Calculus

Problem 14501

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

See Solution

Problem 14502

Evaluate the integral $\int_{-a}^{a}(x+a)(x-a) dx = -40$ .

See Solution

Problem 14503

For which values of $-3 \leq x_{0} \leq 2$ does $\lim _{x \rightarrow x_{0}} g(x)$ not exist?

See Solution

Problem 14504

Tomatenpflanze: Höhe $h(t)$ eines Setzlings mit $v(t)=-0,1 t^{3}+t^{2}$ . Fragen: a) Wachstumsdauer? b) Maximale Höhe? c) Höhe beim schnellsten Wachstum?

See Solution

Problem 14505

Bestimmen Sie das Minimum und Maximum der Funktion $f_{a}(x)=\frac{1}{3 a} x^{3}-x^{2}$ mit $f^{\prime} a(x)=1 a x^{2}-2 x$ .

See Solution

Problem 14506

Berechne die Fläche zwischen den Funktionen $f(x)=\frac{1}{4} x^{2}$ und $g(x)=(x-1)^{2}$ , indem du ihre Schnittpunkte findest.

See Solution

Problem 14507

Aufgabe:
Gegeben ist die Funktion $f(x)=-\frac{5}{16} x^{4}+5 x^{3}$ , die die Änderungsrate des Glyzerin-Tankinhalts beschreibt.
a) Was bedeuten die Koordinaten des Punkts (4/240) im Kontext? b) Ist die Aussage korrekt: Nach 12 Stunden ist die maximale Glyzerinmenge im Tank? c) Wie viel Glyzerin ist nach 20 Stunden im Tank?

See Solution

Problem 14508

Ein Segelflugzeug startet bei 80 m Höhe. Bestimme die Höhenfunktion und beantworte Fragen zu max. Höhe, Änderungszeit und Landeanflug.

See Solution

Problem 14509

Find the values of $x$ where the curve $y=x^{3}-3x+2$ is concave up.

See Solution

Problem 14510

Find the inflection point and extremum of $f(x)=(x-4) \cdot e^{x}$ for $x \in \mathbb{R}$ .

See Solution

Problem 14511

Find the limit: $\lim _{x \rightarrow 6} \frac{x}{x^{2}-36}$ .

See Solution

Problem 14512

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}+7 x+6}{x^{2}-7 x-8}$ .

See Solution

Problem 14513

Bestimme die Wendepunkte der Funktion $f_{t}(x)=x^{3}-3 t x^{2}$ für $t>0$ .

See Solution

Problem 14514

Find the stationary point on the curve given by the equation $y=x^{2}-4 x+3$ .

See Solution

Problem 14515

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{12} x^{3}-\frac{1}{2} x^{2}$ .

See Solution

Problem 14516

Untersuchen Sie das Verhalten der Funktion $f(x)=\frac{2x+1}{x}$ für $x<0$ , wenn $x \rightarrow -\infty$ . Skizzieren Sie den Graphen.

See Solution

Problem 14517

Untersuchen Sie das Verhalten der Funktionen für die Grenzprozesse. Skizzieren Sie die Graphen. a) $f(x)=\frac{2 x+1}{x}, x<0$ , $x \rightarrow-\infty$ b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$

See Solution

Problem 14518

Vulkanprofil: $f(x)=\frac{25}{x^{2}}$ für $2 \leq x \leq 6$ . a) Höhe bei $400 \mathrm{~m}$ Breite? b) Höhe bei $1 \mathrm{~km}$ Durchmesser?

See Solution

Problem 14519

Untersuche die Funktion $f(x)=(x-4) \cdot e^{x}$ auf y-Achsen-Schnittpunkt, x-Achsen-Schnittpunkte, Extrempunkte, Wendepunkte. Zeige, dass $F(x)=(x-5)e^{-x}$ eine Stammfunktion von $f(x)$ ist und bestimme die Fläche im Intervall $[0, 4]$ .

See Solution

Problem 14520

Bestimmen Sie die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ für a) $f_{a}(x)=-x^{2}+a x$ , b) $f_{a}(x)=a x^{3}-3 a x$ , c) $f_{a}(x)=a x^{4}-4 x^{3}+a^{2} x$ . Wann ist die Steigung 1?

See Solution

Problem 14521

Evaluate the double integral $\int_{-2}^{2} \int_{-1}^{1} \sqrt{2\left(\frac{-1}{\sqrt{2}} \sinh (u+v)\right)^{2}+1} \, du \, dv$ .

See Solution

Problem 14522

Find the derivative $\frac{d x}{d t}$ for the function $x=\frac{t^{2}+2}{5 t}$ .

See Solution

Problem 14523

Evaluate the double integral: $\int_{-2}^{2} \int_{-1}^{1} \sqrt{\sinh ^{2}(u+v)+1} \, du \, dv$ .

See Solution

Problem 14524

Berechnen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ , $x_{0}=2$ ; b) $f(x)=2x^{2}$ , $x_{0}=1$ .

See Solution

Problem 14525

A car travels straight with velocity $v(t)$ as a rhombus. Find $\int_{0}^{24} v(t) dt$ and explain its meaning.

See Solution

Problem 14526

Approximate $\int_{0}^{6} f(t) dt$ using right-hand Riemann sum with intervals $[0,2],[2,3],[3,6]$ and values from the table. Choices: A. 204 B. 225 C. 246 D. 252 E. 281

See Solution

Problem 14527

Evaluate the integral $\int_{-2}^{6} f(x) d x$ for the piecewise function: $f(x)=\begin{cases} 2x-6, & x<4 \\ 8, & x \geq 4 \end{cases}$ .

See Solution

Problem 14528

Calculate the derivative $\left.\frac{\mathbb{d} y}{\mathbb{d} x}\right|_{x=1}$ for $y=\frac{4+5 x+9 x^{2}+11 x^{3}+10 x^{4}+5 x^{5}+4 x^{6}}{x^{3}}$ .

See Solution

Problem 14529

Approximate $\int_{1}^{7} \frac{x^{2}+1}{x} dx$ using a Trapezoidal Sum with 3 subintervals. Is it an under or over estimation?

See Solution

Problem 14530

Calculate the integral: $\int_{3}^{5} \frac{2 x^{2}+x+4}{x-1} \, dx$

See Solution

Problem 14531

Determine the horizontal asymptotes of the function $t(x)=\frac{(x+2)(x-1)(x-4)}{5(x+3)(x-1)(x-3)}$ .

See Solution

Problem 14532

Calculate the integral: $\int_{0}^{1} \frac{1}{x^{2}+6 x+10} d x$ .

See Solution

Problem 14533

Find the antiderivative for these integrals: a. $\int \frac{\cos (\theta)}{\sqrt{1+\sin (\theta)}} d \Theta$ b. $\int (x^{3}-3 x) d x$ c. $\int \frac{x}{x^{2}+1} d x$

See Solution

Problem 14534

Evaluate the integral: $\int_{1}^{1} \sin (x) e^{-3 x-6} d x$

See Solution

Problem 14535

Find where the graph of $f(x)=\frac{18}{\sqrt[3]{x-2}}$ is concave down. Choose A, B, C, or D.

See Solution

Problem 14536

Bestimme die Ableitung der Funktionen: a) $f(x)=\frac{2}{x}$ , b) $f(x)=\frac{3}{x}$ , c) $f(x)=x^{2}+1$ , d) $f(x)=2 x^{2}-1$ .

See Solution

Problem 14537

Find intervals where the graph of $f(x)=-\frac{81}{\sqrt[3]{x+5}}$ is concave up. Choose one: (A) $x>\sqrt[3]{5}$ (B) $x<-\sqrt[3]{5}$ (C) $x<-5$ (D) $x>-5$

See Solution

Problem 14538

Berechnen Sie den Flächeninhalt zwischen den Graphen $f(x)=e^{0.5 x}$ und $g(x)=2 e-e^{0.5 x}$ im Bereich $D=\mathbb{R}$ .

See Solution

Problem 14539

Find the minimum slope of the curve given by the equation $y=x^{5}-10 x^{2}$ .

See Solution

Problem 14540

Find the maximum slope of the curve $y=4 x^{3}-24 x^{2}+32 x$ in the interval $[0,5]$ .

See Solution

Problem 14541

What is the shot put's acceleration while falling into the pit, considering gravity?

See Solution

Problem 14542

Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{2}+h\right)-\cos \left(\frac{\pi}{2}\right)}{h}$ . Options: a. 1 b. 0 c. -1 d. does not exist

See Solution

Problem 14543

Find the values of $x$ where the function $f(x)=-\frac{1}{2} x^{4}-6 x^{3}-27 x^{2}$ has points of inflection. Options: A) $x=3$ , B) $x=-3$ , C) $x=\frac{3}{2}$ , D) none.

See Solution

Problem 14544

Calculate the area between the curves $y=16x$ , $y=x^5$ , and the lines $x=0$ , $x=2$ . The area is $\square$ .

See Solution

Problem 14545

Estimate $r^{\prime}(5)$ using values from the table: $r(0)=8$ , $r(3)=10$ , $r(4)=14$ , $r(6)=22$ . Options: a. 8, b. 18, c. 4, d. 16.

See Solution

Problem 14546

What happens at the point $(3,8)$ for the function $f$ given $f(3)=8$ , $f^{\prime}(3)=0$ , and $f^{\prime \prime}(3)=0$ ? Choose: (A) minimum point (B) maximum point (C) not enough information

See Solution

Problem 14547

Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{2}+h\right)-\cos \left(\frac{\pi}{2}\right)}{h}$ . Choices: (a) 1, (b) 0, (c) -1, (d) does not exist.

See Solution

Problem 14548

What happens at the point $(2,3)$ on the graph of the function $f$ given $f(2)=3$ , $f^{\prime}(2)=0$ , and $f^{\prime \prime}(2)=5$ ?
Choose 1 answer: (A) $(2,3)$ is a minimum point. (B) $(2,3)$ is a maximum point. (C) There's not enough information to tell.

See Solution

Problem 14549

What happens to the graph of the twice differentiable function $f$ at the point $(-7,6)$ given $f(-7)=6$ , $f'(-7)=0$ , and $f''(-7)=-5$ ?
Choose 1 answer: (A) $(-7,6)$ is a minimum point. (B) $(-7,6)$ is a maximum point. (C) There's not enough information to tell.

See Solution

Problem 14550

Find the area between $y=4x$ , $y=x^3$ , $x=0$ , and $x=2$ . The area is $\square$ .

See Solution

Problem 14551

Find the integral of the function: $\int\left(\sqrt[4]{x^{3}}+1\right) d x$ .

See Solution

Problem 14552

Calculate the area between the curves $f(x)=x^{4}-8 x^{3}+21 x^{2}$ and $g(x)=20 x+50$ where they intersect.

See Solution

Problem 14553

6 圆 a) Finde die Ableitung von $f(x)=(2 x-3) \cdot e^{x}, g(x)=x \cdot e^{2 x}, h(x)=e^{x} \cdot (x^{2}+2 x)$ . b) Wo haben $f$ und $g$ waagerechte Tangenten? c) Finde die Schnittpunkte von $h$ mit der x-Achse und die Steigung der Tangenten dort.

See Solution

Problem 14554

Find the time $t$ when the particle at position $x(t)=2 t^{3}-21 t^{2}+72 t-53$ is at rest.

See Solution

Problem 14555

Find the average velocity of a particle with position $-5 t^{2}$ from $t=0$ to $t=3$ . Choices: (A) -45 (B) -30 (C) -15 (D) -10 (E) -5

See Solution

Problem 14556

Estimate $f(4.8)$ using local linear approximation at $x=5$ given $f(5)=3$ and $f'(5)=4$ . Options: (A) 2.2 (B) 2.8 (C) 3.4 (D) 3.8 (E) 4.6

See Solution

Problem 14557

Calculate left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum is $\square$ .

See Solution

Problem 14558

Calculate the left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum: $\square$ .

See Solution

Problem 14559

Approximate the displacement of an object with $v=4t+3$ (m/s) from $t=0$ to $t=8$ using $n=2$ subintervals. Answer: $\square$ m.

See Solution

Problem 14560

Find the time $t$ when the particle's velocity $v(t)=t^{3}-6 t^{2}+10 t-4$ changes from right to left for $t \geq 0$ .

See Solution

Problem 14561

Calculate left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum is 20; find right sum.

See Solution

Problem 14562

Approximate the displacement for $v=\frac{1}{4t+1}$ from $t=0$ to $t=8$ using 4 subintervals. Displacement: $\square \mathrm{m}$ .

See Solution

Problem 14563

Calculate the left and right Riemann sums for $f(x)=\frac{1}{x}+1$ on $[1,5]$ with $n=4$ . Left sum: $\square$ .

See Solution

Problem 14564

Determine the end behavior of the polynomial $g(x)=3 x^{4}+2 x^{2}+x-1$ using limits.

See Solution

Problem 14565

Find the tangent line equation to the curve at $t=0$ for $x=\int_{2t}^{at} \sqrt{1+u^{3}} du$ and $y=2+at^{2}+(t+1) \cos^{3}(at)$ , where $a>2$ .

See Solution

Problem 14566

Ein Segelflugzeug hat die Flughöhe $h$ mit der Geschwindigkeit $\mathrm{v}_{1}(\mathrm{t})=-0,15 \mathrm{t}^{2}+0,9 \mathrm{t}$ .
a) Finde die Höhenfunktion. b) Bestimme die maximale Flughöhe. c) Wann ist die Höhenänderung maximal? d) Wann erreicht es wieder 80 m? e) Finde $a$ und $b$ für eine sanfte Landung nach 18 min.

See Solution

Problem 14567

Find points on the curve where the tangent is horizontal or vertical for $x=\log_{a}(\sec(at))$ , $y=\log_{a}(\sin(at))$ , $-\frac{\pi}{2a} \leq t \leq \frac{\pi}{2a}$ , $a>1$ .

See Solution

Problem 14568

Find the instantaneous rate of change of profit at quantity $q=10$ for the profit function $P(q)=300 q-10 q^{2}-200$ .

See Solution

Problem 14569

Find the derivative of $y=6 x^{3}+15 x^{2}$ and evaluate it at $x=3$ .

See Solution

Problem 14570

Calculate the integral from 3 to 4 of the function $\frac{t^{5}-2 t}{t^{3}}$ .

See Solution

Problem 14571

Evaluate the integral $\int_{1}^{\frac{1}{2}} \sqrt{\frac{\pi}{x^{2}}} \mathrm{~d} x$ .

See Solution

Problem 14572

Determine where the function $f(x)=-x^{4}-7 x^{3}-12 x^{2}$ is increasing or decreasing.

See Solution

Problem 14573

Calculate the integral of $f(x)=\left(e^{-\frac{1}{4} x}\right)^{2}$ .

See Solution

Problem 14574

A 400 g isotope decays as $A(t)=400 e^{-0.044 t}$ . Find: (a) amount left after 20 years, (b) time to half decay.

See Solution

Problem 14575

Find the limit: $\lim _{n \rightarrow \infty}\left(1+\frac{\pi}{2^{n+1}}\right)^{2^{n}}$ .

See Solution

Problem 14576

Calculate the integral from 1 to 4 of the function $\sqrt{x}-1$ .

See Solution

Problem 14577

Bestimme die Ableitung $f^{\prime}$ für folgende Funktionen: a) $f(x)=e^{x}+2 x+5$ b) $f(x)=x^{2}+e x+e^{x}+e$ c) $f(x)=e^{x}+2 e^{x}$ d) $f(x)=e x^{3}+e^{3}+e^{x}$ e) $f(x)=x^{2}+2 e+1$ f) $f(x)=2 x+e^{2}+e^{x}+e^{x}$

See Solution

Problem 14578

Analyze the function $f(x)=-x^{4}-7 x^{3}-12 x^{2}$ for end behavior and intervals of increase and decrease.

See Solution

Problem 14579

Find the relative max and min of the function $f(x)=-x^{4}-7x^{3}-12x^{2}$ .

See Solution

Problem 14580

Bestimmen Sie das Krümmungsverhalten und die Wendepunkte für die Funktionen a) $f(x)=x^{3}+3 x^{2}+2$ , b) $f(x)=\frac{1}{2} x^{3}-\frac{3}{2} x$ , c) $f(x)=\frac{1}{4} x^{4}-\frac{1}{3} x^{3}$ , d) $f(x)=1-x$ .

See Solution

Problem 14581

Find the limit: $\lim _{n \rightarrow \infty}\left((\sin (\pi / 2))^{n}+(2 \sin (\pi / 4))^{n}+(1 / 10)^{n}\right)^{1 / n}$

See Solution

Problem 14582

Find the average annual change in the amount $A(t)=1000(1.03)^{t}$ for the first five years.

See Solution

Problem 14583

Find $\frac{d h}{d p}$ when $p=200$ for the equation $20 h^{5}+6,000,000=p^{3}$ . Interpret your answer.

See Solution

Problem 14584

How old is a wooden artifact with 22% carbon-14 left? Use the half-life of carbon-14, which is 5730 years.

See Solution

Problem 14585

Ein Segelflugzeug startet in 80 m Höhe. Bestimme die Höhenfunktion $h(t)$ , maximale Höhe, Zeit der höchsten Änderung und Landeparameter $a, b$ .

See Solution

Problem 14586

Find $f$ where $f'(x) = x^2 + 3$ and $f(0) = 5$ .

See Solution

Problem 14587

Find the second derivative $\frac{d^{2} y}{d x^{2}}$ at $x=-3$ for the function $y=\frac{x+5}{x-2}$ .

See Solution

Problem 14588

Find the limit: $\lim _{n \rightarrow \infty} \sqrt{12 n^{3}+n}-\sqrt{12 n^{3}-5 n}$ .

See Solution

Problem 14589

Berechne die Ableitung von $f(x)=-2 x^{2}$ an der Stelle $x_{0}=3$ .

See Solution

Problem 14590

Find the tangent line equation for the function $f(x)=x^{2}+2x$ at the point where $x=5$ .

See Solution

Problem 14591

Find the total profit from selling the first 80 tickets, given the marginal-profit function $P^{\prime}(x)=7x-1143$ .

See Solution

Problem 14592

Calculate the integral $\int \frac{9}{2+9 x} d x$ , where $x \neq -\frac{2}{9}$ .

See Solution

Problem 14593

Kostenfunktion: Gegeben ist $f(x)=0,01 x^{3}-0,3 x^{2}+9,5 x+100$ . Bestimme $S_{y}$ , Monotonieverhalten und Wendepunkt.

See Solution

Problem 14594

Find the value of $a$ if $\lim _{x \rightarrow-\infty} \frac{\sqrt{a x^{2}+b x+c}}{d-a x}=2$ .

See Solution

Problem 14595

Find the series sum: $\sum_{k=1}^{\infty} \frac{1}{(5 k-2)(5 k+3)}$ using $\frac{n}{15n+9}$ .

See Solution

Problem 14596

Evaluate the integral $\int\left(9+x^{6}\right)^{8} 6 x^{5} d x$ .

See Solution

Problem 14597

Evaluate the integral $\int\left(2+x^{8}\right)^{4} 8 x^{7} d x$ .

See Solution

Problem 14598

Bestimme die Gesamtstrecke eines Autos in einem Geschwindigkeits-Zeit-Diagramm für $t \in[0 ; 30]$ Sekunden.

See Solution

Problem 14599

Find the maximum volume of a box with an open top and square base using 25 square feet of material.

See Solution

Problem 14600

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x$ . Zeigen Sie die Symmetrie, Punkte und Extremstellen der Graphen und finden Sie die Wendetangente mit $m=8$ .

See Solution

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Calculus

Problem 14501

Find the limit: lim⁡x→3−2∣3−x∣=\lim _{x \rightarrow 3^{-}} \frac{2}{|3-x|}=x→3−lim​∣3−x∣2​=

Problem 14502

Evaluate the integral ∫−aa(x+a)(x−a)dx=−40\int_{-a}^{a}(x+a)(x-a) dx = -40∫−aa​(x+a)(x−a)dx=−40.

Problem 14503

For which values of −3≤x0≤2-3 \leq x_{0} \leq 2−3≤x0​≤2 does lim⁡x→x0g(x)\lim _{x \rightarrow x_{0}} g(x)limx→x0​​g(x) not exist?

Problem 14504

Tomatenpflanze: Höhe h(t)h(t)h(t) eines Setzlings mit v(t)=−0,1t3+t2v(t)=-0,1 t^{3}+t^{2}v(t)=−0,1t3+t2. Fragen: a) Wachstumsdauer? b) Maximale Höhe? c) Höhe beim schnellsten Wachstum?

Problem 14505

Bestimmen Sie das Minimum und Maximum der Funktion fa(x)=13ax3−x2f_{a}(x)=\frac{1}{3 a} x^{3}-x^{2}fa​(x)=3a1​x3−x2 mit f′a(x)=1ax2−2xf^{\prime} a(x)=1 a x^{2}-2 xf′a(x)=1ax2−2x.

Problem 14506

Berechne die Fläche zwischen den Funktionen f(x)=14x2f(x)=\frac{1}{4} x^{2}f(x)=41​x2 und g(x)=(x−1)2g(x)=(x-1)^{2}g(x)=(x−1)2, indem du ihre Schnittpunkte findest.

Problem 14507

Problem 14508

Ein Segelflugzeug startet bei 80 m Höhe. Bestimme die Höhenfunktion und beantworte Fragen zu max. Höhe, Änderungszeit und Landeanflug.

Problem 14509

Find the values of xxx where the curve y=x3−3x+2y=x^{3}-3x+2y=x3−3x+2 is concave up.

Problem 14510

Find the inflection point and extremum of f(x)=(x−4)⋅exf(x)=(x-4) \cdot e^{x}f(x)=(x−4)⋅ex for x∈Rx \in \mathbb{R}x∈R.

Problem 14511

Find the limit: lim⁡x→6xx2−36\lim _{x \rightarrow 6} \frac{x}{x^{2}-36}limx→6​x2−36x​.

Problem 14512

Find the limit as xxx approaches -1 for the expression x2+7x+6x2−7x−8\frac{x^{2}+7 x+6}{x^{2}-7 x-8}x2−7x−8x2+7x+6​.

Problem 14513

Bestimme die Wendepunkte der Funktion ft(x)=x3−3tx2f_{t}(x)=x^{3}-3 t x^{2}ft​(x)=x3−3tx2 für t>0t>0t>0.

Problem 14514

Find the stationary point on the curve given by the equation y=x2−4x+3y=x^{2}-4 x+3y=x2−4x+3.

Problem 14515

Bestimmen Sie die Ableitung der Funktion f(x)=112x3−12x2f(x)=\frac{1}{12} x^{3}-\frac{1}{2} x^{2}f(x)=121​x3−21​x2.

Problem 14516

Untersuchen Sie das Verhalten der Funktion f(x)=2x+1xf(x)=\frac{2x+1}{x}f(x)=x2x+1​ für x<0x<0x<0, wenn x→−∞x \rightarrow -\inftyx→−∞. Skizzieren Sie den Graphen.

Problem 14517

Untersuchen Sie das Verhalten der Funktionen für die Grenzprozesse. Skizzieren Sie die Graphen. a) f(x)=2x+1x,x<0f(x)=\frac{2 x+1}{x}, x<0f(x)=x2x+1​,x<0, x→−∞x \rightarrow-\inftyx→−∞ b) f(x)=x+1x2,x>0f(x)=\frac{x+1}{x^{2}}, x>0f(x)=x2x+1​,x>0, x→∞x \rightarrow \inftyx→∞

Problem 14518

Vulkanprofil: f(x)=25x2f(x)=\frac{25}{x^{2}}f(x)=x225​ für 2≤x≤62 \leq x \leq 62≤x≤6. a) Höhe bei 400 m400 \mathrm{~m}400 m Breite? b) Höhe bei 1 km1 \mathrm{~km}1 km Durchmesser?

Problem 14519

Problem 14520

Problem 14521

Evaluate the double integral ∫−22∫−112(−12sinh⁡(u+v))2+1 du dv\int_{-2}^{2} \int_{-1}^{1} \sqrt{2\left(\frac{-1}{\sqrt{2}} \sinh (u+v)\right)^{2}+1} \, du \, dv∫−22​∫−11​2(2​−1​sinh(u+v))2+1​dudv.

Problem 14522

Find the derivative dxdt\frac{d x}{d t}dtdx​ for the function x=t2+25tx=\frac{t^{2}+2}{5 t}x=5tt2+2​.

Problem 14523

Evaluate the double integral: ∫−22∫−11sinh⁡2(u+v)+1 du dv\int_{-2}^{2} \int_{-1}^{1} \sqrt{\sinh ^{2}(u+v)+1} \, du \, dv∫−22​∫−11​sinh2(u+v)+1​dudv.

Problem 14524

Berechnen Sie die Ableitung von fff an x0x_{0}x0​ für: a) f(x)=x2f(x)=x^{2}f(x)=x2, x0=2x_{0}=2x0​=2; b) f(x)=2x2f(x)=2x^{2}f(x)=2x2, x0=1x_{0}=1x0​=1.

Problem 14525

A car travels straight with velocity v(t)v(t)v(t) as a rhombus. Find ∫024v(t)dt\int_{0}^{24} v(t) dt∫024​v(t)dt and explain its meaning.

Problem 14526

Approximate ∫06f(t)dt\int_{0}^{6} f(t) dt∫06​f(t)dt using right-hand Riemann sum with intervals [0,2],[2,3],[3,6][0,2],[2,3],[3,6][0,2],[2,3],[3,6] and values from the table. Choices: A. 204 B. 225 C. 246 D. 252 E. 281

Problem 14527

Evaluate the integral ∫−26f(x)dx\int_{-2}^{6} f(x) d x∫−26​f(x)dx for the piecewise function: f(x)={2x−6,x<48,x≥4f(x)=\begin{cases} 2x-6, & x<4 \\ 8, & x \geq 4 \end{cases}f(x)={2x−6,8,​x<4x≥4​.

Problem 14528

Calculate the derivative dydx∣x=1\left.\frac{\mathbb{d} y}{\mathbb{d} x}\right|_{x=1}dxdy​∣∣​x=1​ for y=4+5x+9x2+11x3+10x4+5x5+4x6x3y=\frac{4+5 x+9 x^{2}+11 x^{3}+10 x^{4}+5 x^{5}+4 x^{6}}{x^{3}}y=x34+5x+9x2+11x3+10x4+5x5+4x6​.

Problem 14529

Approximate ∫17x2+1xdx\int_{1}^{7} \frac{x^{2}+1}{x} dx∫17​xx2+1​dx using a Trapezoidal Sum with 3 subintervals. Is it an under or over estimation?

Problem 14530

Calculate the integral: ∫352x2+x+4x−1 dx\int_{3}^{5} \frac{2 x^{2}+x+4}{x-1} \, dx∫35​x−12x2+x+4​dx

Problem 14531

Determine the horizontal asymptotes of the function t(x)=(x+2)(x−1)(x−4)5(x+3)(x−1)(x−3)t(x)=\frac{(x+2)(x-1)(x-4)}{5(x+3)(x-1)(x-3)}t(x)=5(x+3)(x−1)(x−3)(x+2)(x−1)(x−4)​.

Problem 14532

Calculate the integral: ∫011x2+6x+10dx\int_{0}^{1} \frac{1}{x^{2}+6 x+10} d x∫01​x2+6x+101​dx.

Problem 14533

Find the antiderivative for these integrals: a. ∫cos⁡(θ)1+sin⁡(θ)dΘ\int \frac{\cos (\theta)}{\sqrt{1+\sin (\theta)}} d \Theta∫1+sin(θ)​cos(θ)​dΘ b. ∫(x3−3x)dx\int (x^{3}-3 x) d x∫(x3−3x)dx c. ∫xx2+1dx\int \frac{x}{x^{2}+1} d x∫x2+1x​dx

Problem 14534

Evaluate the integral: ∫11sin⁡(x)e−3x−6dx\int_{1}^{1} \sin (x) e^{-3 x-6} d x∫11​sin(x)e−3x−6dx

Problem 14535

Find where the graph of f(x)=18x−23f(x)=\frac{18}{\sqrt[3]{x-2}}f(x)=3x−2​18​ is concave down. Choose A, B, C, or D.

Problem 14536

Bestimme die Ableitung der Funktionen: a) f(x)=2xf(x)=\frac{2}{x}f(x)=x2​, b) f(x)=3xf(x)=\frac{3}{x}f(x)=x3​, c) f(x)=x2+1f(x)=x^{2}+1f(x)=x2+1, d) f(x)=2x2−1f(x)=2 x^{2}-1f(x)=2x2−1.

Problem 14537

Find intervals where the graph of f(x)=−81x+53f(x)=-\frac{81}{\sqrt[3]{x+5}}f(x)=−3x+5​81​ is concave up. Choose one: (A) x>53x>\sqrt[3]{5}x>35​ (B) x<−53x<-\sqrt[3]{5}x<−35​ (C) x<−5x<-5x<−5 (D) x>−5x>-5x>−5

Problem 14538

Berechnen Sie den Flächeninhalt zwischen den Graphen f(x)=e0.5xf(x)=e^{0.5 x}f(x)=e0.5x und g(x)=2e−e0.5xg(x)=2 e-e^{0.5 x}g(x)=2e−e0.5x im Bereich D=RD=\mathbb{R}D=R.

Problem 14539

Find the minimum slope of the curve given by the equation y=x5−10x2y=x^{5}-10 x^{2}y=x5−10x2.

Problem 14540

Find the maximum slope of the curve y=4x3−24x2+32xy=4 x^{3}-24 x^{2}+32 xy=4x3−24x2+32x in the interval [0,5][0,5][0,5].

Problem 14541

What is the shot put's acceleration while falling into the pit, considering gravity?

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

Evaluate the integral $\int_{-a}^{a}(x+a)(x-a) dx = -40$ .

For which values of $-3 \leq x_{0} \leq 2$ does $\lim _{x \rightarrow x_{0}} g(x)$ not exist?

Tomatenpflanze: Höhe $h(t)$ eines Setzlings mit $v(t)=-0,1 t^{3}+t^{2}$ . Fragen: a) Wachstumsdauer? b) Maximale Höhe? c) Höhe beim schnellsten Wachstum?

Bestimmen Sie das Minimum und Maximum der Funktion $f_{a}(x)=\frac{1}{3 a} x^{3}-x^{2}$ mit $f^{\prime} a(x)=1 a x^{2}-2 x$ .

Berechne die Fläche zwischen den Funktionen $f(x)=\frac{1}{4} x^{2}$ und $g(x)=(x-1)^{2}$ , indem du ihre Schnittpunkte findest.

Find the values of $x$ where the curve $y=x^{3}-3x+2$ is concave up.

Find the inflection point and extremum of $f(x)=(x-4) \cdot e^{x}$ for $x \in \mathbb{R}$ .

Find the limit: $\lim _{x \rightarrow 6} \frac{x}{x^{2}-36}$ .

Find the limit as $x$ approaches -1 for the expression $\frac{x^{2}+7 x+6}{x^{2}-7 x-8}$ .

Bestimme die Wendepunkte der Funktion $f_{t}(x)=x^{3}-3 t x^{2}$ für $t>0$ .

Find the stationary point on the curve given by the equation $y=x^{2}-4 x+3$ .

Bestimmen Sie die Ableitung der Funktion $f(x)=\frac{1}{12} x^{3}-\frac{1}{2} x^{2}$ .

Untersuchen Sie das Verhalten der Funktion $f(x)=\frac{2x+1}{x}$ für $x<0$ , wenn $x \rightarrow -\infty$ . Skizzieren Sie den Graphen.

Untersuchen Sie das Verhalten der Funktionen für die Grenzprozesse. Skizzieren Sie die Graphen. a) $f(x)=\frac{2 x+1}{x}, x<0$ , $x \rightarrow-\infty$ b) $f(x)=\frac{x+1}{x^{2}}, x>0$ , $x \rightarrow \infty$

Vulkanprofil: $f(x)=\frac{25}{x^{2}}$ für $2 \leq x \leq 6$ . a) Höhe bei $400 \mathrm{~m}$ Breite? b) Höhe bei $1 \mathrm{~km}$ Durchmesser?

Evaluate the double integral $\int_{-2}^{2} \int_{-1}^{1} \sqrt{2\left(\frac{-1}{\sqrt{2}} \sinh (u+v)\right)^{2}+1} \, du \, dv$ .

Find the derivative $\frac{d x}{d t}$ for the function $x=\frac{t^{2}+2}{5 t}$ .

Evaluate the double integral: $\int_{-2}^{2} \int_{-1}^{1} \sqrt{\sinh ^{2}(u+v)+1} \, du \, dv$ .

Berechnen Sie die Ableitung von $f$ an $x_{0}$ für: a) $f(x)=x^{2}$ , $x_{0}=2$ ; b) $f(x)=2x^{2}$ , $x_{0}=1$ .

A car travels straight with velocity $v(t)$ as a rhombus. Find $\int_{0}^{24} v(t) dt$ and explain its meaning.

Approximate $\int_{0}^{6} f(t) dt$ using right-hand Riemann sum with intervals $[0,2],[2,3],[3,6]$ and values from the table. Choices: A. 204 B. 225 C. 246 D. 252 E. 281

Evaluate the integral $\int_{-2}^{6} f(x) d x$ for the piecewise function: $f(x)=\begin{cases} 2x-6, & x<4 \\ 8, & x \geq 4 \end{cases}$ .

Calculate the derivative $\left.\frac{\mathbb{d} y}{\mathbb{d} x}\right|_{x=1}$ for $y=\frac{4+5 x+9 x^{2}+11 x^{3}+10 x^{4}+5 x^{5}+4 x^{6}}{x^{3}}$ .

Approximate $\int_{1}^{7} \frac{x^{2}+1}{x} dx$ using a Trapezoidal Sum with 3 subintervals. Is it an under or over estimation?

Calculate the integral: $\int_{3}^{5} \frac{2 x^{2}+x+4}{x-1} \, dx$

Determine the horizontal asymptotes of the function $t(x)=\frac{(x+2)(x-1)(x-4)}{5(x+3)(x-1)(x-3)}$ .

Calculate the integral: $\int_{0}^{1} \frac{1}{x^{2}+6 x+10} d x$ .

Find the antiderivative for these integrals: a. $\int \frac{\cos (\theta)}{\sqrt{1+\sin (\theta)}} d \Theta$ b. $\int (x^{3}-3 x) d x$ c. $\int \frac{x}{x^{2}+1} d x$

Evaluate the integral: $\int_{1}^{1} \sin (x) e^{-3 x-6} d x$

Find where the graph of $f(x)=\frac{18}{\sqrt[3]{x-2}}$ is concave down. Choose A, B, C, or D.

Bestimme die Ableitung der Funktionen: a) $f(x)=\frac{2}{x}$ , b) $f(x)=\frac{3}{x}$ , c) $f(x)=x^{2}+1$ , d) $f(x)=2 x^{2}-1$ .

Find intervals where the graph of $f(x)=-\frac{81}{\sqrt[3]{x+5}}$ is concave up. Choose one: (A) $x>\sqrt[3]{5}$ (B) $x<-\sqrt[3]{5}$ (C) $x<-5$ (D) $x>-5$

Berechnen Sie den Flächeninhalt zwischen den Graphen $f(x)=e^{0.5 x}$ und $g(x)=2 e-e^{0.5 x}$ im Bereich $D=\mathbb{R}$ .

Find the minimum slope of the curve given by the equation $y=x^{5}-10 x^{2}$ .

Find the maximum slope of the curve $y=4 x^{3}-24 x^{2}+32 x$ in the interval $[0,5]$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{2}+h\right)-\cos \left(\frac{\pi}{2}\right)}{h}$ . Options: a. 1 b. 0 c. -1 d. does not exist

Find the values of $x$ where the function $f(x)=-\frac{1}{2} x^{4}-6 x^{3}-27 x^{2}$ has points of inflection. Options: A) $x=3$ , B) $x=-3$ , C) $x=\frac{3}{2}$ , D) none.

Calculate the area between the curves $y=16x$ , $y=x^5$ , and the lines $x=0$ , $x=2$ . The area is $\square$ .

Estimate $r^{\prime}(5)$ using values from the table: $r(0)=8$ , $r(3)=10$ , $r(4)=14$ , $r(6)=22$ . Options: a. 8, b. 18, c. 4, d. 16.

What happens at the point $(3,8)$ for the function $f$ given $f(3)=8$ , $f^{\prime}(3)=0$ , and $f^{\prime \prime}(3)=0$ ? Choose: (A) minimum point (B) maximum point (C) not enough information

Find the limit: $\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{2}+h\right)-\cos \left(\frac{\pi}{2}\right)}{h}$ . Choices: (a) 1, (b) 0, (c) -1, (d) does not exist.

Find the area between $y=4x$ , $y=x^3$ , $x=0$ , and $x=2$ . The area is $\square$ .

Find the integral of the function: $\int\left(\sqrt[4]{x^{3}}+1\right) d x$ .

Calculate the area between the curves $f(x)=x^{4}-8 x^{3}+21 x^{2}$ and $g(x)=20 x+50$ where they intersect.

Find the time $t$ when the particle at position $x(t)=2 t^{3}-21 t^{2}+72 t-53$ is at rest.

Find the average velocity of a particle with position $-5 t^{2}$ from $t=0$ to $t=3$ . Choices: (A) -45 (B) -30 (C) -15 (D) -10 (E) -5

Estimate $f(4.8)$ using local linear approximation at $x=5$ given $f(5)=3$ and $f'(5)=4$ . Options: (A) 2.2 (B) 2.8 (C) 3.4 (D) 3.8 (E) 4.6

Calculate left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum is $\square$ .

Calculate the left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum: $\square$ .

Approximate the displacement of an object with $v=4t+3$ (m/s) from $t=0$ to $t=8$ using $n=2$ subintervals. Answer: $\square$ m.

Find the time $t$ when the particle's velocity $v(t)=t^{3}-6 t^{2}+10 t-4$ changes from right to left for $t \geq 0$ .

Calculate left and right Riemann sums for $f(x)=x+2$ on $[0,5]$ with $n=5$ . Left sum is 20; find right sum.

Approximate the displacement for $v=\frac{1}{4t+1}$ from $t=0$ to $t=8$ using 4 subintervals. Displacement: $\square \mathrm{m}$ .

Calculate the left and right Riemann sums for $f(x)=\frac{1}{x}+1$ on $[1,5]$ with $n=4$ . Left sum: $\square$ .

Determine the end behavior of the polynomial $g(x)=3 x^{4}+2 x^{2}+x-1$ using limits.

Find the tangent line equation to the curve at $t=0$ for $x=\int_{2t}^{at} \sqrt{1+u^{3}} du$ and $y=2+at^{2}+(t+1) \cos^{3}(at)$ , where $a>2$ .

Find points on the curve where the tangent is horizontal or vertical for $x=\log_{a}(\sec(at))$ , $y=\log_{a}(\sin(at))$ , $-\frac{\pi}{2a} \leq t \leq \frac{\pi}{2a}$ , $a>1$ .

Find the instantaneous rate of change of profit at quantity $q=10$ for the profit function $P(q)=300 q-10 q^{2}-200$ .

Find the derivative of $y=6 x^{3}+15 x^{2}$ and evaluate it at $x=3$ .

Calculate the integral from 3 to 4 of the function $\frac{t^{5}-2 t}{t^{3}}$ .

Evaluate the integral $\int_{1}^{\frac{1}{2}} \sqrt{\frac{\pi}{x^{2}}} \mathrm{~d} x$ .

Determine where the function $f(x)=-x^{4}-7 x^{3}-12 x^{2}$ is increasing or decreasing.

Calculate the integral of $f(x)=\left(e^{-\frac{1}{4} x}\right)^{2}$ .

A 400 g isotope decays as $A(t)=400 e^{-0.044 t}$ . Find: (a) amount left after 20 years, (b) time to half decay.