+1}$ is an antiderivative of $x^{n}$ . Choose A, B, C, or D.

Find the limit: $\lim _{x \rightarrow-1} \frac{8 x^{4}+4 x^{3}+6 x+2}{x+1}$ using l'Hôpital's Rule if applicable.

Evaluate the integral: $\int \frac{1}{9 x^{7}} d x=\square($ Type an exact answer.)

Rewrite the limit using l'Hôpital's Rule: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}=\lim _{u \rightarrow \frac{\pi}{4}}(\square)$

Evaluate the limit: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{u-\frac{\pi}{4}}$ using l'Hôpital's Rule.

Evaluate the integral: $\int \frac{9+u}{u} d u = \square$

Calculate the indefinite integral and verify by differentiating: $\int(3 e^{z}+4) dz = \square$

Evaluate the integral: $\int\left(5 x^{4}-\frac{4}{x^{4}}\right) d x=\square$

Find the antiderivative $C(x)$ of $C'(x)=3 x^{2}-4 x$ with $C(0)=1,000$ . What is $C(x)=\square$ ?

Find $f$ where $f^{\prime}(x)=\frac{9}{\sqrt{x}}$ and $f(9)=60$ . What is $f(x)=\square$ ?

Find points $c$ where the Mean Value Theorem applies for $f(x)=x^{3}$ on $[-16,16]$ . Conclusion holds for $\mathrm{c}=\square$ .

Find the antiderivative of $4 x^{-2}+8 x^{-1}-1$ with the condition $y(1)=5$ . What is $y(x)=\square$ ?

Find $f$ where $f'(x)=\frac{4}{\sqrt{x}}$ and $f(9)=39$ . What is $f(x)=\square$ ?

Find the slope of the tangent line to $f(x)=\frac{1}{3+2x}$ at $P=(1,\frac{1}{5})$ and its equation. Options: A, B, C, D.

Find the antiderivative of $\frac{\mathrm{dx}}{\mathrm{dt}}=4 e^{\mathrm{t}}-8$ with $\mathrm{x}(0)=1$ . What is $x(t)=\square$ ?

Find the curve equation passing through (3,4) with slope $\frac{d y}{d x}=3 x-7$ . What is $y(x)=\square$ ?

Find the absolute extreme values of $f(x)=-2x^{3}+27x^{2}-108x$ on $[2,7]$ . Where are the max/min values?

Evaluate the integral $\int_{1/2}^{1} (x^{-3} - 6) \, dx$ using the Fundamental Theorem of Calculus. Find $\int (x^{-3} - 6) \, dx = \square$ .

Find the second derivative $y^{\prime \prime}$ of the function $y=\frac{\sin x}{6 x}$ .

Find the antiderivative $F$ of $f(x)=x^{5}-3 x^{-4}-1$ such that $F(1)=1$ . What is $F(x)=\square$ ?

Find the limits: a. $\lim _{x \rightarrow 8^{+}} \frac{6}{x-8}$ , b. $\lim _{x \rightarrow 8^{-}} \frac{6}{x-8}$ , c. $\lim _{x \rightarrow 8} \frac{6}{x-8}$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=4x-3$ , given $f^{\prime}(-3)=-2$ and $f(-3)=-6$ . Also, find $f(3)$ .

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten von $f_{0}$ und die Eigenschaften von $G_{0}$ .

Find the mean slope of $f(x)=3-6 x^{2}$ on $[-5,7]$ and the value of $c$ where $f'(c)$ equals this slope.

Bestimmen Sie die Punkte mit waagerechter Tangente für die Funktionen: a) $f(x)=x \cdot e^{x}$ , b) $f(x)=x^{2} \cdot e^{2 x}$ , c) $f(x)=(x-2) \cdot e^{-x}$ .

Prove the $(n+1)^{th}$ derivative of $f(x)g(x)$ using Pascal's triangle coefficients $b_k$ for row $n+1$ .

Approximate $\int_{0}^{1} \frac{2}{1+x^{2}} dx$ using the trapezium rule (5 strips) and $\int_{0}^{30} \frac{2}{1+x^{2}} dx$ using Simpson's rule (6 strips). Discuss one advantage and disadvantage of Simpson's rule.

A cell culture starts with 1200 cells and grows as $p(t)=1200+23 t^{3}$ . Find $p^{\prime}(t)$ , its units, and when it's least/greatest on $[0,3]$ .

Find $f(2)$ for the function with $f^{\prime \prime}(x)=5 x+10 \sin (x)$ , given $f(0)=3$ and $f^{\prime}(0)=3$ .

Gegeben ist die Funktion $f_{a}(x)=\frac{1}{8} x^{4}-\frac{a}{12} x^{3}+2 x$ . Untersuchen Sie das Verhalten, lokale Tiefpunkte und Tangenten.

Graph $f(x)=x+\frac{4}{x}$ and the secant line through $(1,5)$ and $(8,8.5)$ . Find $c$ for the Mean Value Theorem on $[1,8]$ : $c=$

Find the maxima, minima, and intervals of increase/decrease for the function $y=4 x^{3}-21 x^{2}-90 x+25$ . Then sketch the graph.

Find the position of a particle at time $t=14$ given $a(t)=24t+18$ , $s(0)=2$ , and $v(0)=10$ .

Find the average slope of $f(x)=\frac{1}{x}$ on $[3,11]$ and values of $c$ in $(3,11)$ where $f^{\prime}(c)$ equals this slope.

Find $c$ using the Mean Value Theorem for $f(x)=6 x^{2}+11 x+11$ on $[9,12]$ . If not applicable, enter DNE.

Check if the Mean Value Theorem applies to $f(x)=10 \ln (x)+2$ on $[1,20]$ . If yes, find $c$ in $[1,20]$ .

Find the mean slope of $f(x)=4 \sqrt{x}+6$ on $[3,7]$ and values of $c$ where $f^{\prime}(c)$ equals this slope.

A turkey cools from $185^{\circ} \mathrm{F}$ to $77^{\circ} \mathrm{F}$ . Find its temp after 45 min and time to reach $100^{\circ} \mathrm{F}$ .

Estimate the time for the world population to double and triple given it was 7.1 billion in 2013 with a growth rate of $1.1\%$ per year.

Trouver les dimensions d'un enclos rectangulaire de 100m de clôture, divisé en deux, pour maximiser l'aire $A$ .

Check if the Mean Value Theorem applies to $f(x)=6x^2+11x+11$ on $[9,12]$ . If yes, find $c$ . If no, enter DNE.