Calculus

Problem 15901

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

See Solution

Problem 15902

Find when the instantaneous velocity $p'(t)$ equals the average velocity over $[1, 16]$ for $p(t)=\sqrt{t}-2$ .

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Problem 15903

Is the claim true or false? Prove your answer: If $\lim _{h \rightarrow 0} \frac{f(a+h)-2 f(a)+f(a-h)}{h^{2}}$ exists, then $f$ is twice differentiable at $x=a$ .

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Problem 15904

Find the limit as $\Delta x$ approaches 0 for $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ where $f(x) = \frac{x^2}{2} + x^2$ .

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Problem 15905

A ball rolls off a $3.2 \mathrm{~m}$ high halfpipe. What is its speed halfway up the other side? Neglect friction.

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Problem 15906

Find the absolute extreme values of $f(x)=-x^{2}+12$ on the interval $[-2,4]$ . What are the max and min values?

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Problem 15907

Find the limit as $\Delta x$ approaches 0 for $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ , where $f(x) = \frac{x^2}{2} + x^2$ .

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Problem 15908

Find the limit of $\frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}$ as $\Delta x$ approaches 0 for $f(x) = \frac{x^2}{2} + x^2$ .

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Problem 15909

Find the power series for $f(x)=\frac{x}{9+x^{2}}$ and its interval of convergence.

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Problem 15910

Find the absolute max/min of $f(x) = x^{2} - 7$ on the interval $[-3, 4]$ .

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Problem 15911

Evaluate the limit using l'Hôpital's Rule:
$\lim _{x \rightarrow -7} \frac{x^{2}+8x+7}{-63-2x+x^{2}}$

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Problem 15912

Find the limit as $x$ approaches -7 for the expression $\frac{x^{2}+8x+7}{-63-2x+x^{2}}$ using IHôpital's Rule.

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Problem 15913

Calculate $\lim _{\Delta x \rightarrow 0} \frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}$ for $f(x) = \frac{x^2}{2} + x^2$ .

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Problem 15914

Find the limit of $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ as $\Delta x \to 0$ to determine $f''(-2)$ .

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Problem 15915

Evaluate the integral: $\int \frac{6}{(t+11)^{5}} d t$

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Problem 15916

Evaluate the integral $\int x^{2}(x^{3}-11)^{28} dx$ using the substitution $u=x^{3}-11$ .

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Problem 15917

Integrate the function from -1 to 1: $\int_{-1}^{1} 10(2 x-5)^{4} d x$ . What is the result?

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Problem 15918

Evaluate the integral from 0 to 1 of $\sqrt{4x + 6} \, dx$ .

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Problem 15919

Find $h^{\prime}(1)$ for $h(x)=f(x) \cdot g(x)$ using values from the chart.

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Problem 15920

Find the Taylor polynomials of degree $n$ for $\frac{3}{4-4x}$ near $x=0$ : $P_{3}(x)$ , $P_{5}(x)$ , $P_{7}(x)$ .

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Problem 15921

Calculate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \cos x \, dx$

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Problem 15922

Evaluate the integral $\int_{0}^{\pi}-2 \cos ^{2} x \sin x \, dx$ . What is the result?

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Problem 15923

Find the value of $h^{\prime}(3)$ for $h(x)=f(x)g(x)$ given $f(3)=4$ , $f'(3)=-1/3$ , $g(3)=3$ , $g'(3)=-2$ .

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Problem 15924

A 140 kg asteroid falls towards Earth. What speed will it impact with, given Earth's mass is $5.97 \times 10^{24} \mathrm{~kg}$ and radius $6400 \mathrm{~km}$ ?

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Problem 15925

Find the optimal landing point $x$ (in km from point $P$ ) for shortest travel time to town, given $a=3$ km, $b=12$ km, boat speed $2.8$ km/h, and walk speed $5$ km/h.

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Problem 15926

Find the relationship between $\frac{d V}{d t}$ , $\frac{d r}{d t}$ , and $\frac{d h}{d t}$ for a cylinder with volume $V = \pi r^2 h$ .

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Problem 15927

A balloon inflates at $100 \pi \mathrm{ft}^{3}/\mathrm{min}$ . Find the radius increase rate at $r=5$ ft.

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Problem 15928

Find the inflection points of the function $f(x)=\frac{x}{e^{6 x}}$ . List values or write DNE if none exist.

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Problem 15929

Find the critical numbers of $f(x)=-\sqrt{3} x+\sin(2x)$ for $0 \leq x \leq \pi$ . List them or enter DNE if none exist.

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Problem 15930

Find the number of infected people after $t=\sqrt{\ln(10)} \approx 1.5$ weeks, given $N^{\prime}(t)=100 t e^{-t^{2}}$ and $N(0)=125$ .

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Problem 15931

Find the rate of change of water depth in a conical tank (10 ft wide, 12 ft deep) when water is 8 ft deep, filling at 10 ft³/min.

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Problem 15932

Find the number of infected people after $t=\sqrt{\ln (10)}$ weeks, given $N^{\prime}(t)=100 t e^{-t^{2}}$ and initial $N(0)=125$ .

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Problem 15933

Find the diagonal change rate of a rectangle with $l=12 \, \text{cm}$ , $w=5 \, \text{cm}$ , $dl/dt=-2$ , $dw/dt=2$ . Use $d = \sqrt{l^2 + w^2}$ .

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Problem 15934

Find where to land a boat ( $x$ km from point $P$ ) to reach a town in the shortest time, given speeds of 2.8 km/h (rowing) and 5 km/h (walking). Use $d=r \cdot t$ and $t=\frac{d}{r}$ . Answer to three decimal places. $x =$ km

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Problem 15935

A 25-ft ladder leans against a wall. The base moves away at 2 ft/sec. Find the speed of the top when the base is 15 ft from the wall.

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Problem 15936

Find the velocity $v(t)$ and acceleration $a(t)$ of a pendulum modeled by $s(t)=0.08 \sin(2t)+3t$ for $0 \leq t \leq \pi$ .

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Problem 15937

Find local extrema of $f(x)=4(x-3)^{6}+2(x-3)^{4}$ . List points as $(x, y)$ or enter DNE if none exist.

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Problem 15938

A 25 ft ladder leans against a wall. Base moves away at 2 ft/s. Find how fast top moves down when base is 15 ft from wall.

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Problem 15939

A ladder leans against a wall, height 25 ft. Find the area change rate when the ladder is 9 ft from the wall. Use $A=\frac{1}{2} x y$ and $\frac{d A}{d t}=\frac{1}{2}\left(x \frac{d x}{d t}+y \frac{d y}{d t}\right)$ . Height $y=\sqrt{25^{2}-9^{2}}$ .

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Problem 15940

A 25 ft ladder leans against a wall. If the base moves away at 2 ft/s, find the rate of change of angle θ when the base is 7 ft out. Use $\tan (\theta) = \frac{y}{x}$ and $\frac{d}{d t}(\tan (\theta))=\frac{x \frac{d y}{d t}-y \frac{d x}{d t}}{x^{2}}$ .

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Problem 15941

Find the derivative of the product of functions $f(x)=-2x$ and $g(x)=8x^2-5x+7$ .

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Problem 15942

Find the value of $C$ that makes the 1-form $\alpha$ a gradient 1-form field.

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Problem 15943

Find the rate of change $\frac{d A}{d t}$ when $A=2$ , $d B/d t=3$ , and $A^3 + B^3 = 9$ .

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Problem 15944

Calculate the work done by the force field $\vec{F}=(3 x^{2}+2 y) \hat{i}+(4 y+2 x) \hat{j}$ along the path $y=x^{2}$ from $x=-1$ to $x=1$ .

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Problem 15945

What is the length of each edge of a cube if its volume increases at $24 \mathrm{in}^{3}/\mathrm{min}$ and edges at $2 \mathrm{in}/\mathrm{min}$ ?

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Problem 15946

An 80-foot building casts a 60-foot shadow. If $\theta$ increases at 0.27 rad/min, find the shadow's decreasing rate.

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Problem 15947

Bestimme die Art der Punkte bei $x=2$ und $x=-2$ , wenn $f''(2) = 4 > 0$ und $f''(-2) = -4 < 0$ .

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Problem 15948

Find how fast Tom's surface area $S$ is decreasing when he weighs $70 \mathrm{~kg}$ , given $h=165 \mathrm{~cm}$ and $w$ is decreasing.

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Problem 15949

Find the rate of change $dA/dt$ when $A=2$ and $dB/dt=3$ for the equation $A^{3}+B^{3}=9$ .

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Problem 15950

Find the derivatives: a. $f(x)=\pi \cdot 5^{x}+e^{14}+\log _{4}\left(\frac{1}{4}\right)$ ; b. $y=\frac{13^{x}}{14}-x^{e}$ .

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Problem 15951

Find $\frac{d y}{d x}$ using the chain rule for: a) $y=u^{2}+3u$ , $u=\sqrt{x}$ , $x=4$ ; b) $y=\sqrt{u}$ , $u=2x^{2}+3x+4$ , $x=-3$ ; c) $y=\frac{1}{u^{2}}$ , $u=x^{3}-5x$ , $x=-2$ ; d) $y=u(2-u^{2})$ , $u=\frac{1}{x}$ , $x=2$ .

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Problem 15952

Ship A moves west at 15 km/hr, Ship B moves north at 10 km/hr. Find:
(a) Distance between ships when $x=4$ km, $y=3$ km. (b) Rate of change of distance at $x=4$ km, $y=3$ km. (c) Rate of change of angle $\theta$ at $x=4$ km, $y=3$ km.

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Problem 15953

Find the tangent line equation for the curve $y=(x^{3}-4x^{2})^{3}$ at the point where $x=3$ .

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Problem 15954

A cone with height $10 \mathrm{~cm}$ and diameter $10 \mathrm{~cm}$ has water evaporating at $-\frac{3}{10}$ cm/hr.
(a) Why can't we use $V=\frac{1}{3} \pi r^{2} h$ to find the volume rate change when $h=5 \mathrm{~cm}$ ? What’s needed?
(b) Calculate the volume change rate when $h=5 \mathrm{~cm}$ . Provide units.
(c) When $h=4 \mathrm{~cm}$ , the surface area changes at $9 \pi \mathrm{cm}^{2}$ /hr. Find the radius change rate. Provide units.

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Problem 15955

Finde die Wendepunkte der Funktionen $f(x)=x^{4}+2 \cdot x^{3}-2$ und $g(x)=x \cdot (x^{3}-1)$ durch Ableiten.

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Problem 15956

Bestimmen Sie die Tangentengleichung der Funktion $f(x)=\frac{1}{2} x^{2}-3 x+1$ an den Punkten: a) $P(4|-3)$ , b) $P(1|-1.5)$ , c) $P(-4|21)$ , d) $P(0|1)$ .

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Problem 15957

Find the area between $f(x)=x^{2}-8x+14$ and $g(x)=-x^{2}+6x-6$ from $x=2$ to $x=5$ .

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Problem 15958

Ein Skifahrer fährt die Strecke $s(t)=1,5 t^{2}$ m. a) Berechne die Geschwindigkeit nach 5 s und die Strecke nach 6 s. b) Wann hat er $36 \frac{\mathrm{m}}{\mathrm{s}}$ ? c) Wann ist er $96 \mathrm{~m}$ gefahren?

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Problem 15959

Gegeben ist die Funktion $f_{a}(x)=-a x^{3}+4 a x$ mit $a \neq 0$ . Zeigen Sie, dass die Graphen punktsymmetrisch sind und durch $P(-2,0)$ und $Q(2,0)$ verlaufen. Bestimmen Sie Hoch- und Tiefpunkte sowie die Wendetangente und den Wert von $a$ für $m=8$ .

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Problem 15960

Find the area between the curves $x=2+\cos \theta$ and $r=5 \cos \theta$ .

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Problem 15961

Finde den Punkt, an dem die Tangente an $f(x)=(x+3)(x-3)^{2}$ die Steigung 6 hat und wo sie die $x$ -Achse schneidet.

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Problem 15962

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

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Problem 15963

Gegeben ist die Funktion $r_{a}(x)=\frac{1}{2 a} x^{4}-\frac{a}{2} x^{2}+3$ .
a) Verhalten von $f_{0}$ für $x \rightarrow+\infty$ und $x \rightarrow-\infty$ ?
b) Nachweis der Achsensymmetrie von $G_{a}$ zur $y$ -Achse?
d) Nachweis, dass $G_{a}$ drei Extrema hat und deren Arten in Abhängigkeit von $a$ ?
e) Existiert ein positives $a$ , sodass $y=4x+5$ Tangente an $G_{a}$ bei $P(2 \mid f(2))$ ist?
f) Ist $f(x)=\frac{1}{6} x^{4}-\frac{3}{2} x^{2}+3$ ein Graph von $G_{a}$ ?
g) Ist $T\left(\frac{3}{2} \sqrt{2}, 1-\frac{3}{8}\right)$ ein lokaler Tiefpunkt von $G_{3}$ ?
h) Berechne den Abstand zwischen den Nullstellen von $G_{3}$ .
i) Länge des roten Teilstücks bei $y=5$ ?
j) Bestimme die Wendetangente mit negativer Steigung und zeige, dass das Dreieck gleichschenklig ist.
k) Warum ist $g(x)$ die Ableitungsfunktion $f^{\prime}$ von $f_{3}$ ?
l) Bedeutung von $\frac{f(1)-f(0)}{1-0}$ im Kontext und berechne seinen Wert.
m) Bestimme die quadratische Funktion für den Lärmschutzwall zwischen den Nullstellen und erkläre die Bedeutung von $a$ .

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Problem 15964

As $x$ approaches infinity, what does the rate of change of $f(x)=\frac{-2 x^{3}}{x^{2}-1}$ approach? A. 0 B. $\infty$ C. $-\infty$ D. 2 E. -2

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Problem 15965

Leiten Sie die Funktion ab: $f(x)=\sin (x) \cdot \cos (x-1)$

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Problem 15966

Determine if these series converge or diverge:
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}$
2. $\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+1}$
3. $\sum_{n=1}^{\infty} \frac{a^{n}}{n !}, a>0$
4. $\sum_{n=1}^{\infty} \frac{4^{n+1}}{7^{2 n+3}}$
5. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n \sqrt{n}}$

See Solution

Problem 15967

A rover's path is defined by $x'(t)=-12 \sin(2t^2)$ and $y'(t)=10 \cos(1+\sqrt{t})$ . Solve for acceleration, speed, distance, $y$ -coordinate, and signal times.

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Problem 15968

Find the limit of the fish population rate $f(t)=\frac{10(2+3 t)}{1+0.03 t}$ as $t$ approaches infinity. Options: A. 0 B. undefined C. 10 D. 100 E. 1000

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Problem 15969

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}$ converges or diverges.

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Problem 15970

Determine if these series converge or diverge:
1. $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}$
2. $\sum_{n=1}^{\infty} \frac{n-1}{n^{3}+1}$
3. $\sum_{n=1}^{\infty} \frac{a^{n}}{n !}, a>0$
4. $\sum_{n=1}^{\infty} \frac{4^{n+1}}{7^{2 n+3}}$
5. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n \sqrt{n}}$

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Problem 15971

Evaluate the integral $\int_{0}^{1} \sqrt{\left(-12 \sin \left(2t^{2}\right)\right)^{2}+(10 \cos (1+\sqrt{t}))^{2}} dt$ using numerical methods.

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Problem 15972

Find the antiderivative of $f(x)=\frac{1}{x^{5}}$ .

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Problem 15973

Evaluate the integral $\int_{\pi / 4}^{\pi / 3}\left(-2 \sec ^{2}(x)-5 \cos (x)\right) d x$ .

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Problem 15974

Given an increasing function $g$ on $[0,4]$ , which could be the value of $\int_{0}^{4} g(x) d x$ : A. 70, B. 80, C. 96?

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Problem 15975

Find the tangent line equation for $f(x)=10 x^{2}-10 x+12$ at the point where $x=10$ .

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Problem 15976

Find the max and min values of $f$ on the interval [0,4] where $f(x)=7+81x-3x^{3}$ .

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Problem 15977

Find the formula for the population change rate, given $P(t)=20-\frac{6}{t+1}$ (in thousands).

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Problem 15978

Find the average value of the function $y=x+\sin x$ on the interval $[0, \frac{3 \pi}{2}]$ .

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Problem 15979

Find the profit rate change $\Pi^{\prime}(p)$ for $\Pi(p) = 2p - 4$ at $p=5€$ . Should the price exceed $5€$ to maximize profit?

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Problem 15980

Messung des Wasserstands: $f(t)=1,5 \sin \left(\frac{\pi}{5}(t-5)\right)+1,5$ .
a) Wasserstand bei $t=0$ . b) Graph skizzieren. c) Steigt oder fällt der Wasserstand bei $t=0$ ? d) Graph der Ableitung identifizieren. e) Maximum, Minimum und Tidenhub berechnen.

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Problem 15981

Approximate the displacement of an object with $v=\frac{1}{4t+1}$ m/s from $t=0$ to $t=8$ using 4 subintervals.

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Problem 15982

Approximate the displacement of an object with $v=\frac{2}{3} t^{2}+6(f t / s)$ on $0 \leq t \leq 18$ using $n=6$ subintervals.

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Problem 15983

Une fenêtre a un rectangle et un demi-cercle. Périmètre total = 10 m. Trouvez les dimensions pour maximiser A = xy + \frac{1}{2} \cdot \pi x^2.

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Problem 15984

L'azote et l'hydrogène forment l'ammoniac avec $Q(t)=100-\frac{1000}{10+t}$ .
a) Trouvez $Q(0)$ et $Q(20)$ . b) Calculez la variation de $Q$ sur [10 s, 20 s]. c) Trouvez le taux de variation moyen sur [10 s, 20 s] et [20 s, 30 s]. d) Évaluez $\lim_{h \to 0^{+}} \frac{Q(h+0)-Q(0)}{h}$ . e) Déterminez la fonction de variation de $Q$ . f) Évaluez $\left.\frac{dQ}{dt}\right|_{t=10 s}$ et $\left.\frac{dQ}{dt}\right|_{t=1 \text{ min}}$ . g) Analysez la tendance de $Q$ et son taux de variation. h) Trouvez $\frac{dQ}{dt}$ pour $Q=70 \mathrm{~g}$ . i) Trouvez $Q$ quand $\frac{dQ}{dt}=1,6 \mathrm{~g/s}$ . j) Évaluez $\lim_{t \to +\infty} Q(t)$ . k) Représentez graphiquement $Q$ et $\frac{dQ}{dt}$ .

See Solution

Problem 15985

Quelle est la dérivée de $f(x)=\log ^{3}(5 x^{2})$ ? A. $\frac{3 \log ^{2}(5 x^{2})}{5 x^{2} \ln 10}$ B. $\frac{2}{x \ln 10}$ C. Aucune de ces réponses D. $3 \log ^{2}(5 x^{2})$ E. $\frac{6 \log ^{2}(5 x^{2})}{x}$ F. $\frac{6 \log ^{2}(5 x^{2})}{x \ln 10}$

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Problem 15986

A baseball player throws a ball at $22 \mathrm{~m/s}$ . Find the maximum height and total time in the air.

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Problem 15987

Approximate the displacement of an object with velocity $v=\frac{2}{3} t^{2}+6$ over $0 \leq t \leq 18$ , using $n=6$ .

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Problem 15988

Find the fish population after one and two breeding seasons using a logistic model with a carrying capacity of 2000 and initial population $p_{0}=200$ . The growth rate is $170\%$ per year. Calculate $p_{1}=$ and $p_{2}=$ .

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Problem 15989

Approximate the displacement of an object with $v=\frac{2}{3} t^{2}+6$ on $0 \leq t \leq 18$ using $n=6$ subintervals.

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Problem 15990

A substance grows at $9\%$ per day. What is its mass after 5 days if it starts at 15 grams? Round to the nearest tenth. $\square$ grams

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Problem 15991

Particle moves in the $xy$ -plane with $v(t)=\left(\cos(t^{2}), e^{0.5 t}\right)$ , starting at $(3,5)$ at $t=1$ .
(a) Find $x(2)$ . (b) When is the slope of the tangent 2 for $0<t<1$ ? (c) When is speed 3? (d) Total distance from $t=0$ to $t=1$ ?

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Problem 15992

Find the rate of change of $f(t)=\frac{94 t}{99 t-94}$ after 1 hour and 10 hours.

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Problem 15993

Show that the function $f$ on $[0,3]$ is Darboux integrable using the partition $\mathrm{P}_{\varepsilon}$ with $0<\varepsilon<\frac{1}{2}$ .

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Problem 15994

Differentiate the function $y=x^{4} \ln x-\frac{1}{3} x^{3}$ . Show your work.

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Problem 15995

Differentiate $y = \frac{x \sqrt{x^{5}+9}}{(x-5)^{2/3}}$ using logarithmic differentiation. Show your work.

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Problem 15996

Find the max and min of $f(t)=1000 e^{3 \cos t}$ and the corresponding $t$ values.

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Problem 15997

Estimate the population in 2010 using $p(t)=38.59(1.014)^{t}$ and find the rate of change at $t=10$ .

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Problem 15998

Find the limit of $\frac{f(a+h)-f(a)}{h}$ for $f(x)=3x^{2}+8x-2$ .

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Problem 15999

Given $f(t)=1000 e^{3 \cos t}$ , find the max/min of $f(t)$ and the corresponding $t$ values.

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Problem 16000

Find the limit of $\frac{f(a+h)-f(a)}{h}$ for $f(x)=\frac{4}{x-1}$ .

See Solution

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Calculus

Problem 15901

Evaluate the limit using l'Hôpital's Rule: 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​ Find the limit and provide the exact answer.

Problem 15902

Find when the instantaneous velocity p′(t)p'(t)p′(t) equals the average velocity over [1,16][1, 16][1,16] for p(t)=t−2p(t)=\sqrt{t}-2p(t)=t​−2.

Problem 15903

Is the claim true or false? Prove your answer: If lim⁡h→0f(a+h)−2f(a)+f(a−h)h2\lim _{h \rightarrow 0} \frac{f(a+h)-2 f(a)+f(a-h)}{h^{2}}limh→0​h2f(a+h)−2f(a)+f(a−h)​ exists, then fff is twice differentiable at x=ax=ax=a.

Problem 15904

Find the limit as Δx\Delta xΔx approaches 0 for f′(−2+Δx)−f′(−2)Δx\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}Δxf′(−2+Δx)−f′(−2)​ where f(x)=x22+x2f(x) = \frac{x^2}{2} + x^2f(x)=2x2​+x2.

Problem 15905

A ball rolls off a 3.2 m3.2 \mathrm{~m}3.2 m high halfpipe. What is its speed halfway up the other side? Neglect friction.

Problem 15906

Find the absolute extreme values of f(x)=−x2+12f(x)=-x^{2}+12f(x)=−x2+12 on the interval [−2,4][-2,4][−2,4]. What are the max and min values?

Problem 15907

Find the limit as Δx\Delta xΔx approaches 0 for f′(−2+Δx)−f′(−2)Δx\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}Δxf′(−2+Δx)−f′(−2)​, where f(x)=x22+x2f(x) = \frac{x^2}{2} + x^2f(x)=2x2​+x2.

Problem 15908

Find the limit of f′(−2+Δx)−f′(−2)Δx\frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}Δxf′(−2+Δx)−f′(−2)​ as Δx\Delta xΔx approaches 0 for f(x)=x22+x2f(x) = \frac{x^2}{2} + x^2f(x)=2x2​+x2.

Problem 15909

Find the power series for f(x)=x9+x2f(x)=\frac{x}{9+x^{2}}f(x)=9+x2x​ and its interval of convergence.

Problem 15910

Find the absolute max/min of f(x)=x2−7f(x) = x^{2} - 7f(x)=x2−7 on the interval [−3,4][-3, 4][−3,4].

Problem 15911

Evaluate the limit using l'Hôpital's Rule: lim⁡x→−7x2+8x+7−63−2x+x2 \lim _{x \rightarrow -7} \frac{x^{2}+8x+7}{-63-2x+x^{2}} x→−7lim​−63−2x+x2x2+8x+7​

Problem 15912

Find the limit as xxx approaches -7 for the expression x2+8x+7−63−2x+x2\frac{x^{2}+8x+7}{-63-2x+x^{2}}−63−2x+x2x2+8x+7​ using IHôpital's Rule.

Problem 15913

Calculate lim⁡Δx→0f′(−2+Δx)−f′(−2)Δx\lim _{\Delta x \rightarrow 0} \frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}limΔx→0​Δxf′(−2+Δx)−f′(−2)​ for f(x)=x22+x2f(x) = \frac{x^2}{2} + x^2f(x)=2x2​+x2.

Problem 15914

Find the limit of f′(−2+Δx)−f′(−2)Δx\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}Δxf′(−2+Δx)−f′(−2)​ as Δx→0\Delta x \to 0Δx→0 to determine f′′(−2)f''(-2)f′′(−2).

Problem 15915

Evaluate the integral: ∫6(t+11)5dt\int \frac{6}{(t+11)^{5}} d t∫(t+11)56​dt

Problem 15916

Evaluate the integral ∫x2(x3−11)28dx\int x^{2}(x^{3}-11)^{28} dx∫x2(x3−11)28dx using the substitution u=x3−11u=x^{3}-11u=x3−11.

Problem 15917

Integrate the function from -1 to 1: ∫−1110(2x−5)4dx\int_{-1}^{1} 10(2 x-5)^{4} d x∫−11​10(2x−5)4dx. What is the result?

Problem 15918

Evaluate the integral from 0 to 1 of 4x+6 dx\sqrt{4x + 6} \, dx4x+6​dx.

Problem 15919

Find h′(1)h^{\prime}(1)h′(1) for h(x)=f(x)⋅g(x)h(x)=f(x) \cdot g(x)h(x)=f(x)⋅g(x) using values from the chart.

Problem 15920

Find the Taylor polynomials of degree nnn for 34−4x\frac{3}{4-4x}4−4x3​ near x=0x=0x=0: P3(x)P_{3}(x)P3​(x), P5(x)P_{5}(x)P5​(x), P7(x)P_{7}(x)P7​(x).

Problem 15921

Calculate the integral: ∫−π2π2sin⁡2xcos⁡x dx\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \cos x \, dx∫−2π​2π​​sin2xcosxdx

Problem 15922

Evaluate the integral ∫0π−2cos⁡2xsin⁡x dx\int_{0}^{\pi}-2 \cos ^{2} x \sin x \, dx∫0π​−2cos2xsinxdx. What is the result?

Problem 15923

Find the value of h′(3)h^{\prime}(3)h′(3) for h(x)=f(x)g(x)h(x)=f(x)g(x)h(x)=f(x)g(x) given f(3)=4f(3)=4f(3)=4, f′(3)=−1/3f'(3)=-1/3f′(3)=−1/3, g(3)=3g(3)=3g(3)=3, g′(3)=−2g'(3)=-2g′(3)=−2.

Problem 15924

A 140 kg asteroid falls towards Earth. What speed will it impact with, given Earth's mass is 5.97×1024 kg5.97 \times 10^{24} \mathrm{~kg}5.97×1024 kg and radius 6400 km6400 \mathrm{~km}6400 km?

Problem 15925

Find the optimal landing point xxx (in km from point PPP) for shortest travel time to town, given a=3a=3a=3 km, b=12b=12b=12 km, boat speed 2.82.82.8 km/h, and walk speed 555 km/h.

Problem 15926

Find the relationship between dVdt\frac{d V}{d t}dtdV​, drdt\frac{d r}{d t}dtdr​, and dhdt\frac{d h}{d t}dtdh​ for a cylinder with volume V=πr2hV = \pi r^2 hV=πr2h.

Problem 15927

A balloon inflates at 100πft3/min100 \pi \mathrm{ft}^{3}/\mathrm{min}100πft3/min. Find the radius increase rate at r=5r=5r=5 ft.

Problem 15928

Find the inflection points of the function f(x)=xe6xf(x)=\frac{x}{e^{6 x}}f(x)=e6xx​. List values or write DNE if none exist.

Problem 15929

Find the critical numbers of f(x)=−3x+sin⁡(2x)f(x)=-\sqrt{3} x+\sin(2x)f(x)=−3​x+sin(2x) for 0≤x≤π0 \leq x \leq \pi0≤x≤π. List them or enter DNE if none exist.

Problem 15930

Find the number of infected people after t=ln⁡(10)≈1.5t=\sqrt{\ln(10)} \approx 1.5t=ln(10)​≈1.5 weeks, given N′(t)=100te−t2N^{\prime}(t)=100 t e^{-t^{2}}N′(t)=100te−t2 and N(0)=125N(0)=125N(0)=125.

Problem 15931

Find the rate of change of water depth in a conical tank (10 ft wide, 12 ft deep) when water is 8 ft deep, filling at 10 ft³/min.

Problem 15932

Find the number of infected people after t=ln⁡(10)t=\sqrt{\ln (10)}t=ln(10)​ weeks, given N′(t)=100te−t2N^{\prime}(t)=100 t e^{-t^{2}}N′(t)=100te−t2 and initial N(0)=125N(0)=125N(0)=125.

Problem 15933

Find the diagonal change rate of a rectangle with l=12 cml=12 \, \text{cm}l=12cm, w=5 cmw=5 \, \text{cm}w=5cm, dl/dt=−2dl/dt=-2dl/dt=−2, dw/dt=2dw/dt=2dw/dt=2. Use d=l2+w2d = \sqrt{l^2 + w^2}d=l2+w2​.

Problem 15934

Find where to land a boat (xxx km from point PPP) to reach a town in the shortest time, given speeds of 2.8 km/h (rowing) and 5 km/h (walking). Use d=r⋅td=r \cdot td=r⋅t and t=drt=\frac{d}{r}t=rd​. Answer to three decimal places. x= x = x= km

Problem 15935

A 25-ft ladder leans against a wall. The base moves away at 2 ft/sec. Find the speed of the top when the base is 15 ft from the wall.

Problem 15936

Find the velocity v(t)v(t)v(t) and acceleration a(t)a(t)a(t) of a pendulum modeled by s(t)=0.08sin⁡(2t)+3ts(t)=0.08 \sin(2t)+3ts(t)=0.08sin(2t)+3t for 0≤t≤π0 \leq t \leq \pi0≤t≤π.

Problem 15937

Find local extrema of f(x)=4(x−3)6+2(x−3)4f(x)=4(x-3)^{6}+2(x-3)^{4}f(x)=4(x−3)6+2(x−3)4. List points as (x,y)(x, y)(x,y) or enter DNE if none exist.

Problem 15938

A 25 ft ladder leans against a wall. Base moves away at 2 ft/s. Find how fast top moves down when base is 15 ft from wall.

Problem 15939

Problem 15940

Problem 15941

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

Find when the instantaneous velocity $p'(t)$ equals the average velocity over $[1, 16]$ for $p(t)=\sqrt{t}-2$ .

Is the claim true or false? Prove your answer: If $\lim _{h \rightarrow 0} \frac{f(a+h)-2 f(a)+f(a-h)}{h^{2}}$ exists, then $f$ is twice differentiable at $x=a$ .

Find the limit as $\Delta x$ approaches 0 for $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ where $f(x) = \frac{x^2}{2} + x^2$ .

A ball rolls off a $3.2 \mathrm{~m}$ high halfpipe. What is its speed halfway up the other side? Neglect friction.

Find the absolute extreme values of $f(x)=-x^{2}+12$ on the interval $[-2,4]$ . What are the max and min values?

Find the limit as $\Delta x$ approaches 0 for $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ , where $f(x) = \frac{x^2}{2} + x^2$ .

Find the limit of $\frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}$ as $\Delta x$ approaches 0 for $f(x) = \frac{x^2}{2} + x^2$ .

Find the power series for $f(x)=\frac{x}{9+x^{2}}$ and its interval of convergence.

Find the absolute max/min of $f(x) = x^{2} - 7$ on the interval $[-3, 4]$ .

Evaluate the limit using l'Hôpital's Rule:
$\lim _{x \rightarrow -7} \frac{x^{2}+8x+7}{-63-2x+x^{2}}$

Find the limit as $x$ approaches -7 for the expression $\frac{x^{2}+8x+7}{-63-2x+x^{2}}$ using IHôpital's Rule.

Calculate $\lim _{\Delta x \rightarrow 0} \frac{f^{\prime}(-2+\Delta x)-f^{\prime}(-2)}{\Delta x}$ for $f(x) = \frac{x^2}{2} + x^2$ .

Find the limit of $\frac{f'(-2+\Delta x)-f'(-2)}{\Delta x}$ as $\Delta x \to 0$ to determine $f''(-2)$ .

Evaluate the integral: $\int \frac{6}{(t+11)^{5}} d t$

Evaluate the integral $\int x^{2}(x^{3}-11)^{28} dx$ using the substitution $u=x^{3}-11$ .

Integrate the function from -1 to 1: $\int_{-1}^{1} 10(2 x-5)^{4} d x$ . What is the result?

Evaluate the integral from 0 to 1 of $\sqrt{4x + 6} \, dx$ .

Find $h^{\prime}(1)$ for $h(x)=f(x) \cdot g(x)$ using values from the chart.

Find the Taylor polynomials of degree $n$ for $\frac{3}{4-4x}$ near $x=0$ : $P_{3}(x)$ , $P_{5}(x)$ , $P_{7}(x)$ .

Calculate the integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \cos x \, dx$

Evaluate the integral $\int_{0}^{\pi}-2 \cos ^{2} x \sin x \, dx$ . What is the result?

Find the value of $h^{\prime}(3)$ for $h(x)=f(x)g(x)$ given $f(3)=4$ , $f'(3)=-1/3$ , $g(3)=3$ , $g'(3)=-2$ .

A 140 kg asteroid falls towards Earth. What speed will it impact with, given Earth's mass is $5.97 \times 10^{24} \mathrm{~kg}$ and radius $6400 \mathrm{~km}$ ?

Find the optimal landing point $x$ (in km from point $P$ ) for shortest travel time to town, given $a=3$ km, $b=12$ km, boat speed $2.8$ km/h, and walk speed $5$ km/h.

Find the relationship between $\frac{d V}{d t}$ , $\frac{d r}{d t}$ , and $\frac{d h}{d t}$ for a cylinder with volume $V = \pi r^2 h$ .

A balloon inflates at $100 \pi \mathrm{ft}^{3}/\mathrm{min}$ . Find the radius increase rate at $r=5$ ft.

Find the inflection points of the function $f(x)=\frac{x}{e^{6 x}}$ . List values or write DNE if none exist.

Find the critical numbers of $f(x)=-\sqrt{3} x+\sin(2x)$ for $0 \leq x \leq \pi$ . List them or enter DNE if none exist.

Find the number of infected people after $t=\sqrt{\ln(10)} \approx 1.5$ weeks, given $N^{\prime}(t)=100 t e^{-t^{2}}$ and $N(0)=125$ .

Find the number of infected people after $t=\sqrt{\ln (10)}$ weeks, given $N^{\prime}(t)=100 t e^{-t^{2}}$ and initial $N(0)=125$ .

Find the diagonal change rate of a rectangle with $l=12 \, \text{cm}$ , $w=5 \, \text{cm}$ , $dl/dt=-2$ , $dw/dt=2$ . Use $d = \sqrt{l^2 + w^2}$ .

Find where to land a boat ( $x$ km from point $P$ ) to reach a town in the shortest time, given speeds of 2.8 km/h (rowing) and 5 km/h (walking). Use $d=r \cdot t$ and $t=\frac{d}{r}$ . Answer to three decimal places. $x =$ km

Find the velocity $v(t)$ and acceleration $a(t)$ of a pendulum modeled by $s(t)=0.08 \sin(2t)+3t$ for $0 \leq t \leq \pi$ .

Find local extrema of $f(x)=4(x-3)^{6}+2(x-3)^{4}$ . List points as $(x, y)$ or enter DNE if none exist.

Find the derivative of the product of functions $f(x)=-2x$ and $g(x)=8x^2-5x+7$ .

Find the value of $C$ that makes the 1-form $\alpha$ a gradient 1-form field.

Find the rate of change $\frac{d A}{d t}$ when $A=2$ , $d B/d t=3$ , and $A^3 + B^3 = 9$ .

Calculate the work done by the force field $\vec{F}=(3 x^{2}+2 y) \hat{i}+(4 y+2 x) \hat{j}$ along the path $y=x^{2}$ from $x=-1$ to $x=1$ .

What is the length of each edge of a cube if its volume increases at $24 \mathrm{in}^{3}/\mathrm{min}$ and edges at $2 \mathrm{in}/\mathrm{min}$ ?

An 80-foot building casts a 60-foot shadow. If $\theta$ increases at 0.27 rad/min, find the shadow's decreasing rate.

Bestimme die Art der Punkte bei $x=2$ und $x=-2$ , wenn $f''(2) = 4 > 0$ und $f''(-2) = -4 < 0$ .

Find how fast Tom's surface area $S$ is decreasing when he weighs $70 \mathrm{~kg}$ , given $h=165 \mathrm{~cm}$ and $w$ is decreasing.

Find the rate of change $dA/dt$ when $A=2$ and $dB/dt=3$ for the equation $A^{3}+B^{3}=9$ .

Find the derivatives: a. $f(x)=\pi \cdot 5^{x}+e^{14}+\log _{4}\left(\frac{1}{4}\right)$ ; b. $y=\frac{13^{x}}{14}-x^{e}$ .

Ship A moves west at 15 km/hr, Ship B moves north at 10 km/hr. Find:
(a) Distance between ships when $x=4$ km, $y=3$ km. (b) Rate of change of distance at $x=4$ km, $y=3$ km. (c) Rate of change of angle $\theta$ at $x=4$ km, $y=3$ km.

Find the tangent line equation for the curve $y=(x^{3}-4x^{2})^{3}$ at the point where $x=3$ .

Finde die Wendepunkte der Funktionen $f(x)=x^{4}+2 \cdot x^{3}-2$ und $g(x)=x \cdot (x^{3}-1)$ durch Ableiten.

Bestimmen Sie die Tangentengleichung der Funktion $f(x)=\frac{1}{2} x^{2}-3 x+1$ an den Punkten: a) $P(4|-3)$ , b) $P(1|-1.5)$ , c) $P(-4|21)$ , d) $P(0|1)$ .

Find the area between $f(x)=x^{2}-8x+14$ and $g(x)=-x^{2}+6x-6$ from $x=2$ to $x=5$ .

Ein Skifahrer fährt die Strecke $s(t)=1,5 t^{2}$ m. a) Berechne die Geschwindigkeit nach 5 s und die Strecke nach 6 s. b) Wann hat er $36 \frac{\mathrm{m}}{\mathrm{s}}$ ? c) Wann ist er $96 \mathrm{~m}$ gefahren?

Find the area between the curves $x=2+\cos \theta$ and $r=5 \cos \theta$ .

Finde den Punkt, an dem die Tangente an $f(x)=(x+3)(x-3)^{2}$ die Steigung 6 hat und wo sie die $x$ -Achse schneidet.

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

As $x$ approaches infinity, what does the rate of change of $f(x)=\frac{-2 x^{3}}{x^{2}-1}$ approach? A. 0 B. $\infty$ C. $-\infty$ D. 2 E. -2

Leiten Sie die Funktion ab: $f(x)=\sin (x) \cdot \cos (x-1)$

A rover's path is defined by $x'(t)=-12 \sin(2t^2)$ and $y'(t)=10 \cos(1+\sqrt{t})$ . Solve for acceleration, speed, distance, $y$ -coordinate, and signal times.

Find the limit of the fish population rate $f(t)=\frac{10(2+3 t)}{1+0.03 t}$ as $t$ approaches infinity. Options: A. 0 B. undefined C. 10 D. 100 E. 1000

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1}$ converges or diverges.

Evaluate the integral $\int_{0}^{1} \sqrt{\left(-12 \sin \left(2t^{2}\right)\right)^{2}+(10 \cos (1+\sqrt{t}))^{2}} dt$ using numerical methods.

Find the antiderivative of $f(x)=\frac{1}{x^{5}}$ .

Evaluate the integral $\int_{\pi / 4}^{\pi / 3}\left(-2 \sec ^{2}(x)-5 \cos (x)\right) d x$ .

Given an increasing function $g$ on $[0,4]$ , which could be the value of $\int_{0}^{4} g(x) d x$ : A. 70, B. 80, C. 96?