Calculus

Problem 7801

Find the critical points of $f(x)=\frac{x}{x^{2}+49}$ and calculate its derivative $f'(x)$ .

See Solution

Problem 7802

Find the horizontal asymptote $y=a$ for the velocity $v(t)=-185(1-e^{-\frac{t}{6}})$ feet/sec. What is $a$ ?

See Solution

Problem 7803

Find the derivative of $f(x) = 15x(3x+4)^{4} + (3x+4)^{5}$ .

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Problem 7804

Find the critical points of the function $f(x)=x^{2} \sqrt{x+18}$ . A. $x=$ (comma-separated) B. No critical points.

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Problem 7805

Calculate the average rate of change of $g(x) = 6 \sqrt{x}$ from $x = 1$ to $x = 5$ .

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Problem 7806

Find the velocity $v(5)$ and acceleration $a(5)$ for the position function $s(t)=t^{2}-3t$ .

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Problem 7807

Find the acceleration of a particle at $t=\pi$ given its position $x(t)=5 \sin 2t + 2 \cos 3t$ .

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Problem 7808

Calculate the average rate of change of $g(x) = -14 x^{3}$ from $x = -1$ to $x = 1$ .

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Problem 7809

Calculate the average rate of change of $k(x) = 2x^2 - 15$ from $x = 4$ to $x = 5$ .

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Problem 7810

Calculate the average rate of change of $k(x) = \frac{15}{x}$ from $x = 7$ to $x = 17$ .

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Problem 7811

Calculate velocity and acceleration at $t=5$ s for $s(t)=t^{4}-2t$ . Find $v(5)=?$

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Problem 7812

Calculate the average rate of change of $k(x) = \frac{18}{x-18}$ from $x=6$ to $x=8$ .

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Problem 7813

Find the velocity and acceleration of the body at $t=\frac{\pi}{4}$ sec for the position function $s=11 \sin t+14 \cos t$ .

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Problem 7814

Show that $f^{\prime \prime}(0)>0$ for $m=2$ in Hill's equation: $f(P)=\frac{P^{2}}{k^{2}+P^{2}}$ . Simplify.

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Problem 7815

A 5 ft ladder leans against a wall. If it slides away at $12 \mathrm{ft/s}$ , how fast is the top falling when the foot is 3 ft from the wall?

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Problem 7816

Calculate the average rate of change of $g(x) = 2x^{4}$ from $x = -1$ to $x = 1$ .

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Problem 7817

Calculate the average rate of change of $g(x) = -3 \sqrt{x} + 12$ from $x = 5$ to $x = 7$ .

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Problem 7818

Find the smaller critical point, $t_1$ , of the function $F(t)=\frac{1}{3} t^{3}-5 t^{2}+16 t+29$ .

See Solution

Problem 7819

Calculate the average rate of change of $g(x) = \frac{18}{x+10}$ from $x=4$ to $x=9$ .

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Problem 7820

Find the absolute extreme values of $f(x)=8 x^{\frac{3}{4}}-x$ on $[0,4096]$ . Where are the max values?

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Problem 7821

Calculate the average rate of change of $f(x) = 2 \sqrt{x} + 2$ on the interval $[18, 20]$ .

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Problem 7822

Why can't we use Rolle's Theorem for $f(x)=|x|$ on $[-a, a]$ where $a>0$ ? Choose: A, B, C, or D.

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Problem 7823

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2 x^{2}+3 x-2$ , simplify your answer.

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Problem 7824

Calculate the average rate of change of $f(x) = 7 \sqrt{x + 17}$ from $x = -16$ to $x = -12$ .

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Problem 7825

Find $d y / d t$ at $x=3$ for $y=x^{2}-3 x$ with $d x / d t=3$ .

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Problem 7826

A rock creates a ripple in a pond. If the radius grows at $2 \mathrm{ft} / \mathrm{sec}$ , how fast is the area increasing at 6 ft?

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Problem 7827

Find the rate of change of the area of a circle as its radius increases at $7 \mathrm{ft/sec}$ , in terms of circumference $C$ .

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Problem 7828

A point moves on the curve $5 x^{3}+6 y^{3}=x y$ . At $P=\left(\frac{1}{11}, \frac{1}{11}\right)$ , $y$ increases at 6 units/sec. Find the speed and direction of $x$ .

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Problem 7829

Find the population change in 2013 using the model $P(t)=400(1.065)^{t}$ (thousands) since 2000.

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Problem 7830

Find the derivative of $f(x)=5 \cdot 2^{x}+5 e^{x}+5 x^{2}$ . Which option is correct?

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Problem 7831

Find the derivative of the function $y=e^{\left(x^{2}-2\right)}$ : $\mathrm{dy} / \mathbf{d x}$ .

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Problem 7832

Find the limits of $s(x)=3 x^{3}-5 x^{2}+x-9$ as $x \to \pm \infty$ . What do they approach?

See Solution

Problem 7833

Find points $c$ where the Mean Value Theorem holds for $f(x)=x^{3}$ on $[-2,2]$ . What is $\mathrm{c}=$ ?

See Solution

Problem 7834

Find the volume decrease rate of a snowball with diameter 3 inches if its radius shrinks at $1 \mathrm{in} / \mathrm{min}$ .

See Solution

Problem 7835

Determine if the Mean Value Theorem applies to $f(x)=-3-x^{2}$ on $[-2,1]$ and find any guaranteed points.

See Solution

Problem 7836

A 10 ft pole casts Joe's shadow as he walks away at 4 ft/s. Find the speed of the shadow's tip when Joe is 11 ft away.

See Solution

Problem 7837

Check if the Mean Value Theorem applies to $f(x)=|x-5|$ on $[-5,11]$ and find guaranteed points if it does.

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Problem 7838

Two cyclists leave at the same time: one goes north at $24 \mathrm{mph}$ , the other east at $10 \mathrm{mph}$ . Find the rate of distance change after 2 hours.

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Problem 7839

Given $x^{4}+y^{4}=82$ , find $\frac{d y}{d x}$ and the tangent line at $(3,-1)$ .

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Problem 7840

Graph $f(x)=\frac{-16 x}{x^{2}+9}$ . Identify intervals of increase, decrease, max/min values, domain, and range.

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Problem 7841

A 48 cm wire is cut into two pieces to form squares.
(a) Find $A(x)$ , the total area of the squares in terms of $x$ . (b) Determine the side length $x$ that minimizes the area. (c) What is the minimum area of the squares?

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Problem 7842

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-x^{2}+3x-2$ , where $h \neq 0$ .

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Problem 7843

Calculate the average rate of change of $f(x)=x^{3}-2 x^{2}+3 x$ between $x=1$ and $x=2$ . Simplify your answer.

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Problem 7844

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-7 x+7$ , where $h \neq 0$ .

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Problem 7845

Find the rate of change of the balance $A=5000 e^{0.08 t}$ at $t=1$ , $t=10$ , and $t=50$ years. Round to two decimal places.

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Problem 7846

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-2x+8$ , where $h \neq 0$ .

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Problem 7847

Find the slope $m$ of the tangent to $y=4+5x^{2}-2x^{3}$ at $x=a$ where $m=1$ . Also, find tangent line equations at specified points.

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Problem 7848

Solve the integral $\int_{-\pi / 2}^{\pi / 3} \sin (4 x) \cos ^{4}(4 x) d x$ using $u=\cos(4x)$ . Find $u$ values at $x=-\pi/2$ and $x=\pi/3$ .

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Problem 7849

Berechne den Flächeninhalt unter $f(x)=x^{3}$ von 0 bis 1.

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Problem 7850

Berechnen Sie den Flächeninhalt unter $f(x)=-\frac{1}{3} x^{2}+\frac{4}{3} x+\frac{5}{3}$ im Intervall $[-1,6]$ .

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Problem 7851

Berechne den Flächeninhalt unter $f(x)=-x^{2}+2 x+3$ und über der x-Achse von $x=0.5$ bis $x=3$ .

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Problem 7852

Berechnen Sie den Flächeninhalt unter der Funktion $f(x)=x^{3}-4 x^{2}+3 x$ im Intervall [0,3].

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Problem 7853

Find the derivative of $z=\left(\frac{\tan^{2} t \cos^{2} t}{\sin t}+\frac{1}{2} \csc t \sin 2 t\right)^{2}$ w.r.t. $t$ .

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Problem 7854

Analysiere die Besucherzahlen eines Vergnügungsparks mit $f(x)=100(x-10)e^{-0,05x}+10000$ . Beantworte a) bis e).

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Problem 7855

Modellierung der Besucherzahlen im Freizeitpark: $f(x)=100(x-10)e^{-0,05 x}+10000$ . Analysiere Verlauf, Maximum, Abnahme, Zunahme und Rentabilität.

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Problem 7856

Modelliere die Besucherzahlen $f(x)=100(x-10)e^{-0,05x}+10000$ . Beantworte Fragen zu Verlauf, Maximum, Abnahme und Rentabilität.

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Problem 7857

Bestimme die Ableitungen der Funktionen: e) $j(x)=\frac{\sin (x)}{\sqrt{3 x-2}}$ , f) $k(x)=(2 x+3)^{3} \cdot \sqrt{x}$ .

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Problem 7858

Ein Behälter hat zu Beginn 2 m³ Öl. Bestimmen Sie die Ölmenge $g(t)$ für $t>0$ und die Zeit bis 2,5 m³ erreicht sind.

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Problem 7859

A rock is thrown on Mars with $H=11t-1.86t^{2}$ . Find its velocity after 2s, at $t=a$ , when it hits ground, and impact velocity.

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Problem 7860

Skizzieren und berechnen Sie die Integrale: a) $\int_{2}^{4} 1,5 \mathrm{dt}$ b) $\int_{0}^{3} \frac{1}{4} x d x$

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Problem 7861

Bestimmen Sie die Wendepunkte und die Steigung der Tangenten für die Funktionen: a) $f_{a}(x)=x^{3}-a x^{2}$ , b) $f_{a}(x)=\frac{a^{3}}{x}-x^{2}$ , c) $f_{a}(x)=x^{4}-2 a x^{2}+1$ .

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Problem 7862

Find the local extrema of the function $f a(x)=-a \cdot x^{3}+4 a x$ across its entire domain.

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Problem 7863

Find the second derivative of $f_{a}(x)=-a \cdot x^{3}+4 a x$ .

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Problem 7864

Untersuchen Sie die Funktion $A(t)=100-85 \cdot e^{-\lambda-t}$ auf den Grenzwert für $t \to \infty$ . Bestimmen Sie $\lambda$ für $A(2)=80\%$ und die Zeit für $A(t)=90\%$ .

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Problem 7865

Find the value of \ $300 invested at 9% interest compounded continuously after 3 years. Use$ A(t)=P \bullet e^{r t}$.

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Problem 7866

Find the derivative of the function $y=2 \sqrt[5]{x^{3}}$ , expressed in radical form without negative exponents.

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Problem 7867

Find the derivative of the function $f(x)=\frac{3}{2 \sqrt{x}}$ and express it in radical form.

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Problem 7868

A particle's position is given by $s(t)=t^{3}-3t+1$ . Find velocity, acceleration, and describe motion at $t=0$ and $t=2$ .

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Problem 7869

A ball's height is given by $s(t)=12t-t^{3}+1$ . Find its release height, max height, velocity at $t=0,1,3$ , and total distance before hitting ground. A stone's height is $s(t)=15t-5t^{2}$ . Find its max height and time to reach it.

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Problem 7870

1. A ball's height is given by $s(t)=12t-t^{3}+1$ . Find the release height, max height, and velocity at $t=0, 1, 3$ . Calculate total distance before hitting ground.
2. A stone's height is $s(t)=15t-5t^{2}$ . Find the time for max height and the max height.
3. A diver's height is $s(t)=10+5t-t^{2}$ . Find the board height, time to hit water, and velocity/acceleration at impact.
4. A rocket's height is $h(t)=h_{0}+v_{0}t-4.9t^{2}$ . With $v_{0}=50$ , find max height and time to hit ground.
5. Analyze a velocity graph for walking east: find when still, moving east/west, moving quickly, and speeding up/slowing down.

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Problem 7871

Maximize monthly profit $P=14x-0.1x^{2}-200$ . Find production level and maximum profit in \$ for (a) and (b).

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Problem 7872

A stone's height is given by $s(t)=15t-5t^{2}$ . Find when it reaches max height and the max height.
A diver's height is $s(t)=10+5t-t^{2}$ . Find: a) height of the board, b) time to hit water, c) velocity and acceleration on impact.

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Problem 7873

Find the limit as $x$ approaches 2 for $\frac{4 \tan (3 x-6)}{3 \ln (3 x-5)}$ in simplest form.

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Problem 7874

Bestimme die Steigung der Kurve $y=x^{3}-5 x^{2}+6 x$ an ihrem Schnittpunkt mit der $y$ -Achse.

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Problem 7875

Find the derivative of $f(x)=(4 x^{8}+6)^{4}(12 x^{6}+5)^{3}$ .

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Problem 7876

Find the time when a stone projected with $s(t)=15t-5t^{2}$ reaches its maximum height and the height itself.

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Problem 7877

Find the intervals where the function $f(x)=-5 x^{3}-120 x^{2}+9 x-1$ is concave up or down.

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Problem 7878

Bestimme den Anstieg der Kurve $y=x^{3}-5 x^{2}+6 x$ an den Schnittpunkten mit der $x$ -Achse.

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Problem 7879

Find points of inflection for the function $f(x)=-5 x^{3}-120 x^{2}+9 x-1$ . Provide answers as $(x, y)$ -pairs.

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Problem 7880

Bestimmen Sie die Ableitung von $f(t)=\sqrt{\frac{2}{t}}$ .

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Problem 7881

Find $f^{\prime}(1)$ if $f(x+h)-f(x)=2 x h+h^{2}+2 h$ .

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Problem 7882

Find the function $f$ and the number $a$ for these limits representing derivatives: (a) $\lim _{h \rightarrow 0} \frac{(1+h)^{5}-1}{h}$ , (b) $\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$ .

See Solution

Problem 7883

Find the tangent equation to the curve $y=x^{4}+2 e^{x}$ at the point where $x=0$ .

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Problem 7884

Find the derivative of $y=\left(\frac{x^{2}+1}{x^{2}-1}\right)^{3}$ .

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Problem 7885

Bestimmen Sie die Ableitung von $f(t)=\sqrt{2} \cdot t^{-\frac{1}{2}}$ .

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Problem 7886

Find the derivative of the function $y=8 x^{3}+x-4$ with respect to $x$ : $\frac{d y}{d x}$ .

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Problem 7887

Find the derivative of $f(g(x))$ where $f(x)=\frac{2}{x}$ and $g(x)=2-x^{2}$ .

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Problem 7888

Find the derivative of these functions: (a) $y=\left(\frac{x^{2}+1}{x^{2}-1}\right)^{3}$ , (b) $y=e^{x \cos x}$ , (c) $\left[e^{(2 x)} \tan \left(x^{2}+1\right)\right]^{1 / 3}$ .

See Solution

Problem 7889

Find the derivative of $f(g(x))$ where $f(x)=x^{4}$ and $g(x)=3 x-9$ . Compute $\frac{d}{d x} f(g(x))=$

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Problem 7890

Find the derivative $\frac{d y}{d x}$ for $y=x^{3} \sqrt{x+1}$ and simplify it.

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Problem 7891

Find the percentage of doctors prescribing the medication after 2 months using $P(t)=100(1-e^{-0.39 t})$ . Also, calculate $P^{\prime}(2)$ and interpret it. Choose the correct interpretation from the options provided.

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Problem 7892

Given the formula $P(t)=100\left(1-e^{-0.39 t}\right)$ , find the percentage of doctors prescribing after 2 months and $P^{\prime}(2)$ . Interpret $P^{\prime}(2)$ .

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Problem 7893

Find $h^{\prime}(1)$ if $h(x)=\sqrt{4+3 f(x)}$ , given $f(1)=7$ and $f^{\prime}(1)=4$ .

See Solution

Problem 7894

Bestimme die Ableitung von $f(t)=\frac{1}{\sin (t)}$ .

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Problem 7895

Find the average rate of change of $f(x)$ from $x=-6$ (15) to $x=-1$ (0). What is it over the interval $[-6, -1]$ ?

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Problem 7896

Find $f^{\prime \prime}(1)$ , $f^{\prime \prime}(-8)$ , and $f^{\prime \prime}(-9)$ for $f(x)=3x^{3}-4x^{2}-8x$ .

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Problem 7897

Find points of inflection for the function $f(x)=-5 x^{3}-120 x^{2}+9 x-1$ . Provide your answer as $(x, y)$ -pairs.

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Problem 7898

Finde die Stammfunktion von $f(x)=3^{x}$ im Intervall $I=[-2, 0]$ .

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Problem 7899

Find the derivative of $f(x)=-8 x^{3}-5 x^{2}-4 x$ .

See Solution

Problem 7900

Find the second derivative of the function $f(x)=\sqrt{6 x-2}$ .

See Solution

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Calculus

Problem 7801

Find the critical points of f(x)=xx2+49f(x)=\frac{x}{x^{2}+49}f(x)=x2+49x​ and calculate its derivative f′(x)f'(x)f′(x).

Problem 7802

Find the horizontal asymptote y=ay=ay=a for the velocity v(t)=−185(1−e−t6)v(t)=-185(1-e^{-\frac{t}{6}})v(t)=−185(1−e−6t​) feet/sec. What is aaa?

Problem 7803

Find the derivative of f(x)=15x(3x+4)4+(3x+4)5f(x) = 15x(3x+4)^{4} + (3x+4)^{5}f(x)=15x(3x+4)4+(3x+4)5.

Problem 7804

Find the critical points of the function f(x)=x2x+18f(x)=x^{2} \sqrt{x+18}f(x)=x2x+18​. A. x=x=x= (comma-separated) B. No critical points.

Problem 7805

Calculate the average rate of change of g(x)=6xg(x) = 6 \sqrt{x}g(x)=6x​ from x=1x = 1x=1 to x=5x = 5x=5.

Problem 7806

Find the velocity v(5)v(5)v(5) and acceleration a(5)a(5)a(5) for the position function s(t)=t2−3ts(t)=t^{2}-3ts(t)=t2−3t.

Problem 7807

Find the acceleration of a particle at t=πt=\pit=π given its position x(t)=5sin⁡2t+2cos⁡3tx(t)=5 \sin 2t + 2 \cos 3tx(t)=5sin2t+2cos3t.

Problem 7808

Calculate the average rate of change of g(x)=−14x3g(x) = -14 x^{3}g(x)=−14x3 from x=−1x = -1x=−1 to x=1x = 1x=1.

Problem 7809

Calculate the average rate of change of k(x)=2x2−15k(x) = 2x^2 - 15k(x)=2x2−15 from x=4x = 4x=4 to x=5x = 5x=5.

Problem 7810

Calculate the average rate of change of k(x)=15xk(x) = \frac{15}{x}k(x)=x15​ from x=7x = 7x=7 to x=17x = 17x=17.

Problem 7811

Calculate velocity and acceleration at t=5t=5t=5 s for s(t)=t4−2ts(t)=t^{4}-2ts(t)=t4−2t. Find v(5)=?v(5)=?v(5)=?

Problem 7812

Calculate the average rate of change of k(x)=18x−18k(x) = \frac{18}{x-18}k(x)=x−1818​ from x=6x=6x=6 to x=8x=8x=8.

Problem 7813

Find the velocity and acceleration of the body at t=π4t=\frac{\pi}{4}t=4π​ sec for the position function s=11sin⁡t+14cos⁡ts=11 \sin t+14 \cos ts=11sint+14cost.

Problem 7814

Show that f′′(0)>0f^{\prime \prime}(0)>0f′′(0)>0 for m=2m=2m=2 in Hill's equation: f(P)=P2k2+P2f(P)=\frac{P^{2}}{k^{2}+P^{2}}f(P)=k2+P2P2​. Simplify.

Problem 7815

A 5 ft ladder leans against a wall. If it slides away at 12ft/s12 \mathrm{ft/s}12ft/s, how fast is the top falling when the foot is 3 ft from the wall?

Problem 7816

Calculate the average rate of change of g(x)=2x4g(x) = 2x^{4}g(x)=2x4 from x=−1x = -1x=−1 to x=1x = 1x=1.

Problem 7817

Calculate the average rate of change of g(x)=−3x+12g(x) = -3 \sqrt{x} + 12g(x)=−3x​+12 from x=5x = 5x=5 to x=7x = 7x=7.

Problem 7818

Find the smaller critical point, t1t_1t1​, of the function F(t)=13t3−5t2+16t+29F(t)=\frac{1}{3} t^{3}-5 t^{2}+16 t+29F(t)=31​t3−5t2+16t+29.

Problem 7819

Calculate the average rate of change of g(x)=18x+10g(x) = \frac{18}{x+10}g(x)=x+1018​ from x=4x=4x=4 to x=9x=9x=9.

Problem 7820

Find the absolute extreme values of f(x)=8x34−xf(x)=8 x^{\frac{3}{4}}-xf(x)=8x43​−x on [0,4096][0,4096][0,4096]. Where are the max values?

Problem 7821

Calculate the average rate of change of f(x)=2x+2f(x) = 2 \sqrt{x} + 2f(x)=2x​+2 on the interval [18,20][18, 20][18,20].

Problem 7822

Why can't we use Rolle's Theorem for f(x)=∣x∣f(x)=|x|f(x)=∣x∣ on [−a,a][-a, a][−a,a] where a>0a>0a>0? Choose: A, B, C, or D.

Problem 7823

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=−2x2+3x−2f(x)=-2 x^{2}+3 x-2f(x)=−2x2+3x−2, simplify your answer.

Problem 7824

Calculate the average rate of change of f(x)=7x+17f(x) = 7 \sqrt{x + 17}f(x)=7x+17​ from x=−16x = -16x=−16 to x=−12x = -12x=−12.

Problem 7825

Find dy/dtd y / d tdy/dt at x=3x=3x=3 for y=x2−3xy=x^{2}-3 xy=x2−3x with dx/dt=3d x / d t=3dx/dt=3.

Problem 7826

A rock creates a ripple in a pond. If the radius grows at 2ft/sec2 \mathrm{ft} / \mathrm{sec}2ft/sec, how fast is the area increasing at 6 ft?

Problem 7827

Find the rate of change of the area of a circle as its radius increases at 7ft/sec7 \mathrm{ft/sec}7ft/sec, in terms of circumference CCC.

Problem 7828

A point moves on the curve 5x3+6y3=xy5 x^{3}+6 y^{3}=x y5x3+6y3=xy. At P=(111,111)P=\left(\frac{1}{11}, \frac{1}{11}\right)P=(111​,111​), yyy increases at 6 units/sec. Find the speed and direction of xxx.

Problem 7829

Find the population change in 2013 using the model P(t)=400(1.065)tP(t)=400(1.065)^{t}P(t)=400(1.065)t (thousands) since 2000.

Problem 7830

Find the derivative of f(x)=5⋅2x+5ex+5x2f(x)=5 \cdot 2^{x}+5 e^{x}+5 x^{2}f(x)=5⋅2x+5ex+5x2. Which option is correct?

Problem 7831

Find the derivative of the function y=e(x2−2)y=e^{\left(x^{2}-2\right)}y=e(x2−2): dy/dx\mathrm{dy} / \mathbf{d x}dy/dx.

Problem 7832

Find the limits of s(x)=3x3−5x2+x−9s(x)=3 x^{3}-5 x^{2}+x-9s(x)=3x3−5x2+x−9 as x→±∞x \to \pm \inftyx→±∞. What do they approach?

Problem 7833

Find points ccc where the Mean Value Theorem holds for f(x)=x3f(x)=x^{3}f(x)=x3 on [−2,2][-2,2][−2,2]. What is c=\mathrm{c}=c=?

Problem 7834

Find the volume decrease rate of a snowball with diameter 3 inches if its radius shrinks at 1in/min1 \mathrm{in} / \mathrm{min}1in/min.

Problem 7835

Determine if the Mean Value Theorem applies to f(x)=−3−x2f(x)=-3-x^{2}f(x)=−3−x2 on [−2,1][-2,1][−2,1] and find any guaranteed points.

Problem 7836

A 10 ft pole casts Joe's shadow as he walks away at 4 ft/s. Find the speed of the shadow's tip when Joe is 11 ft away.

Problem 7837

Check if the Mean Value Theorem applies to f(x)=∣x−5∣f(x)=|x-5|f(x)=∣x−5∣ on [−5,11][-5,11][−5,11] and find guaranteed points if it does.

Problem 7838

Two cyclists leave at the same time: one goes north at 24mph24 \mathrm{mph}24mph, the other east at 10mph10 \mathrm{mph}10mph. Find the rate of distance change after 2 hours.

Problem 7839

Given x4+y4=82x^{4}+y^{4}=82x4+y4=82, find dydx\frac{d y}{d x}dxdy​ and the tangent line at (3,−1)(3,-1)(3,−1).

Problem 7840

Find the critical points of $f(x)=\frac{x}{x^{2}+49}$ and calculate its derivative $f'(x)$ .

Find the horizontal asymptote $y=a$ for the velocity $v(t)=-185(1-e^{-\frac{t}{6}})$ feet/sec. What is $a$ ?

Find the derivative of $f(x) = 15x(3x+4)^{4} + (3x+4)^{5}$ .

Find the critical points of the function $f(x)=x^{2} \sqrt{x+18}$ . A. $x=$ (comma-separated) B. No critical points.

Calculate the average rate of change of $g(x) = 6 \sqrt{x}$ from $x = 1$ to $x = 5$ .

Find the velocity $v(5)$ and acceleration $a(5)$ for the position function $s(t)=t^{2}-3t$ .

Find the acceleration of a particle at $t=\pi$ given its position $x(t)=5 \sin 2t + 2 \cos 3t$ .

Calculate the average rate of change of $g(x) = -14 x^{3}$ from $x = -1$ to $x = 1$ .

Calculate the average rate of change of $k(x) = 2x^2 - 15$ from $x = 4$ to $x = 5$ .

Calculate the average rate of change of $k(x) = \frac{15}{x}$ from $x = 7$ to $x = 17$ .

Calculate velocity and acceleration at $t=5$ s for $s(t)=t^{4}-2t$ . Find $v(5)=?$

Calculate the average rate of change of $k(x) = \frac{18}{x-18}$ from $x=6$ to $x=8$ .

Find the velocity and acceleration of the body at $t=\frac{\pi}{4}$ sec for the position function $s=11 \sin t+14 \cos t$ .

Show that $f^{\prime \prime}(0)>0$ for $m=2$ in Hill's equation: $f(P)=\frac{P^{2}}{k^{2}+P^{2}}$ . Simplify.

A 5 ft ladder leans against a wall. If it slides away at $12 \mathrm{ft/s}$ , how fast is the top falling when the foot is 3 ft from the wall?

Calculate the average rate of change of $g(x) = 2x^{4}$ from $x = -1$ to $x = 1$ .

Calculate the average rate of change of $g(x) = -3 \sqrt{x} + 12$ from $x = 5$ to $x = 7$ .

Find the smaller critical point, $t_1$ , of the function $F(t)=\frac{1}{3} t^{3}-5 t^{2}+16 t+29$ .

Calculate the average rate of change of $g(x) = \frac{18}{x+10}$ from $x=4$ to $x=9$ .

Find the absolute extreme values of $f(x)=8 x^{\frac{3}{4}}-x$ on $[0,4096]$ . Where are the max values?

Calculate the average rate of change of $f(x) = 2 \sqrt{x} + 2$ on the interval $[18, 20]$ .

Why can't we use Rolle's Theorem for $f(x)=|x|$ on $[-a, a]$ where $a>0$ ? Choose: A, B, C, or D.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2 x^{2}+3 x-2$ , simplify your answer.

Calculate the average rate of change of $f(x) = 7 \sqrt{x + 17}$ from $x = -16$ to $x = -12$ .

Find $d y / d t$ at $x=3$ for $y=x^{2}-3 x$ with $d x / d t=3$ .

A rock creates a ripple in a pond. If the radius grows at $2 \mathrm{ft} / \mathrm{sec}$ , how fast is the area increasing at 6 ft?

Find the rate of change of the area of a circle as its radius increases at $7 \mathrm{ft/sec}$ , in terms of circumference $C$ .

A point moves on the curve $5 x^{3}+6 y^{3}=x y$ . At $P=\left(\frac{1}{11}, \frac{1}{11}\right)$ , $y$ increases at 6 units/sec. Find the speed and direction of $x$ .

Find the population change in 2013 using the model $P(t)=400(1.065)^{t}$ (thousands) since 2000.

Find the derivative of $f(x)=5 \cdot 2^{x}+5 e^{x}+5 x^{2}$ . Which option is correct?

Find the derivative of the function $y=e^{\left(x^{2}-2\right)}$ : $\mathrm{dy} / \mathbf{d x}$ .

Find the limits of $s(x)=3 x^{3}-5 x^{2}+x-9$ as $x \to \pm \infty$ . What do they approach?

Find points $c$ where the Mean Value Theorem holds for $f(x)=x^{3}$ on $[-2,2]$ . What is $\mathrm{c}=$ ?

Find the volume decrease rate of a snowball with diameter 3 inches if its radius shrinks at $1 \mathrm{in} / \mathrm{min}$ .

Determine if the Mean Value Theorem applies to $f(x)=-3-x^{2}$ on $[-2,1]$ and find any guaranteed points.

Check if the Mean Value Theorem applies to $f(x)=|x-5|$ on $[-5,11]$ and find guaranteed points if it does.

Two cyclists leave at the same time: one goes north at $24 \mathrm{mph}$ , the other east at $10 \mathrm{mph}$ . Find the rate of distance change after 2 hours.

Given $x^{4}+y^{4}=82$ , find $\frac{d y}{d x}$ and the tangent line at $(3,-1)$ .

Graph $f(x)=\frac{-16 x}{x^{2}+9}$ . Identify intervals of increase, decrease, max/min values, domain, and range.

A 48 cm wire is cut into two pieces to form squares.
(a) Find $A(x)$ , the total area of the squares in terms of $x$ . (b) Determine the side length $x$ that minimizes the area. (c) What is the minimum area of the squares?

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-x^{2}+3x-2$ , where $h \neq 0$ .

Calculate the average rate of change of $f(x)=x^{3}-2 x^{2}+3 x$ between $x=1$ and $x=2$ . Simplify your answer.

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-7 x+7$ , where $h \neq 0$ .

Find the rate of change of the balance $A=5000 e^{0.08 t}$ at $t=1$ , $t=10$ , and $t=50$ years. Round to two decimal places.

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=2x^{2}-2x+8$ , where $h \neq 0$ .

Find the slope $m$ of the tangent to $y=4+5x^{2}-2x^{3}$ at $x=a$ where $m=1$ . Also, find tangent line equations at specified points.

Solve the integral $\int_{-\pi / 2}^{\pi / 3} \sin (4 x) \cos ^{4}(4 x) d x$ using $u=\cos(4x)$ . Find $u$ values at $x=-\pi/2$ and $x=\pi/3$ .

Berechne den Flächeninhalt unter $f(x)=x^{3}$ von 0 bis 1.

Berechnen Sie den Flächeninhalt unter $f(x)=-\frac{1}{3} x^{2}+\frac{4}{3} x+\frac{5}{3}$ im Intervall $[-1,6]$ .

Berechne den Flächeninhalt unter $f(x)=-x^{2}+2 x+3$ und über der x-Achse von $x=0.5$ bis $x=3$ .

Berechnen Sie den Flächeninhalt unter der Funktion $f(x)=x^{3}-4 x^{2}+3 x$ im Intervall [0,3].

Find the derivative of $z=\left(\frac{\tan^{2} t \cos^{2} t}{\sin t}+\frac{1}{2} \csc t \sin 2 t\right)^{2}$ w.r.t. $t$ .

Analysiere die Besucherzahlen eines Vergnügungsparks mit $f(x)=100(x-10)e^{-0,05x}+10000$ . Beantworte a) bis e).

Modellierung der Besucherzahlen im Freizeitpark: $f(x)=100(x-10)e^{-0,05 x}+10000$ . Analysiere Verlauf, Maximum, Abnahme, Zunahme und Rentabilität.

Modelliere die Besucherzahlen $f(x)=100(x-10)e^{-0,05x}+10000$ . Beantworte Fragen zu Verlauf, Maximum, Abnahme und Rentabilität.

Bestimme die Ableitungen der Funktionen: e) $j(x)=\frac{\sin (x)}{\sqrt{3 x-2}}$ , f) $k(x)=(2 x+3)^{3} \cdot \sqrt{x}$ .

Ein Behälter hat zu Beginn 2 m³ Öl. Bestimmen Sie die Ölmenge $g(t)$ für $t>0$ und die Zeit bis 2,5 m³ erreicht sind.

A rock is thrown on Mars with $H=11t-1.86t^{2}$ . Find its velocity after 2s, at $t=a$ , when it hits ground, and impact velocity.

Skizzieren und berechnen Sie die Integrale: a) $\int_{2}^{4} 1,5 \mathrm{dt}$ b) $\int_{0}^{3} \frac{1}{4} x d x$

Bestimmen Sie die Wendepunkte und die Steigung der Tangenten für die Funktionen: a) $f_{a}(x)=x^{3}-a x^{2}$ , b) $f_{a}(x)=\frac{a^{3}}{x}-x^{2}$ , c) $f_{a}(x)=x^{4}-2 a x^{2}+1$ .

Find the local extrema of the function $f a(x)=-a \cdot x^{3}+4 a x$ across its entire domain.

Find the second derivative of $f_{a}(x)=-a \cdot x^{3}+4 a x$ .

Untersuchen Sie die Funktion $A(t)=100-85 \cdot e^{-\lambda-t}$ auf den Grenzwert für $t \to \infty$ . Bestimmen Sie $\lambda$ für $A(2)=80\%$ und die Zeit für $A(t)=90\%$ .

Find the value of \ $300 invested at 9% interest compounded continuously after 3 years. Use$ A(t)=P \bullet e^{r t}$.

Find the derivative of the function $y=2 \sqrt[5]{x^{3}}$ , expressed in radical form without negative exponents.

Find the derivative of the function $f(x)=\frac{3}{2 \sqrt{x}}$ and express it in radical form.

A particle's position is given by $s(t)=t^{3}-3t+1$ . Find velocity, acceleration, and describe motion at $t=0$ and $t=2$ .

A ball's height is given by $s(t)=12t-t^{3}+1$ . Find its release height, max height, velocity at $t=0,1,3$ , and total distance before hitting ground. A stone's height is $s(t)=15t-5t^{2}$ . Find its max height and time to reach it.

Maximize monthly profit $P=14x-0.1x^{2}-200$ . Find production level and maximum profit in \$ for (a) and (b).

A stone's height is given by $s(t)=15t-5t^{2}$ . Find when it reaches max height and the max height.
A diver's height is $s(t)=10+5t-t^{2}$ . Find: a) height of the board, b) time to hit water, c) velocity and acceleration on impact.

Find the limit as $x$ approaches 2 for $\frac{4 \tan (3 x-6)}{3 \ln (3 x-5)}$ in simplest form.

Bestimme die Steigung der Kurve $y=x^{3}-5 x^{2}+6 x$ an ihrem Schnittpunkt mit der $y$ -Achse.