Calculus

Problem 18801

Calculate the average value $f_{\text{ave}}$ of the function $f(x)=\frac{x^{2}}{(x^{3}+10)^{2}}$ over the interval [-2, 2].

See Solution

Problem 18802

Use Euler's method with step size 0.25 to find $y_{1}, y_{2}, y_{3}, y_{4}$ for $y' = 2 - 2x - 2y$ , $y(0) = -2$ .

See Solution

Problem 18803

Find the initial temperature and the temperature after 18 minutes for $T(x)=-4+26 e^{-0.028 x}$ . Round to the nearest degree.

See Solution

Problem 18804

Given $\dot{y}=4 t e^{-y}$ with $y(0)=-1$ , use Euler's Method ( $h=0.2$ ) to find $y(0.2)$ to $y(1.0)$ , then find $y(t)$ exactly and compute errors for $y(0.2)$ , $y(0.6)$ , and $y(1)$ .

See Solution

Problem 18805

Evaluate the limit: $\lim _{N \rightarrow \infty} \sum_{s=1}^{N} \frac{3 s^{2}-2 s-5}{N^{3}}$ as a whole number.

See Solution

Problem 18806

Find $g'(x)$ if $g(x)=\int_{0}^{x} \sqrt{t+t^{3}} dt$ .

See Solution

Problem 18807

Differentiate the integral $\int_{1}^{x} e^{-t^{2}} dt$ with respect to $x$ .

See Solution

Problem 18808

A girl flies a kite at 300 ft high, moving horizontally at 25 ft/sec. How fast must she pay out string when the kite is 500 ft away? $20 \mathrm{ft/s}$

See Solution

Problem 18809

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

See Solution

Problem 18810

Find $g'(s)$ for the function defined by $g(s)=\int_{5}^{s}(t-t^{2})^{8} dt$ .

See Solution

Problem 18811

Find the total area between the curve $f(x)=x^{3}+x^{2}-2 x$ and the $x$ -axis, considering where it is above and below.

See Solution

Problem 18812

Trouvez les nombres critiques de $f(x) = \sqrt[3]{x^{2}}(2-x)$ où $f'(c)=0$ ou $f'(c)$ n'existe pas.

See Solution

Problem 18813

Water is in an inverted cone tank (8 ft diameter, 10 ft deep). Rate of water removal is $5 \mathrm{ft}^{3}/\mathrm{min}$ . Find the water level change rate at 6 ft depth and time to empty the tank.

See Solution

Problem 18814

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

See Solution

Problem 18815

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

See Solution

Problem 18816

Find the limit of $f(x)=\frac{3}{2} x+4$ on $[0,4]$ and verify using geometry. Provide a whole number answer.

See Solution

Problem 18817

A balloon is 200 ft up, rising at 15 ft/sec. A car below moves at 45 mph. Find the rate of distance change after 1 sec.

See Solution

Problem 18818

Find the area of the shaded region between $y=\sqrt[3]{x}$ , $y=\frac{1}{x}$ , and the line $x=27$ .

See Solution

Problem 18819

Which two series converge by the root test? Choose from:
1. $\sum_{n=1}^{\infty}\left(\frac{2 n}{2 n+1}\right)^{2 n}$
2. $\sum_{n=1}^{\infty}\left(\frac{n^{4} \ln ^{2}\left(\cos \frac{1}{n}\right)}{n+1}\right)^{n}$
3. $\sum_{n=1}^{\infty}\left(\frac{3 n}{3 n-1}\right)^{n^{2}}$
4. $\sum_{n=1}^{\infty}\left(\frac{2 \arctan n}{\pi}\right)^{n}$
5. $\sum_{n=1}^{\infty}\left(\frac{n}{\ln n}\right)^{n / 2}$
6. $\sum_{n=1}^{\infty}\left(\frac{n+1}{2 n-1}\right)^{n}$

See Solution

Problem 18820

Which two series diverge according to the ratio test? Choose from the following:
1. $\sum_{n=1}^{\infty} \frac{(n+1)^{2 n}}{(n+1) !}$
2. $\sum_{n=1}^{\infty} \frac{\left(n^{2}\right) !}{(2 n) !}$
3. $\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(2 n) !}$
4. $\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 5}+\cdots$
5. $\frac{3}{1}+\frac{3 \cdot 5}{1 \cdot 3}+\cdots$
6. $\sum_{n=1}^{\infty} \frac{n !}{2^{n !}}$

See Solution

Problem 18821

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

See Solution

Problem 18822

Find the limit of $f(x)=27-9x$ over $[2,6]$ and confirm with geometry. Provide a whole number answer. $\lim _{N \rightarrow \infty} M_{N}=$

See Solution

Problem 18823

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

See Solution

Problem 18824

Find the volume of the solid formed by rotating the area between $x=-3y^{2}+15y-12$ and $x=0$ about the $x$ -axis using cylindrical shells.

See Solution

Problem 18825

Find the volume of the solid formed by rotating the region $x^{2}+(y-2)^{2}=4$ around the $y$ -axis.

See Solution

Problem 18826

Rewrite the limit using l'Hôpital's Rule to resolve the indeterminate form:
$\lim _{x \rightarrow 1} \frac{\ln x}{11 x-x^{2}-10}$

See Solution

Problem 18827

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{3 x^{2}}$ using l'Hôpital's Rule when applicable.

See Solution

Problem 18828

Find how $\frac{dA}{dt}$ relates to $\frac{da}{dt}$ , $\frac{db}{dt}$ , and $\frac{d\theta}{dt}$ for area $A=(1/2)ab\sin\theta$ .

See Solution

Problem 18829

Evaluate the integral: $\int \frac{3 d t}{(t^{2}-9)^{2}}$ (include absolute values and use $C$ for the constant).

See Solution

Problem 18830

Rewrite the limit using l'Hôpital's Rule to eliminate the indeterminate form:
$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{2 x^{2}+2 x}=\lim _{x \rightarrow 0}(\square)$

See Solution

Problem 18831

Find the derivative of the function defined by $f(x)=\int_{x}^{15} t^{2} d t$ . What is $f^{\prime}(x)=?$

See Solution

Problem 18832

Find the distance traveled by an object with velocity $v(t)=14 t$ m/s over intervals $[0,2]$ and $[2,5]$ .

See Solution

Problem 18833

Compute $R_{6}, L_{6}, M_{3}$ to estimate distance over $[0,3]$ with given velocity at half-second intervals.

See Solution

Problem 18834

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

See Solution

Problem 18835

Find the displacement formula and total distance traveled by a particle with velocity $v(t)=8-1 t$ at $t=4$ seconds.

See Solution

Problem 18836

Calculate the integral $\int_{-98}^{-21}[7 f(x)+5 g(x)-h(x)] d x$ using given values for $f(x)$ , $g(x)$ , and $h(x)$ .

See Solution

Problem 18837

A dinghy is pulled to a dock with a rope $6 \mathrm{ft}$ above the bow, hauled in at $2 \mathrm{ft/sec}$ . Find: (a) Speed of the boat when $10 \mathrm{ft}$ of rope is out. (b) Rate of change of angle $\theta$ at that moment.

See Solution

Problem 18838

Find $f^{\prime}(-1)$ given that $f(x)=x^{10} h(x)$ , $h(-1)=2$ , and $h^{\prime}(-1)=5$ .

See Solution

Problem 18839

Find $R_{N}$ and calculate the area under $f(x)=7x^{2}+49x$ from $x=6$ to $x=11$ as a limit. $\lim_{N \rightarrow \infty} R_{N} =$

See Solution

Problem 18840

Solve using Laplace transforms: $y^{\prime \prime}+y=u(t-\pi)-u(t-3\pi)$ , with $y(0)=0$ , $y^{\prime}(0)=0$ .

See Solution

Problem 18841

Approximate the area $A$ under $f(x)=2x^2+12x$ from $x=1$ to $x=3$ using a left-endpoint sum: $A=\lim_{N \to \infty} L_N$ .

See Solution

Problem 18842

Inge flies a kite at $300 \mathrm{ft}$ high, moving away at $25 \mathrm{ft/sec}$ . How fast to let out string when $500 \mathrm{ft}$ away?

See Solution

Problem 18843

Approximate the area $A$ under $f(x)=2x^2+12x$ from $x=1$ to $x=3$ using left-endpoint sums. Find $A=\lim_{N \to \infty} L_N$ .

See Solution

Problem 18844

Approximate the area $A$ under $f(x) = 2x^2 + 12x$ over $[1,3]$ using a left-endpoint method. Find $A = \lim_{N \to \infty} L_N$ . Enter exact value.

See Solution

Problem 18845

Find the local maximum of $y=-x^{3}-11x^{2}-40x-54$ and choose a suitable graph window from the options provided.

See Solution

Problem 18846

Find the limit of $f(x)=27-9x$ over $[2,6]$ and verify geometrically. The limit is $\lim _{N \rightarrow \infty} M_{N}=7$ .

See Solution

Problem 18847

Find the limit of $f(x)=27-9x$ as $N$ approaches infinity over $[2,6]$ and verify using geometry.

See Solution

Problem 18848

Evaluate the integral of $\sqrt[4]{u}$ with respect to $u$ : $\int \sqrt[4]{u} \, du$ .

See Solution

Problem 18849

Calculate the integral: $\int \frac{d w}{\sqrt{2}}$ .

See Solution

Problem 18850

Calculate the integral of the function: $\int 4 x^{3} dx$

See Solution

Problem 18851

Find $\frac{dV}{dt}$ in terms of $R$ and $\frac{dR}{dt}$ for $V= \frac{400}{280} (R^2 - r^2)$ with $L=70$ , $p=400$ , $v=0.003$ .

See Solution

Problem 18852

A 26 ft ladder leans against a wall. If the base moves away at 5 ft/s, how fast is the top lowering when 10 ft from the wall?

See Solution

Problem 18853

Given constants $p=500$ , $L=80 \mathrm{~mm}$ , and $v=0.003$ , find $d V / d t$ in terms of $R$ and $d R / d t$ . If $\frac{d R}{d t}=-0.0002 \mathrm{~mm/sec}$ and $R=0.075 \mathrm{~mm}$ , compute $d V / d t$ .

See Solution

Problem 18854

A cylinder has a radius decreasing at 2 in/sec and height increasing at 5 in/sec. Find the volume change rate when $r=4$ in and $h=6$ in. Use $V=\pi r^{2} h$ .

See Solution

Problem 18855

Two cars start together; one goes north at $25 \mathrm{mph}$ , the other east at $60 \mathrm{mph}$ . Find the distance increase rate after $1 \mathrm{hr}$ . Use $D^{2}=x^{2}+y^{2}$ to solve.

See Solution

Problem 18856

Berechne für $v_{0}=34 \mathrm{~m/s}$ die mittlere Geschwindigkeit in den ersten 2 Sekunden und die Geschwindigkeit bei $t=0,1,2,3,4,5$ s. Was bedeutet negative Geschwindigkeit?

See Solution

Problem 18857

Ein Auto fährt gegen eine Mauer. Berechne die kinetische Energie $E(t)=5000 \cdot t^{2}$ in den Intervallen $[0; 5]$ und $[5; 10]$ . Vergleiche die Zunahmen und die mittlere Zunahme. Bestimme die Zunahmegeschwindigkeit $E^{\prime}(t)$ und den Zeitpunkt, an dem $E(t)$ am schnellsten wächst. Berechne die Zunahmegeschwindigkeit zu Beginn, nach 5 s und nach 10 s.

See Solution

Problem 18858

Bestimme den Flächeninhalt A, der zwischen den Schnittpunkten der Funktionen $f(x)=1 x-x^{2}$ und $g(x)=x^{2}-2 x+1$ liegt.

See Solution

Problem 18859

Show that the function $f(x)=\sqrt{x}$ is uniformly continuous on the interval $[a,+\infty)$ for $a>0$ .

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Problem 18860

A lamppost is 26 ft tall. Andy, 6 ft tall, walks away at 4 ft/sec. When 20 ft away, find the rate of change of his shadow's length.

See Solution

Problem 18861

Bestimme die Ableitung von $f_a(x)=a x^{4}-4 x^{3}+a^{2} x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

See Solution

Problem 18862

At noon, ship A is 150 km east of ship B. A sails west at 35 km/h, B sails north at 25 km/h. Find the distance change rate at 4 p.m.

See Solution

Problem 18863

Find the derivative of $f_a(x)=\frac{1}{a} x^{2}+\frac{2}{a}+a$ at $x=0$ and the value of $a$ for slope = 1.

See Solution

Problem 18864

Find the formula for the area $A(y)$ of a square cross-section of a pyramid 12 inches tall with a 10-inch base. Show $A(12)=100$ and evaluate $\int_{0}^{12} A(y) dy$ . Explain the result.

See Solution

Problem 18865

A blue car heads north from 15 miles south, and a red car heads east from 20 miles west. Find rates of separation and closest distance.

See Solution

Problem 18866

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 18867

Calculate the area under the normal curve with mean $\mu = 100$ and standard deviation $15$ from scores 112 to 114.

See Solution

Problem 18868

Find the slope of the curve $7 y^{6}+8 x^{5}=5 y+10 x$ at the point (1,1).

See Solution

Problem 18869

Ein Ball wird mit 30 m/s in 35 m Höhe geworfen. Bestimme die Höhenfunktion $h(t)$ und die Aufprallgeschwindigkeit.

See Solution

Problem 18870

Find the slope of the curve $7 y^{9}+8 x^{3}=2 y+13 x$ at the point (1,1).

See Solution

Problem 18871

Find the derivative of $y=(2t-1)(6t-5)^{-1}$ .

See Solution

Problem 18872

Find the derivative of the function $y=3 e^{-x}+e^{4 x}$ .

See Solution

Problem 18873

Finde die Stammfunktion von $f(x)=2 \cdot x$ für die Bedingungen: a) Ursprung, b) Punkt P(1|4), c) Schnitt bei 3, d) $W=\{y \in \mathbb{R}, y \geq 4\}$ , e) Berührung bei Q(1|2).

See Solution

Problem 18874

Bestimmen Sie $a$ und $b$ für die Funktion $f(x)=\frac{1}{2} x^{4}-2 x^{2}+4$ . Untersuchen Sie Symmetrie, Nullstellen, Extrema, Wendepunkte und zeichnen Sie den Graphen für $-2 \leq x \leq 2$ .

See Solution

Problem 18875

Find the derivative of $f(x) = x^{2} + x \sqrt{x} + 3x - 2$ and its domain.

See Solution

Problem 18876

Find the derivative of $f(x) = x^{4}-3 x^{3}+4 x^{2}-1$ and state its domain.

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Problem 18877

Untersuche die Funktion $f(x)=-\frac{1}{3} x^{3}+3 x$ : a) Ist sie punktsymmetrisch? b) Finde Nullstellen, Hoch-, Tief- und Wendepunkt. c) Berechne die Steigung im Wendepunkt.

See Solution

Problem 18878

Determine where the function $f$ is concave down based on its rate of change: (A) $0<x<1$ (B) $1<x<2$ (C) $2<x<3$ (D) $3<x<4$

See Solution

Problem 18879

Let $f_n(x) = \frac{x}{1 + n x^2}$ . Show: 1) $f_n \to f$ pointwise, 2) $f_n$ converges uniformly to $f$ , 3) $f'(x) = \lim_{n \to +\infty} f_n'(x)$ iff $x \in \mathbb{R}^*$ .

See Solution

Problem 18880

Find the limit: $\lim _{n \rightarrow \infty} 3 n - \sqrt{9 n^{2}+1}$ .

See Solution

Problem 18881

Find the average rate of change of $f(x)=10(2)^{x}$ from $x=3$ to $x=5$ . Enter your answer in the box.

See Solution

Problem 18882

Compute the integral: $\int \frac{1}{(1+9 x^{2}) \sqrt{8-2(\tan^{-1}(3 x))^{2}}} dx$

See Solution

Problem 18883

Find the extreme points of $f_{a}(x)=x^{3}-3 a^{2} x+2$ as a function of the parameter $a$ .

See Solution

Problem 18884

Find the area to the right of $x=1$ for the given curve. Choose all that apply: $P(x<1)$ , $1-P(x>1)$ , $1+P(x<1)$ , $1-P(x<1)$ , $P(x>1)$ .

See Solution

Problem 18885

Find the average rate of change of $f(x)=4-2 x^{2}$ over the interval $[-1,3]$ .

See Solution

Problem 18886

Calculate the average rate of change of $g(x)=2x-1$ on the interval $[2, 2+b]$ .

See Solution

Problem 18887

Find the average rate of change of $p(t)=\frac{(t^{2}-2)(t+2)}{t^{2}+4}$ on the interval $[-3,-1]$ .

See Solution

Problem 18888

Consider the functions $f_{n}(x)=\frac{x}{1+n x^{2}}$ . Prove: 1. $f_{n} \rightarrow f$ pointwise. 2. $f_{n}$ converges uniformly. 3. $f^{\prime}(x)=\lim_{n \rightarrow+\infty} f_{n}^{\prime}(x)$ iff $x \in \mathbb{R}^{*}$ .

See Solution

Problem 18889

Find the derivative $f^{\prime}(x)$ for $f(x)=-3 e^{2}-5 x^{-2}-\ln (\tan x)$ .

See Solution

Problem 18890

Calculate the average rate of change of $g(x)=2x^{2}-4$ over the interval $[x, x+h]$ .

See Solution

Problem 18891

Find the tangent line equation for the curve $f(x)=-3 x^{2}+1$ at the point where $x=-1$ .

See Solution

Problem 18892

Find the average rate of change of $f(x)=\frac{1}{x+2}$ on the interval $[0,0+h]$ for real numbers $h$ .

See Solution

Problem 18893

Find the derivative $d y$ for the function $y=\cot^{-1}\left(\frac{1}{x^{2}}\right)+\sin^{-1}(5x)$ .

See Solution

Problem 18894

Find dy for the function $y=\cot^{-1}\left(\frac{1}{x^2}\right)+\sin^{-1}(5x)$ .

See Solution

Problem 18895

Find the acceleration of the object at $t=5$ seconds, given $s(t)=4 t^{3}-2 t+1$ .

See Solution

Problem 18896

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

See Solution

Problem 18897

Approximate the area under $f(x)=x^{2}-2 x+3$ from $x=1$ to $x=3$ using Right Hand Riemann Rule with $n=4$ .

See Solution

Problem 18898

Find the derivative of $f(x) \cot x$ . Which option is correct?
1) $f^{\prime}(x) \cot x + f(x) \csc^2 x$ 2) $-f^{\prime}(x) \csc^2 x$ 3) $f^{\prime}(x) \tan x - f(x) \csc^2 x$ 4) $f^{\prime}(x) \cot x - f(x) \csc^2 x$

See Solution

Problem 18899

Find the derivative of $f(x) \cot x$ . Choose the correct option from:
1. $f^{\prime}(x) \cot x + f(x) \csc^2 x$
2. $-f^{\prime}(x) \csc^2 x$
3. $f^{\prime}(x) \tan x - f(x) \csc^2 x$
4. $f^{\prime}(x) \cot x - f(x) \csc^2 x$

See Solution

Problem 18900

Find $f^{\prime}\left(\frac{\pi}{3}\right)$ for $f(x)=\frac{\cot x}{\sin x}$ .

See Solution

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Calculus

Problem 18801

Calculate the average value favef_{\text{ave}}fave​ of the function f(x)=x2(x3+10)2f(x)=\frac{x^{2}}{(x^{3}+10)^{2}}f(x)=(x3+10)2x2​ over the interval [-2, 2].

Problem 18802

Use Euler's method with step size 0.25 to find y1,y2,y3,y4y_{1}, y_{2}, y_{3}, y_{4}y1​,y2​,y3​,y4​ for y′=2−2x−2yy' = 2 - 2x - 2yy′=2−2x−2y, y(0)=−2y(0) = -2y(0)=−2.

Problem 18803

Find the initial temperature and the temperature after 18 minutes for T(x)=−4+26e−0.028xT(x)=-4+26 e^{-0.028 x}T(x)=−4+26e−0.028x. Round to the nearest degree.

Problem 18804

Given y˙=4te−y\dot{y}=4 t e^{-y}y˙​=4te−y with y(0)=−1y(0)=-1y(0)=−1, use Euler's Method (h=0.2h=0.2h=0.2) to find y(0.2)y(0.2)y(0.2) to y(1.0)y(1.0)y(1.0), then find y(t)y(t)y(t) exactly and compute errors for y(0.2)y(0.2)y(0.2), y(0.6)y(0.6)y(0.6), and y(1)y(1)y(1).

Problem 18805

Evaluate the limit: lim⁡N→∞∑s=1N3s2−2s−5N3\lim _{N \rightarrow \infty} \sum_{s=1}^{N} \frac{3 s^{2}-2 s-5}{N^{3}}N→∞lim​s=1∑N​N33s2−2s−5​ as a whole number.

Problem 18806

Find g′(x)g'(x)g′(x) if g(x)=∫0xt+t3dtg(x)=\int_{0}^{x} \sqrt{t+t^{3}} dtg(x)=∫0x​t+t3​dt.

Problem 18807

Differentiate the integral ∫1xe−t2dt\int_{1}^{x} e^{-t^{2}} dt∫1x​e−t2dt with respect to xxx.

Problem 18808

A girl flies a kite at 300 ft high, moving horizontally at 25 ft/sec. How fast must she pay out string when the kite is 500 ft away? 20ft/s20 \mathrm{ft/s}20ft/s

Problem 18809

Evaluate the integral ∫−111x dx\int_{-1}^{1} \frac{1}{x} \, dx∫−11​x1​dx.

Problem 18810

Find g′(s)g'(s)g′(s) for the function defined by g(s)=∫5s(t−t2)8dtg(s)=\int_{5}^{s}(t-t^{2})^{8} dtg(s)=∫5s​(t−t2)8dt.

Problem 18811

Find the total area between the curve f(x)=x3+x2−2xf(x)=x^{3}+x^{2}-2 xf(x)=x3+x2−2x and the xxx-axis, considering where it is above and below.

Problem 18812

Trouvez les nombres critiques de f(x)=x23(2−x)f(x) = \sqrt[3]{x^{2}}(2-x)f(x)=3x2​(2−x) où f′(c)=0f'(c)=0f′(c)=0 ou f′(c)f'(c)f′(c) n'existe pas.

Problem 18813

Water is in an inverted cone tank (8 ft diameter, 10 ft deep). Rate of water removal is 5ft3/min5 \mathrm{ft}^{3}/\mathrm{min}5ft3/min. Find the water level change rate at 6 ft depth and time to empty the tank.

Problem 18814

Find the limit: lim⁡x→∞ex−e−x\lim _{x \rightarrow \infty} e^{x}-e^{-x}limx→∞​ex−e−x.

Problem 18815

Find the limit: lim⁡x→∞xex−e−x\lim _{x \rightarrow \infty} \frac{x}{e^{x}-e^{-x}}limx→∞​ex−e−xx​.

Problem 18816

Find the limit of f(x)=32x+4f(x)=\frac{3}{2} x+4f(x)=23​x+4 on [0,4][0,4][0,4] and verify using geometry. Provide a whole number answer.

Problem 18817

A balloon is 200 ft up, rising at 15 ft/sec. A car below moves at 45 mph. Find the rate of distance change after 1 sec.

Problem 18818

Find the area of the shaded region between y=x3y=\sqrt[3]{x}y=3x​, y=1xy=\frac{1}{x}y=x1​, and the line x=27x=27x=27.

Problem 18819

Problem 18820

Problem 18821

Find the volume VVV of the solid formed by rotating the area between y=x−1y=\sqrt{x-1}y=x−1​, y=0y=0y=0, and x=6x=6x=6 around the xxx-axis.

Problem 18822

Find the limit of f(x)=27−9xf(x)=27-9xf(x)=27−9x over [2,6][2,6][2,6] and confirm with geometry. Provide a whole number answer. lim⁡N→∞MN=\lim _{N \rightarrow \infty} M_{N}=N→∞lim​MN​=

Problem 18823

Find the volume VVV of the solid formed by rotating the area between y=x−1y=\sqrt{x-1}y=x−1​, y=0y=0y=0, x=6x=6x=6 around the xxx-axis. V= V= V= Sketch the region.

Problem 18824

Find the volume of the solid formed by rotating the area between x=−3y2+15y−12x=-3y^{2}+15y-12x=−3y2+15y−12 and x=0x=0x=0 about the xxx-axis using cylindrical shells.

Problem 18825

Find the volume of the solid formed by rotating the region x2+(y−2)2=4x^{2}+(y-2)^{2}=4x2+(y−2)2=4 around the yyy-axis.

Problem 18826

Rewrite the limit using l'Hôpital's Rule to resolve the indeterminate form: lim⁡x→1ln⁡x11x−x2−10 \lim _{x \rightarrow 1} \frac{\ln x}{11 x-x^{2}-10} x→1lim​11x−x2−10lnx​

Problem 18827

Evaluate the limit: lim⁡x→01−cos⁡3x3x2\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{3 x^{2}}limx→0​3x21−cos3x​ using l'Hôpital's Rule when applicable.

Problem 18828

Find how dAdt\frac{dA}{dt}dtdA​ relates to dadt\frac{da}{dt}dtda​, dbdt\frac{db}{dt}dtdb​, and dθdt\frac{d\theta}{dt}dtdθ​ for area A=(1/2)absin⁡θA=(1/2)ab\sin\thetaA=(1/2)absinθ.

Problem 18829

Evaluate the integral: ∫3dt(t2−9)2\int \frac{3 d t}{(t^{2}-9)^{2}}∫(t2−9)23dt​ (include absolute values and use CCC for the constant).

Problem 18830

Rewrite the limit using l'Hôpital's Rule to eliminate the indeterminate form: lim⁡x→0e4x−12x2+2x=lim⁡x→0(□) \lim _{x \rightarrow 0} \frac{e^{4 x}-1}{2 x^{2}+2 x}=\lim _{x \rightarrow 0}(\square) x→0lim​2x2+2xe4x−1​=x→0lim​(□)

Problem 18831

Find the derivative of the function defined by f(x)=∫x15t2dtf(x)=\int_{x}^{15} t^{2} d tf(x)=∫x15​t2dt. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 18832

Find the distance traveled by an object with velocity v(t)=14tv(t)=14 tv(t)=14t m/s over intervals [0,2][0,2][0,2] and [2,5][2,5][2,5].

Problem 18833

Compute R6,L6,M3R_{6}, L_{6}, M_{3}R6​,L6​,M3​ to estimate distance over [0,3][0,3][0,3] with given velocity at half-second intervals.

Problem 18834

Problem 18835

Find the displacement formula and total distance traveled by a particle with velocity v(t)=8−1tv(t)=8-1 tv(t)=8−1t at t=4t=4t=4 seconds.

Problem 18836

Calculate the integral ∫−98−21[7f(x)+5g(x)−h(x)]dx\int_{-98}^{-21}[7 f(x)+5 g(x)-h(x)] d x∫−98−21​[7f(x)+5g(x)−h(x)]dx using given values for f(x)f(x)f(x), g(x)g(x)g(x), and h(x)h(x)h(x).

Problem 18837

A dinghy is pulled to a dock with a rope 6ft6 \mathrm{ft}6ft above the bow, hauled in at 2ft/sec2 \mathrm{ft/sec}2ft/sec. Find: (a) Speed of the boat when 10ft10 \mathrm{ft}10ft of rope is out. (b) Rate of change of angle θ\thetaθ at that moment.

Problem 18838

Find f′(−1)f^{\prime}(-1)f′(−1) given that f(x)=x10h(x)f(x)=x^{10} h(x)f(x)=x10h(x), h(−1)=2h(-1)=2h(−1)=2, and h′(−1)=5h^{\prime}(-1)=5h′(−1)=5.

Problem 18839

Find RNR_{N}RN​ and calculate the area under f(x)=7x2+49xf(x)=7x^{2}+49xf(x)=7x2+49x from x=6x=6x=6 to x=11x=11x=11 as a limit. lim⁡N→∞RN=\lim_{N \rightarrow \infty} R_{N} =N→∞lim​RN​=

Problem 18840

Solve using Laplace transforms: y′′+y=u(t−π)−u(t−3π)y^{\prime \prime}+y=u(t-\pi)-u(t-3\pi)y′′+y=u(t−π)−u(t−3π), with y(0)=0y(0)=0y(0)=0, y′(0)=0y^{\prime}(0)=0y′(0)=0.

Problem 18841

Approximate the area AAA under f(x)=2x2+12xf(x)=2x^2+12xf(x)=2x2+12x from x=1x=1x=1 to x=3x=3x=3 using a left-endpoint sum: A=lim⁡N→∞LNA=\lim_{N \to \infty} L_NA=limN→∞​LN​.

Calculate the average value $f_{\text{ave}}$ of the function $f(x)=\frac{x^{2}}{(x^{3}+10)^{2}}$ over the interval [-2, 2].

Use Euler's method with step size 0.25 to find $y_{1}, y_{2}, y_{3}, y_{4}$ for $y' = 2 - 2x - 2y$ , $y(0) = -2$ .

Find the initial temperature and the temperature after 18 minutes for $T(x)=-4+26 e^{-0.028 x}$ . Round to the nearest degree.

Given $\dot{y}=4 t e^{-y}$ with $y(0)=-1$ , use Euler's Method ( $h=0.2$ ) to find $y(0.2)$ to $y(1.0)$ , then find $y(t)$ exactly and compute errors for $y(0.2)$ , $y(0.6)$ , and $y(1)$ .

Evaluate the limit: $\lim _{N \rightarrow \infty} \sum_{s=1}^{N} \frac{3 s^{2}-2 s-5}{N^{3}}$ as a whole number.

Find $g'(x)$ if $g(x)=\int_{0}^{x} \sqrt{t+t^{3}} dt$ .

Differentiate the integral $\int_{1}^{x} e^{-t^{2}} dt$ with respect to $x$ .

A girl flies a kite at 300 ft high, moving horizontally at 25 ft/sec. How fast must she pay out string when the kite is 500 ft away? $20 \mathrm{ft/s}$

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

Find $g'(s)$ for the function defined by $g(s)=\int_{5}^{s}(t-t^{2})^{8} dt$ .

Find the total area between the curve $f(x)=x^{3}+x^{2}-2 x$ and the $x$ -axis, considering where it is above and below.

Trouvez les nombres critiques de $f(x) = \sqrt[3]{x^{2}}(2-x)$ où $f'(c)=0$ ou $f'(c)$ n'existe pas.

Water is in an inverted cone tank (8 ft diameter, 10 ft deep). Rate of water removal is $5 \mathrm{ft}^{3}/\mathrm{min}$ . Find the water level change rate at 6 ft depth and time to empty the tank.

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

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

Find the limit of $f(x)=\frac{3}{2} x+4$ on $[0,4]$ and verify using geometry. Provide a whole number answer.

Find the area of the shaded region between $y=\sqrt[3]{x}$ , $y=\frac{1}{x}$ , and the line $x=27$ .

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

Find the limit of $f(x)=27-9x$ over $[2,6]$ and confirm with geometry. Provide a whole number answer. $\lim _{N \rightarrow \infty} M_{N}=$

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

Find the volume of the solid formed by rotating the area between $x=-3y^{2}+15y-12$ and $x=0$ about the $x$ -axis using cylindrical shells.

Find the volume of the solid formed by rotating the region $x^{2}+(y-2)^{2}=4$ around the $y$ -axis.

Rewrite the limit using l'Hôpital's Rule to resolve the indeterminate form:
$\lim _{x \rightarrow 1} \frac{\ln x}{11 x-x^{2}-10}$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{3 x^{2}}$ using l'Hôpital's Rule when applicable.

Find how $\frac{dA}{dt}$ relates to $\frac{da}{dt}$ , $\frac{db}{dt}$ , and $\frac{d\theta}{dt}$ for area $A=(1/2)ab\sin\theta$ .

Evaluate the integral: $\int \frac{3 d t}{(t^{2}-9)^{2}}$ (include absolute values and use $C$ for the constant).

Rewrite the limit using l'Hôpital's Rule to eliminate the indeterminate form:
$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{2 x^{2}+2 x}=\lim _{x \rightarrow 0}(\square)$

Find the derivative of the function defined by $f(x)=\int_{x}^{15} t^{2} d t$ . What is $f^{\prime}(x)=?$

Find the distance traveled by an object with velocity $v(t)=14 t$ m/s over intervals $[0,2]$ and $[2,5]$ .

Compute $R_{6}, L_{6}, M_{3}$ to estimate distance over $[0,3]$ with given velocity at half-second intervals.

Find the displacement formula and total distance traveled by a particle with velocity $v(t)=8-1 t$ at $t=4$ seconds.

Calculate the integral $\int_{-98}^{-21}[7 f(x)+5 g(x)-h(x)] d x$ using given values for $f(x)$ , $g(x)$ , and $h(x)$ .

A dinghy is pulled to a dock with a rope $6 \mathrm{ft}$ above the bow, hauled in at $2 \mathrm{ft/sec}$ . Find: (a) Speed of the boat when $10 \mathrm{ft}$ of rope is out. (b) Rate of change of angle $\theta$ at that moment.

Find $f^{\prime}(-1)$ given that $f(x)=x^{10} h(x)$ , $h(-1)=2$ , and $h^{\prime}(-1)=5$ .

Find $R_{N}$ and calculate the area under $f(x)=7x^{2}+49x$ from $x=6$ to $x=11$ as a limit. $\lim_{N \rightarrow \infty} R_{N} =$

Solve using Laplace transforms: $y^{\prime \prime}+y=u(t-\pi)-u(t-3\pi)$ , with $y(0)=0$ , $y^{\prime}(0)=0$ .

Approximate the area $A$ under $f(x)=2x^2+12x$ from $x=1$ to $x=3$ using a left-endpoint sum: $A=\lim_{N \to \infty} L_N$ .

Inge flies a kite at $300 \mathrm{ft}$ high, moving away at $25 \mathrm{ft/sec}$ . How fast to let out string when $500 \mathrm{ft}$ away?

Approximate the area $A$ under $f(x)=2x^2+12x$ from $x=1$ to $x=3$ using left-endpoint sums. Find $A=\lim_{N \to \infty} L_N$ .

Approximate the area $A$ under $f(x) = 2x^2 + 12x$ over $[1,3]$ using a left-endpoint method. Find $A = \lim_{N \to \infty} L_N$ . Enter exact value.

Find the local maximum of $y=-x^{3}-11x^{2}-40x-54$ and choose a suitable graph window from the options provided.

Find the limit of $f(x)=27-9x$ over $[2,6]$ and verify geometrically. The limit is $\lim _{N \rightarrow \infty} M_{N}=7$ .

Find the limit of $f(x)=27-9x$ as $N$ approaches infinity over $[2,6]$ and verify using geometry.

Evaluate the integral of $\sqrt[4]{u}$ with respect to $u$ : $\int \sqrt[4]{u} \, du$ .

Calculate the integral: $\int \frac{d w}{\sqrt{2}}$ .

Calculate the integral of the function: $\int 4 x^{3} dx$

Find $\frac{dV}{dt}$ in terms of $R$ and $\frac{dR}{dt}$ for $V= \frac{400}{280} (R^2 - r^2)$ with $L=70$ , $p=400$ , $v=0.003$ .

A cylinder has a radius decreasing at 2 in/sec and height increasing at 5 in/sec. Find the volume change rate when $r=4$ in and $h=6$ in. Use $V=\pi r^{2} h$ .

Two cars start together; one goes north at $25 \mathrm{mph}$ , the other east at $60 \mathrm{mph}$ . Find the distance increase rate after $1 \mathrm{hr}$ . Use $D^{2}=x^{2}+y^{2}$ to solve.

Berechne für $v_{0}=34 \mathrm{~m/s}$ die mittlere Geschwindigkeit in den ersten 2 Sekunden und die Geschwindigkeit bei $t=0,1,2,3,4,5$ s. Was bedeutet negative Geschwindigkeit?

Bestimme den Flächeninhalt A, der zwischen den Schnittpunkten der Funktionen $f(x)=1 x-x^{2}$ und $g(x)=x^{2}-2 x+1$ liegt.

Show that the function $f(x)=\sqrt{x}$ is uniformly continuous on the interval $[a,+\infty)$ for $a>0$ .

Bestimme die Ableitung von $f_a(x)=a x^{4}-4 x^{3}+a^{2} x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

Find the derivative of $f_a(x)=\frac{1}{a} x^{2}+\frac{2}{a}+a$ at $x=0$ and the value of $a$ for slope = 1.

Find the formula for the area $A(y)$ of a square cross-section of a pyramid 12 inches tall with a 10-inch base. Show $A(12)=100$ and evaluate $\int_{0}^{12} A(y) dy$ . Explain the result.

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

Calculate the area under the normal curve with mean $\mu = 100$ and standard deviation $15$ from scores 112 to 114.

Find the slope of the curve $7 y^{6}+8 x^{5}=5 y+10 x$ at the point (1,1).

Ein Ball wird mit 30 m/s in 35 m Höhe geworfen. Bestimme die Höhenfunktion $h(t)$ und die Aufprallgeschwindigkeit.

Find the slope of the curve $7 y^{9}+8 x^{3}=2 y+13 x$ at the point (1,1).

Find the derivative of $y=(2t-1)(6t-5)^{-1}$ .

Find the derivative of the function $y=3 e^{-x}+e^{4 x}$ .

Finde die Stammfunktion von $f(x)=2 \cdot x$ für die Bedingungen: a) Ursprung, b) Punkt P(1|4), c) Schnitt bei 3, d) $W=\{y \in \mathbb{R}, y \geq 4\}$ , e) Berührung bei Q(1|2).

Bestimmen Sie $a$ und $b$ für die Funktion $f(x)=\frac{1}{2} x^{4}-2 x^{2}+4$ . Untersuchen Sie Symmetrie, Nullstellen, Extrema, Wendepunkte und zeichnen Sie den Graphen für $-2 \leq x \leq 2$ .