Calculus

Problem 21401

Find the growth rate $\frac{d L}{d t}$ at $t=10, 20, 25$ weeks for $L=-37.41+3.66 t-6.32 \times 10^{-4} t^{3}$ . What happens as $t$ increases?

See Solution

Problem 21402

(a) What does $\frac{d H}{d x}$ mean? (b) Prove that $\frac{d H}{d x}$ is a constant for the formula $H(x)=205-0.65 x$ .

See Solution

Problem 21403

Find points on the curve $y=x^{3}+2x+9$ where the tangent line is parallel to $7x-y=-5$ . Are there multiple points?

See Solution

Problem 21404

Find the tangent line to $f(x)=a x^{2}$ at $x=-2$ , where $a$ is a positive constant.

See Solution

Problem 21405

Find critical numbers of $f(x)=x^{6}-x^{4}+5 x^{3}-4 x$ using Newton's method, correct to six decimal places. Also, find the absolute minimum value of $f$ , correct to four decimal places.

See Solution

Problem 21406

Evaluate the integral: $\int_{1}^{\infty} x e^{-x} \, dx$

See Solution

Problem 21407

Differentiate $R(T)=\frac{3 \pi^{5}}{16} \frac{k^{2}}{c^{6} h^{7}} T^{5}$ with respect to $T$ . Constants: $k, c, h > 0$ .

See Solution

Problem 21408

Differentiate $f(x)=a x^{20}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 21409

Differentiate the function $f(s)=s^{4} e^{5}+6 e$ with respect to $s$ .

See Solution

Problem 21410

Berechnen Sie die Steigung von $f$ bei $A(2 \mid f(2))$ für $f(x)=\frac{3}{2} x^{2}$ .

See Solution

Problem 21411

Differentiate $g(N)=r N(1-\frac{N}{K})$ with respect to $N$ , where $K$ and $r$ are positive constants.

See Solution

Problem 21412

Differentiate the function $h(t)=\frac{11}{2} t^{2}-2 t+8$ . Find $h^{\prime}(t)=\square$ .

See Solution

Problem 21413

Bestimmen Sie die Ableitungen von den Funktionen: a) $f(x)=x^{3}$ , b) $f(x)=x^{5}$ , c) $f(x)=x^{2 n}$ , d) $f(x)=x$ , e) $f(x)=x^{n+4}$ , f) $f(x)=x^{2016}$ .

See Solution

Problem 21414

Bestimmen Sie den Grad von $f^{\prime}$ und $f^{\prime \prime}$ . Leiten Sie $f(t)=t^{4}-\frac{8}{3} t^{3}-t^{2}-4 t+16$ zweimal ab.

See Solution

Problem 21415

Formen Sie $f(x)=x \cdot\left(x^{4}+4 x-7\right)$ um und leiten Sie die Funktion ab.

See Solution

Problem 21416

Calculate the integral from $-\infty$ to $\infty$ of $\frac{1}{1+x^{2}}$ dx.

See Solution

Problem 21417

A rancher has 600 feet of fencing for a rectangular field divided into 2 plots. Find dimensions for maximum area.

See Solution

Problem 21418

Explain why $\frac{d x}{d t}=k([A]_0-x)([B]_0-x)$ for the reaction $A+B \rightarrow A B$ .

See Solution

Problem 21419

Bestimme die Stellen, an denen $f'(x)=5$ für $f(x)=4 x^{2}-7 x+6$ . Berechne $f'(5)$ .

See Solution

Problem 21420

Beweisen Sie die Potenzregel für $n=5$ und für beliebiges $n \in \mathbb{N}$ , basierend auf dem Beweis für $n=4$ .

See Solution

Problem 21421

Given the population size $\mathrm{N}(t)$ satisfies $\frac{dN}{dt}=rN$ :
(a) What is the per capita growth rate? (b) If $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ , is $\mathrm{N}(1)$ greater or less than 20? Explain.

See Solution

Problem 21422

Bestimme die Punkte, an denen der Graph von $f(x)=\frac{1}{8} x^{4}-4 x+3$ eine waagerechte Tangente hat.

See Solution

Problem 21423

Bestimme die Punkte, an denen der Graph von $f(x)=\frac{1}{8} x^{4}-4 x+3$ eine waagerechte Tangente hat.

See Solution

Problem 21424

Given the equation $\frac{dN}{dt}=rN$ , find the per capita growth rate $r$ . If $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ , is $\mathrm{N}(1)$ > 20 or < 20? Explain.

See Solution

Problem 21425

Find the derivative: $\frac{d}{d x} \int_{\frac{\pi}{2}}^{x^{2}} -\frac{1}{2} \cos (u) du = ?$ a) $x \cos \left(x^{2}\right)$ b) $-x \cos \left(x^{2}\right)$ c) $-2 x \sin \left(x^{2}\right)$ d) $2 x \sin \left(x^{2}\right)$ e) None of the above

See Solution

Problem 21426

Find $\int_{-4}^{4} f(x) dx$ for the piecewise function: $f(x)=3x^2$ for $-4 \leq x<0$ and $f(x)=4x^3$ for $0 \leq x \leq 4$ .

See Solution

Problem 21427

Bestimmen Sie die positive Zahl z für: a) $\int_{0}^{z} x \, dx=18$ , b) $\int_{1}^{z} 4x \, dx=30$ .

See Solution

Problem 21428

Evaluate the integral $\int_{-1}^{0} \frac{1}{x^{5}} dx$ .

See Solution

Problem 21429

Find all antiderivatives of $f(x)=e^{x}+\frac{1}{x^{2}}+\cos (x)$ .

See Solution

Problem 21430

Calculate the integral $\int_{-4}^{4} f(x) dx$ where $f(x)=\begin{cases}3x^2 & \text{if } -4 \leq x < 0 \\ 4x^3 & \text{if } 0 \leq x \leq 4\end{cases}$ $. Choose a) 8, b) 31, c) 320, d) 768, or e) None.

See Solution

Problem 21431

Bestimme die Tangentengleichung der Funktion $f(x)=2x^3-6x$ bei $x_0=-1$ .

See Solution

Problem 21432

Is it true that $\int_{5}^{8} f(x) dx = \int_{5}^{30} f(x) dx - \int_{8}^{30} f(x) dx$ for $f$ on $(5,30]$ ? a) True b) False

See Solution

Problem 21433

Berechne die Integrale: a) $\int_{0}^{4}-x \, dx$ , b) $\int_{-1}^{1}-2x \, dx$ , c) $\int_{-2}^{2}-x^{2} \, dx$ .

See Solution

Problem 21434

Determine if the series converges: $\sum_{k=1}^{\infty} \frac{5 k^{5}+k}{7 k^{5}-6}$ Justify your answer.

See Solution

Problem 21435

Zwei der folgenden vier Aussagen über Ableitungen sind falsch. Welche sind es? (1) $f(x)=x^{3} \Rightarrow f^{\prime}(x)=3 \cdot x^{2}$ (2) $f(x)=x^{x} \Rightarrow f^{\prime}(x)=x \cdot x^{x-1}$ (3) $f(x)=x^{2 a} \Rightarrow f^{\prime}(x)=2 \cdot x^{a}$ (4) $f(x)=x^{a+1} \Rightarrow f^{\prime}(x)=(a+1) \cdot x^{a}$

See Solution

Problem 21436

Calculate the total energy requirements (in $\mathrm{kJ} / \mathrm{W}^{0.67}$ ) for a female Collie from age 730 to 1460 days using $E(t)=5 t^{-0.2}$ . Options: a) 904.515 b) 1000 c) 1085.418 d) 723.612

See Solution

Problem 21437

Calculate the integral from 0 to infinity of $e^{-x} \, dx$ .

See Solution

Problem 21438

Determine if the series $\sum_{k=1}^{\infty} \frac{5^{k}}{k !+4}$ converges. Justify your answer.

See Solution

Problem 21439

Determine if the series $\sum_{k=1}^{\infty} \frac{(-14)^{k}}{k !}$ converges. Justify your answer.

See Solution

Problem 21440

Analysez la fonction $f(x)$ : croissance, décroissance, minimums, maximums, concavité, points d'inflexion, asymptotes.

See Solution

Problem 21441

Find local extrema of $f(x)=x^{c / 3}(x-3)^{2}$ using the first derivative test. Choose the correct option.

See Solution

Problem 21442

Determine if the series $\sum_{k=1}^{\infty}\left(1+\frac{a}{k}\right)^{13 k}$ converges. Justify your answer.

See Solution

Problem 21443

Berechnen Sie die Integrale a) $\int_{0}^{4} 2 x d x$ und b) $\int_{-1}^{3} \frac{1}{2} x^{2} d x$ mit dem Hauptsatz.

See Solution

Problem 21444

Find local extrema of $f(x)=x^{2/3}(x-3)^{2}$ using the first derivative test. Choose the correct option.

See Solution

Problem 21445

Determine if the series $\sum_{k=1}^{\infty} \frac{5 k^{5}+k}{7 k^{5}-6}$ converges and justify your answer.

See Solution

Problem 21446

Evaluate the series $\sum_{k=1}^{\infty} \frac{(-k)^{5}}{5 k^{5}+3}$ and select the correct choice regarding its convergence.

See Solution

Problem 21447

Bestimme die Steigung von $f(x) = x^{2} + 2x$ an der Stelle $x_{0} = 3$ .

See Solution

Problem 21448

Determine if the series $\sum_{k=1}^{\infty} \frac{1}{\sqrt{k} e^{\sqrt{k}}}$ converges and justify your answer.

See Solution

Problem 21449

Determine if the series $\sum_{k=1}^{\infty} \frac{9(4 k) !}{(k !)^{4}}$ converges and justify your answer.

See Solution

Problem 21450

Find values of $p$ such that the integral $\int_{0}^{1} \frac{1}{x^{p}} d x$ converges.

See Solution

Problem 21451

Find the absolute extrema of $f(x)=x^{4}-2x^{2}+5$ on $[-2, 1]$ . Choose: a) min at 1, max at -1; b) no max/min; c) min at -2, max at 0; d) min at 1 and -1, max at -2.

See Solution

Problem 21452

Find the absolute extrema of $f(x)=x^{4}-2x^{2}+5$ for $-2 \leq x \leq 1$ . Options: a) min at 1, max at -1; b) no max/min; c) min at -2, max at 0; d) min at 1 and -1, max at -2.

See Solution

Problem 21453

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 21454

Determine if the series $\sum_{k=1}^{\infty} \frac{k !}{k^{k}+20}$ converges or diverges. Justify your answer.

See Solution

Problem 21455

Find the value of $\int_{2}^{\infty} g^{\prime}(x) \, dx$ given $g(2)=1$ and $\lim_{x \to \infty} g(x)=8$ .

See Solution

Problem 21456

Find the domain and local minimum points of $P(x)=-x^{6}+6x^{4}-9x^{2}+4$ . Domain: $(-\infty, \infty)$ . Local min: \square, absolute min: \square.

See Solution

Problem 21457

Determine if the series $\sum_{k=1}^{\infty}\left(-\frac{10}{k}\right)^{k}$ converges and justify your answer.

See Solution

Problem 21458

Snowboarder auf Hang:
a) Berechne $s(1)$ und $s(5)$ für $\mathrm{s}(\mathrm{t})=1,5 \mathrm{t}^{2}$ . b) Finde die mittlere Geschwindigkeit in 5 Sek. c) Bestimme die Momentangeschwindigkeit nach 5 Sek.

See Solution

Problem 21459

Determine if the series converges: $\sum_{k=0}^{\infty} \frac{3 k}{\sqrt[5]{k^{5}+5}}$ . Justify your answer.

See Solution

Problem 21460

Determine if the series $\sum_{k=1}^{\infty}\left(\frac{1}{k^{7}}+\frac{1}{k^{8}}\right)$ converges. Justify your answer.

See Solution

Problem 21461

Find intervals of $x$ where $f(x)=x^{2}+\frac{1}{x^{2}}$ is concave up or down.

See Solution

Problem 21462

Determine if the series $\sum_{k=1}^{\infty}\left(\frac{2 k}{3 k+1}\right)^{k}$ converges using a convergence test.

See Solution

Problem 21463

Determine if $f(x)=\sec x$ is concave up or down at $x=\frac{21 \pi}{4}$ . Options: a) up b) down c) neither d) unknown.

See Solution

Problem 21464

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k}\left(\frac{7 k}{6 k+3}\right)^{k}$ converges. Justify your answer.

See Solution

Problem 21465

Find the area of the region $R$ between the curve $y=\frac{x}{(1+x^{2})^{2}}$ and the $x$ -axis for $x \geq 0$ .

See Solution

Problem 21466

Determine if the series $\sum_{k=1}^{\infty} \frac{3+(-1)^{k}}{2^{k}}$ converges or diverges. Justify your answer.

See Solution

Problem 21467

Determine if the series $\sum_{k=1}^{\infty} \frac{6^{k}}{7^{k}-6^{k}}$ converges or diverges. Justify your answer.

See Solution

Problem 21468

Graph the function $P(x)=-x^{6}+6 x^{4}-9 x^{2}+4$ and find local/absolute max points and the range.

See Solution

Problem 21469

Graph $f(x)=x^{3}-2x$ and the secant line through $(-2,-4)$ and $(2,4)$ . Estimate where the tangent is parallel to the secant. Find $c$ for the mean value theorem on $[-2,2]$ .

See Solution

Problem 21470

What values of $p$ make $\int_{1}^{\infty} \frac{1}{x^{7p-3}} dx$ converge?

See Solution

Problem 21471

Find the critical numbers of the function $f(x) = x^{3} + 3x^{2} - 45x$ .

See Solution

Problem 21472

Find the mean slope of $f(x)=4-3 x^{2}$ on $[-2,6]$ and the value of $c$ where $f'(c)$ equals this slope.

See Solution

Problem 21473

Find the average slope of $f(x)=\frac{1}{x}$ on $[1,6]$ and values of $c$ where $f'(c)$ equals this slope.

See Solution

Problem 21474

Find all values of $p$ for which the integral $\int_{0}^{1} \frac{1}{x^{p}} dx$ converges.

See Solution

Problem 21475

Find the critical number(s) of the function $h(p)=\frac{p-2}{p^{2}+2}$ .

See Solution

Problem 21476

Find critical points of $f$ from $f^{\prime}(x)=\frac{x^{2}(x-5)}{x+8}$ , and determine intervals of increase/decrease.

See Solution

Problem 21477

Rewrite the integral $I=\int_{0}^{3}\left|x^{2}-1\right| d x$ as a sum of integrals and evaluate it.

See Solution

Problem 21478

Find the critical numbers of the function $g(y)=\frac{y-1}{y^{2}-3y+3}$ .

See Solution

Problem 21479

Find the critical points of $f$ for $f^{\prime}(x)=(8 \sin x-8)(\sqrt{2} \cos x+1)$ on $0 \leq x \leq 2 \pi$ .

See Solution

Problem 21480

Find the derivative of $5 e^{\wedge}(7 x^{3}-3 x^{9})$ using the chain rule.

See Solution

Problem 21481

Sketch the curve $x=t \cos t, y=t \sin t$ for $0 \leq t \leq 4\pi$ . At $t=4$ , is the particle moving down $\quad \vee$ right? Estimate speed at $t=4$ using positions $t=4$ and $t=4.01$ . Calculate speed using derivatives: $\text{speed}=\sqrt{(\cos 4-4 \sin 4)^{2}+(\sin 4+4 \cos 4)^{2}}$ .

See Solution

Problem 21482

Calculate the derivative of $5^{x}$ . Use $\ln(x)$ for the natural logarithm.

See Solution

Problem 21483

Analyze the function with derivative $f^{\prime}(x)=(8 \sin x-8)(\sqrt{2} \cos x+1), 0 \leq x \leq 2 \pi$ for critical points, intervals of increase/decrease, and local extrema.

See Solution

Problem 21484

Given the function $f(x)=8 x-6 \ln (x)$ for $x>0$ , find critical numbers, intervals of increase/decrease, local maxima/minima, concavity, and inflection points.

See Solution

Problem 21485

Find the absolute max and min of $f(x)=2 x^{3}-6 x^{2}-18 x+3$ on the interval $[-2,4]$ .

See Solution

Problem 21486

Given $f(x)=(11-2 x) e^{x}$ , find critical values, intervals of increase/decrease, local max/min, concavity, and inflection points.

See Solution

Problem 21487

Calculate the arc length of the curve given by $x=\cos(\sqrt{t})$ , $y=\sin(\sqrt{t})$ for $0 \leq t \leq 1.$

See Solution

Problem 21488

Evaluate the integral: $\int_{b}^{a} x^{14} d x =$ in terms of constants.

See Solution

Problem 21489

Find the limit as $x$ approaches -7 from the left for the expression $\frac{-5 x}{x+7}$ .

See Solution

Problem 21490

Polonium-210 has a half-life of 138 days.
(a) Find the equation $P(d)$ for the percent left after $d$ days.
(b) Estimate the percent left after 310 days (round to one decimal place).

See Solution

Problem 21491

Find the limit as $x$ approaches -2 from the right: $\lim _{x \rightarrow-2^{+}} \frac{-4 x}{x+2}$ .

See Solution

Problem 21492

Find the inflection points of $f(x)$ given $f'(x)=(x+7)(11-x)(x-15)$ , with local extrema at $x=-7$ , $x=11$ , $x=15$ .

See Solution

Problem 21493

Find the inflection points of $f(x)$ given $f'(x)=(x+7)(11-x)(x-15)$ . Local extrema occur at $x=-7, 11, 15$ .

See Solution

Problem 21494

Find the average rate of change of $g(x)=2^{x}-1$ from $x=3$ to $x=7$ .

See Solution

Problem 21495

Find the value(s) of $A$ that make the function $f(x)$ continuous at $x=10$ using limits. Define $f(x)$ as: $f(x)=\left\{\begin{array}{ll} x^{2}-60, & \text { if } x<10 \\ A x, & \text { if } x \geq 10 \end{array}\right.$

See Solution

Problem 21496

Find critical points, intervals of increase/decrease, and local extrema for $f^{\prime}(x)=(8 \sin x-8)(\sqrt{2} \cos x+1)$ , $0 \leq x \leq 2 \pi$ .

See Solution

Problem 21497

Find the derivative $f'(x)$ of $f(x)=-5 x^{3} \ln x$ and evaluate $f'(e^{3})$ .

See Solution

Problem 21498

A train's position is $s(t)=\frac{60}{t}$ for $3 \leq t \leq 9$ . Find its average velocity over this interval.

See Solution

Problem 21499

A stone is dropped from a $45.1 \mathrm{~m}$ cliff. Find its velocity just before hitting the ground.

See Solution

Problem 21500

A pumpkin falls from $3,351 \mathrm{~m}$ . With gravity at $9.81 \mathrm{m/s^2}$ , how long until it hits the ground?

See Solution

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Calculus

Problem 21401

Find the growth rate dLdt\frac{d L}{d t}dtdL​ at t=10,20,25t=10, 20, 25t=10,20,25 weeks for L=−37.41+3.66t−6.32×10−4t3L=-37.41+3.66 t-6.32 \times 10^{-4} t^{3}L=−37.41+3.66t−6.32×10−4t3. What happens as ttt increases?

Problem 21402

(a) What does dHdx\frac{d H}{d x}dxdH​ mean? (b) Prove that dHdx\frac{d H}{d x}dxdH​ is a constant for the formula H(x)=205−0.65xH(x)=205-0.65 xH(x)=205−0.65x.

Problem 21403

Find points on the curve y=x3+2x+9y=x^{3}+2x+9y=x3+2x+9 where the tangent line is parallel to 7x−y=−57x-y=-57x−y=−5. Are there multiple points?

Problem 21404

Find the tangent line to f(x)=ax2f(x)=a x^{2}f(x)=ax2 at x=−2x=-2x=−2, where aaa is a positive constant.

Problem 21405

Find critical numbers of f(x)=x6−x4+5x3−4xf(x)=x^{6}-x^{4}+5 x^{3}-4 xf(x)=x6−x4+5x3−4x using Newton's method, correct to six decimal places. Also, find the absolute minimum value of fff, correct to four decimal places.

Problem 21406

Evaluate the integral: ∫1∞xe−x dx\int_{1}^{\infty} x e^{-x} \, dx∫1∞​xe−xdx

Problem 21407

Differentiate R(T)=3π516k2c6h7T5R(T)=\frac{3 \pi^{5}}{16} \frac{k^{2}}{c^{6} h^{7}} T^{5}R(T)=163π5​c6h7k2​T5 with respect to TTT. Constants: k,c,h>0k, c, h > 0k,c,h>0.

Problem 21408

Differentiate f(x)=ax20f(x)=a x^{20}f(x)=ax20 with respect to xxx, where aaa is a constant.

Problem 21409

Differentiate the function f(s)=s4e5+6ef(s)=s^{4} e^{5}+6 ef(s)=s4e5+6e with respect to sss.

Problem 21410

Berechnen Sie die Steigung von fff bei A(2∣f(2))A(2 \mid f(2))A(2∣f(2)) für f(x)=32x2f(x)=\frac{3}{2} x^{2}f(x)=23​x2.

Problem 21411

Differentiate g(N)=rN(1−NK)g(N)=r N(1-\frac{N}{K})g(N)=rN(1−KN​) with respect to NNN, where KKK and rrr are positive constants.

Problem 21412

Differentiate the function h(t)=112t2−2t+8h(t)=\frac{11}{2} t^{2}-2 t+8h(t)=211​t2−2t+8. Find h′(t)=□h^{\prime}(t)=\squareh′(t)=□.

Problem 21413

Bestimmen Sie die Ableitungen von den Funktionen: a) f(x)=x3f(x)=x^{3}f(x)=x3, b) f(x)=x5f(x)=x^{5}f(x)=x5, c) f(x)=x2nf(x)=x^{2 n}f(x)=x2n, d) f(x)=xf(x)=xf(x)=x, e) f(x)=xn+4f(x)=x^{n+4}f(x)=xn+4, f) f(x)=x2016f(x)=x^{2016}f(x)=x2016.

Problem 21414

Bestimmen Sie den Grad von f′f^{\prime}f′ und f′′f^{\prime \prime}f′′. Leiten Sie f(t)=t4−83t3−t2−4t+16f(t)=t^{4}-\frac{8}{3} t^{3}-t^{2}-4 t+16f(t)=t4−38​t3−t2−4t+16 zweimal ab.

Problem 21415

Formen Sie f(x)=x⋅(x4+4x−7)f(x)=x \cdot\left(x^{4}+4 x-7\right)f(x)=x⋅(x4+4x−7) um und leiten Sie die Funktion ab.

Problem 21416

Calculate the integral from −∞-\infty−∞ to ∞\infty∞ of 11+x2\frac{1}{1+x^{2}}1+x21​ dx.

Problem 21417

A rancher has 600 feet of fencing for a rectangular field divided into 2 plots. Find dimensions for maximum area.

Problem 21418

Explain why dxdt=k([A]0−x)([B]0−x)\frac{d x}{d t}=k([A]_0-x)([B]_0-x)dtdx​=k([A]0​−x)([B]0​−x) for the reaction A+B→ABA+B \rightarrow A BA+B→AB.

Problem 21419

Bestimme die Stellen, an denen f′(x)=5f'(x)=5f′(x)=5 für f(x)=4x2−7x+6f(x)=4 x^{2}-7 x+6f(x)=4x2−7x+6. Berechne f′(5)f'(5)f′(5).

Problem 21420

Beweisen Sie die Potenzregel für n=5n=5n=5 und für beliebiges n∈Nn \in \mathbb{N}n∈N, basierend auf dem Beweis für n=4n=4n=4.

Problem 21421

Given the population size N(t)\mathrm{N}(t)N(t) satisfies dNdt=rN\frac{dN}{dt}=rNdtdN​=rN:(a) What is the per capita growth rate? (b) If r<0\mathrm{r}<0r<0 and N(0)=20\mathrm{N}(0)=20N(0)=20, is N(1)\mathrm{N}(1)N(1) greater or less than 20? Explain.

Problem 21422

Bestimme die Punkte, an denen der Graph von f(x)=18x4−4x+3f(x)=\frac{1}{8} x^{4}-4 x+3f(x)=81​x4−4x+3 eine waagerechte Tangente hat.

Problem 21423

Bestimme die Punkte, an denen der Graph von f(x)=18x4−4x+3f(x)=\frac{1}{8} x^{4}-4 x+3f(x)=81​x4−4x+3 eine waagerechte Tangente hat.

Problem 21424

Given the equation dNdt=rN\frac{dN}{dt}=rNdtdN​=rN, find the per capita growth rate rrr. If r<0\mathrm{r}<0r<0 and N(0)=20\mathrm{N}(0)=20N(0)=20, is N(1)\mathrm{N}(1)N(1) > 20 or < 20? Explain.

Problem 21425

Problem 21426

Find ∫−44f(x)dx\int_{-4}^{4} f(x) dx∫−44​f(x)dx for the piecewise function: f(x)=3x2f(x)=3x^2f(x)=3x2 for −4≤x<0-4 \leq x<0−4≤x<0 and f(x)=4x3f(x)=4x^3f(x)=4x3 for 0≤x≤40 \leq x \leq 40≤x≤4.

Problem 21427

Bestimmen Sie die positive Zahl z für: a) ∫0zx dx=18\int_{0}^{z} x \, dx=18∫0z​xdx=18, b) ∫1z4x dx=30\int_{1}^{z} 4x \, dx=30∫1z​4xdx=30.

Problem 21428

Evaluate the integral ∫−101x5dx\int_{-1}^{0} \frac{1}{x^{5}} dx∫−10​x51​dx.

Problem 21429

Find all antiderivatives of f(x)=ex+1x2+cos⁡(x)f(x)=e^{x}+\frac{1}{x^{2}}+\cos (x)f(x)=ex+x21​+cos(x).

Problem 21430

Problem 21431

Bestimme die Tangentengleichung der Funktion f(x)=2x3−6xf(x)=2x^3-6xf(x)=2x3−6x bei x0=−1x_0=-1x0​=−1.

Problem 21432

Is it true that ∫58f(x)dx=∫530f(x)dx−∫830f(x)dx\int_{5}^{8} f(x) dx = \int_{5}^{30} f(x) dx - \int_{8}^{30} f(x) dx∫58​f(x)dx=∫530​f(x)dx−∫830​f(x)dx for fff on (5,30](5,30](5,30]? a) True b) False

Problem 21433

Berechne die Integrale: a) ∫04−x dx\int_{0}^{4}-x \, dx∫04​−xdx, b) ∫−11−2x dx\int_{-1}^{1}-2x \, dx∫−11​−2xdx, c) ∫−22−x2 dx\int_{-2}^{2}-x^{2} \, dx∫−22​−x2dx.

Problem 21434

Determine if the series converges: ∑k=1∞5k5+k7k5−6\sum_{k=1}^{\infty} \frac{5 k^{5}+k}{7 k^{5}-6}k=1∑∞​7k5−65k5+k​ Justify your answer.

Problem 21435

Problem 21436

Calculate the total energy requirements (in kJ/W0.67\mathrm{kJ} / \mathrm{W}^{0.67}kJ/W0.67) for a female Collie from age 730 to 1460 days using E(t)=5t−0.2E(t)=5 t^{-0.2}E(t)=5t−0.2. Options: a) 904.515 b) 1000 c) 1085.418 d) 723.612

Problem 21437

Calculate the integral from 0 to infinity of e−x dxe^{-x} \, dxe−xdx.

Problem 21438

Determine if the series ∑k=1∞5kk!+4\sum_{k=1}^{\infty} \frac{5^{k}}{k !+4}∑k=1∞​k!+45k​ converges. Justify your answer.

Problem 21439

Determine if the series ∑k=1∞(−14)kk!\sum_{k=1}^{\infty} \frac{(-14)^{k}}{k !}∑k=1∞​k!(−14)k​ converges. Justify your answer.

Problem 21440

Analysez la fonction f(x)f(x)f(x) : croissance, décroissance, minimums, maximums, concavité, points d'inflexion, asymptotes.

Problem 21441

Find local extrema of f(x)=xc/3(x−3)2f(x)=x^{c / 3}(x-3)^{2}f(x)=xc/3(x−3)2 using the first derivative test. Choose the correct option.

Find the growth rate $\frac{d L}{d t}$ at $t=10, 20, 25$ weeks for $L=-37.41+3.66 t-6.32 \times 10^{-4} t^{3}$ . What happens as $t$ increases?

(a) What does $\frac{d H}{d x}$ mean? (b) Prove that $\frac{d H}{d x}$ is a constant for the formula $H(x)=205-0.65 x$ .

Find points on the curve $y=x^{3}+2x+9$ where the tangent line is parallel to $7x-y=-5$ . Are there multiple points?

Find the tangent line to $f(x)=a x^{2}$ at $x=-2$ , where $a$ is a positive constant.

Find critical numbers of $f(x)=x^{6}-x^{4}+5 x^{3}-4 x$ using Newton's method, correct to six decimal places. Also, find the absolute minimum value of $f$ , correct to four decimal places.

Evaluate the integral: $\int_{1}^{\infty} x e^{-x} \, dx$

Differentiate $R(T)=\frac{3 \pi^{5}}{16} \frac{k^{2}}{c^{6} h^{7}} T^{5}$ with respect to $T$ . Constants: $k, c, h > 0$ .

Differentiate $f(x)=a x^{20}$ with respect to $x$ , where $a$ is a constant.

Differentiate the function $f(s)=s^{4} e^{5}+6 e$ with respect to $s$ .

Berechnen Sie die Steigung von $f$ bei $A(2 \mid f(2))$ für $f(x)=\frac{3}{2} x^{2}$ .

Differentiate $g(N)=r N(1-\frac{N}{K})$ with respect to $N$ , where $K$ and $r$ are positive constants.

Differentiate the function $h(t)=\frac{11}{2} t^{2}-2 t+8$ . Find $h^{\prime}(t)=\square$ .

Bestimmen Sie die Ableitungen von den Funktionen: a) $f(x)=x^{3}$ , b) $f(x)=x^{5}$ , c) $f(x)=x^{2 n}$ , d) $f(x)=x$ , e) $f(x)=x^{n+4}$ , f) $f(x)=x^{2016}$ .

Bestimmen Sie den Grad von $f^{\prime}$ und $f^{\prime \prime}$ . Leiten Sie $f(t)=t^{4}-\frac{8}{3} t^{3}-t^{2}-4 t+16$ zweimal ab.

Formen Sie $f(x)=x \cdot\left(x^{4}+4 x-7\right)$ um und leiten Sie die Funktion ab.

Calculate the integral from $-\infty$ to $\infty$ of $\frac{1}{1+x^{2}}$ dx.

Explain why $\frac{d x}{d t}=k([A]_0-x)([B]_0-x)$ for the reaction $A+B \rightarrow A B$ .

Bestimme die Stellen, an denen $f'(x)=5$ für $f(x)=4 x^{2}-7 x+6$ . Berechne $f'(5)$ .

Beweisen Sie die Potenzregel für $n=5$ und für beliebiges $n \in \mathbb{N}$ , basierend auf dem Beweis für $n=4$ .

Given the population size $\mathrm{N}(t)$ satisfies $\frac{dN}{dt}=rN$ :
(a) What is the per capita growth rate? (b) If $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ , is $\mathrm{N}(1)$ greater or less than 20? Explain.

Bestimme die Punkte, an denen der Graph von $f(x)=\frac{1}{8} x^{4}-4 x+3$ eine waagerechte Tangente hat.

Bestimme die Punkte, an denen der Graph von $f(x)=\frac{1}{8} x^{4}-4 x+3$ eine waagerechte Tangente hat.

Given the equation $\frac{dN}{dt}=rN$ , find the per capita growth rate $r$ . If $\mathrm{r}<0$ and $\mathrm{N}(0)=20$ , is $\mathrm{N}(1)$ > 20 or < 20? Explain.

Find $\int_{-4}^{4} f(x) dx$ for the piecewise function: $f(x)=3x^2$ for $-4 \leq x<0$ and $f(x)=4x^3$ for $0 \leq x \leq 4$ .

Bestimmen Sie die positive Zahl z für: a) $\int_{0}^{z} x \, dx=18$ , b) $\int_{1}^{z} 4x \, dx=30$ .

Evaluate the integral $\int_{-1}^{0} \frac{1}{x^{5}} dx$ .

Find all antiderivatives of $f(x)=e^{x}+\frac{1}{x^{2}}+\cos (x)$ .

Bestimme die Tangentengleichung der Funktion $f(x)=2x^3-6x$ bei $x_0=-1$ .

Is it true that $\int_{5}^{8} f(x) dx = \int_{5}^{30} f(x) dx - \int_{8}^{30} f(x) dx$ for $f$ on $(5,30]$ ? a) True b) False

Berechne die Integrale: a) $\int_{0}^{4}-x \, dx$ , b) $\int_{-1}^{1}-2x \, dx$ , c) $\int_{-2}^{2}-x^{2} \, dx$ .

Determine if the series converges: $\sum_{k=1}^{\infty} \frac{5 k^{5}+k}{7 k^{5}-6}$ Justify your answer.

Calculate the total energy requirements (in $\mathrm{kJ} / \mathrm{W}^{0.67}$ ) for a female Collie from age 730 to 1460 days using $E(t)=5 t^{-0.2}$ . Options: a) 904.515 b) 1000 c) 1085.418 d) 723.612

Calculate the integral from 0 to infinity of $e^{-x} \, dx$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{5^{k}}{k !+4}$ converges. Justify your answer.

Determine if the series $\sum_{k=1}^{\infty} \frac{(-14)^{k}}{k !}$ converges. Justify your answer.

Analysez la fonction $f(x)$ : croissance, décroissance, minimums, maximums, concavité, points d'inflexion, asymptotes.

Find local extrema of $f(x)=x^{c / 3}(x-3)^{2}$ using the first derivative test. Choose the correct option.

Determine if the series $\sum_{k=1}^{\infty}\left(1+\frac{a}{k}\right)^{13 k}$ converges. Justify your answer.

Berechnen Sie die Integrale a) $\int_{0}^{4} 2 x d x$ und b) $\int_{-1}^{3} \frac{1}{2} x^{2} d x$ mit dem Hauptsatz.

Find local extrema of $f(x)=x^{2/3}(x-3)^{2}$ using the first derivative test. Choose the correct option.

Determine if the series $\sum_{k=1}^{\infty} \frac{5 k^{5}+k}{7 k^{5}-6}$ converges and justify your answer.

Evaluate the series $\sum_{k=1}^{\infty} \frac{(-k)^{5}}{5 k^{5}+3}$ and select the correct choice regarding its convergence.

Bestimme die Steigung von $f(x) = x^{2} + 2x$ an der Stelle $x_{0} = 3$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{1}{\sqrt{k} e^{\sqrt{k}}}$ converges and justify your answer.

Determine if the series $\sum_{k=1}^{\infty} \frac{9(4 k) !}{(k !)^{4}}$ converges and justify your answer.

Find values of $p$ such that the integral $\int_{0}^{1} \frac{1}{x^{p}} d x$ converges.

Find the absolute extrema of $f(x)=x^{4}-2x^{2}+5$ on $[-2, 1]$ . Choose: a) min at 1, max at -1; b) no max/min; c) min at -2, max at 0; d) min at 1 and -1, max at -2.

Find the absolute extrema of $f(x)=x^{4}-2x^{2}+5$ for $-2 \leq x \leq 1$ . Options: a) min at 1, max at -1; b) no max/min; c) min at -2, max at 0; d) min at 1 and -1, max at -2.

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) $f(x)=2 x+1 ; x_{0}=3$

Determine if the series $\sum_{k=1}^{\infty} \frac{k !}{k^{k}+20}$ converges or diverges. Justify your answer.

Find the value of $\int_{2}^{\infty} g^{\prime}(x) \, dx$ given $g(2)=1$ and $\lim_{x \to \infty} g(x)=8$ .

Find the domain and local minimum points of $P(x)=-x^{6}+6x^{4}-9x^{2}+4$ . Domain: $(-\infty, \infty)$ . Local min: \square, absolute min: \square.

Determine if the series $\sum_{k=1}^{\infty}\left(-\frac{10}{k}\right)^{k}$ converges and justify your answer.

Snowboarder auf Hang:
a) Berechne $s(1)$ und $s(5)$ für $\mathrm{s}(\mathrm{t})=1,5 \mathrm{t}^{2}$ . b) Finde die mittlere Geschwindigkeit in 5 Sek. c) Bestimme die Momentangeschwindigkeit nach 5 Sek.

Determine if the series converges: $\sum_{k=0}^{\infty} \frac{3 k}{\sqrt[5]{k^{5}+5}}$ . Justify your answer.

Determine if the series $\sum_{k=1}^{\infty}\left(\frac{1}{k^{7}}+\frac{1}{k^{8}}\right)$ converges. Justify your answer.

Find intervals of $x$ where $f(x)=x^{2}+\frac{1}{x^{2}}$ is concave up or down.

Determine if the series $\sum_{k=1}^{\infty}\left(\frac{2 k}{3 k+1}\right)^{k}$ converges using a convergence test.

Determine if $f(x)=\sec x$ is concave up or down at $x=\frac{21 \pi}{4}$ . Options: a) up b) down c) neither d) unknown.

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k}\left(\frac{7 k}{6 k+3}\right)^{k}$ converges. Justify your answer.

Find the area of the region $R$ between the curve $y=\frac{x}{(1+x^{2})^{2}}$ and the $x$ -axis for $x \geq 0$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{3+(-1)^{k}}{2^{k}}$ converges or diverges. Justify your answer.

Determine if the series $\sum_{k=1}^{\infty} \frac{6^{k}}{7^{k}-6^{k}}$ converges or diverges. Justify your answer.

Graph the function $P(x)=-x^{6}+6 x^{4}-9 x^{2}+4$ and find local/absolute max points and the range.

Graph $f(x)=x^{3}-2x$ and the secant line through $(-2,-4)$ and $(2,4)$ . Estimate where the tangent is parallel to the secant. Find $c$ for the mean value theorem on $[-2,2]$ .

What values of $p$ make $\int_{1}^{\infty} \frac{1}{x^{7p-3}} dx$ converge?

Find the critical numbers of the function $f(x) = x^{3} + 3x^{2} - 45x$ .

Find the mean slope of $f(x)=4-3 x^{2}$ on $[-2,6]$ and the value of $c$ where $f'(c)$ equals this slope.

Find the average slope of $f(x)=\frac{1}{x}$ on $[1,6]$ and values of $c$ where $f'(c)$ equals this slope.

Find all values of $p$ for which the integral $\int_{0}^{1} \frac{1}{x^{p}} dx$ converges.