Calculus

Problem 5001

Find the derivative of the function $y=3 x^{2} \sin x$ , i.e., compute $\frac{d y}{d x}$ .

See Solution

Problem 5002

Find $h^{\prime}(2)$ given that $h(x)=f(x) \cdot g(x)$ and the values for $f$ and $g$ at $x=2$ .

See Solution

Problem 5003

Find the derivative of $y=f\left(g\left(2 x^{8}\right)\right)$ , where $f$ and $g$ are differentiable functions.

See Solution

Problem 5004

Find the derivative of $y=\sqrt{f(x)}$ , given that $f$ is differentiable and nonnegative at $x$ .

See Solution

Problem 5005

Approximate the root of $2 x^{3}+3 x+3=0$ using Newton's method starting with $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

See Solution

Problem 5006

Approximate the root of $4 x^{7}+6 x^{4}+4=0$ using Newton's method with initial guess $x_{1}=1$ . Find $x_{2}$ and $x_{3}$ .

See Solution

Problem 5007

For the function $f(x)=x^{3}-3 x+4$ , write Newton's formula as $x \_ {n+1} = x \_ n - \left[\frac{f\left(x \_ n\right)}{f^{\prime}\left(x \_ n\right)}\right]$ .

See Solution

Problem 5008

Find the $\mathrm{x}$ where the max of $y=\sin x-3 x^{2}$ occurs. Use Newton's method, starting with $x=0$ . Give the second guess as a fraction.

See Solution

Problem 5009

Approximate a root of $x^{3}+x+3=0$ using Newton's method with $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

See Solution

Problem 5010

Find the object's height function $s=f(t)$ given $f(2)=90$ and $f'(2)=15$ . Fill in: (a) $f($ ) $=$ (b) $f^{\prime}($ ) $=$

See Solution

Problem 5011

Tumor size $S=2^{t}$ (mm³).
(a) Find the change in size in 4 months.
(b) Calculate the average rate of change in 4 months.
(c) Estimate growth rate at $t=4$ .

See Solution

Problem 5012

Find approximate values for $f^{\prime}(x)$ at $x = 0, 4, 8, 12, 16$ using right or left-hand approximations.

See Solution

Problem 5013

Find the limit: $\lim _{x \rightarrow 3} e^{x} \cos \left(\frac{\pi x}{3}\right)$ .

See Solution

Problem 5014

Find approximate values for $f^{\prime}(x)$ at $x = 0, 2, 4, 6, 8$ using right-hand or left-hand approximations.

See Solution

Problem 5015

Find the limit: $\lim _{x \rightarrow-3^{+}} \frac{2 x^{2}-9 x+9}{x^{2}-9}$ .

See Solution

Problem 5016

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

See Solution

Problem 5017

Find the limit as $x$ approaches 0 for the expression $\frac{3 \sin 3 x}{3 x}$ .

See Solution

Problem 5018

Find the limit: $\lim _{\theta \rightarrow 0} \frac{3-3 \cos \theta}{\theta}$ .

See Solution

Problem 5019

Find the population size of a species modeled by $P(t)=\frac{340}{1+9 e^{-0.1 t}}$ after 4 and 10 years.

See Solution

Problem 5020

How long does it take for a bacteria population to double with a growth rate of $5.6\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 5021

A radioactive substance starts at $393 \mathrm{~kg}$ and decays at $17 \%$ daily. Find its mass after 4 days. Round to $\square \mathrm{kg}$ .

See Solution

Problem 5022

Find the hourly growth rate of a bacteria population that grows from 3000 to 3131 in 2 hours. Express as a percentage. $|n \%$

See Solution

Problem 5023

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 \sqrt{x-4}$ using the limit definition.

See Solution

Problem 5024

Determine if the cereal consumption rate was higher in 2001 or 2009 using $C(t)=-0.0036 t^{3}+0.096 t^{2}-0.349 t+13$ . Calculate $C'(11)$ and $C'(19)$ .

See Solution

Problem 5025

Find the first and second derivatives of the function $f(x)=-3 x^{3}+5 x-2$ using the limit definition.

See Solution

Problem 5026

Given the position $s=f(t)=6 t^{3}+4 t+9$ , find:
(a) Velocity $v(t)$ ,
(b) Velocity at $t=3$ seconds,
(c) Acceleration $a(t)$ ,
(d) Acceleration at $t=3$ seconds.

See Solution

Problem 5027

Find the price elasticity of demand $E$ for tissues at $p = \$ 24$ using $q=9,409-194 p+p^{2}$ . Round to three decimals.

See Solution

Problem 5028

Find the derivative of the function $f(x)=4 x^{7}-5 x^{6}$ . What is $f^{\prime}(x)=?$

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Problem 5029

A fried chicken franchise has a demand equation $q=\frac{39}{p^{0.85}}$ . Find the price elasticity $E(p)$ at $p=\$4.00$ . What is the % decrease in demand for a 1% price increase?

See Solution

Problem 5030

Find the price elasticity of demand $E$ for tissues at $p=\$ 24$ using $q=9,409-194 p+p^{2}$ . Round to three decimals. Interpret $E$ .

See Solution

Problem 5031

A tank has $20 \mathrm{~kg}$ of salt in $5000 \mathrm{~L}$ of water. Brine with $0.03 \mathrm{~kg/L}$ enters at $25 \mathrm{~L/min}$ . How much salt after 30 minutes?

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Problem 5032

Untersuchen Sie die Funktion $f(x)=5x \cdot e^{-\frac{1}{2}x}$ durch Differenzieren oder Integrieren.

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Problem 5033

Berechnen Sie die Integrale: a) $\int(2 x+1)^{2} d x$ , b) $\int 6 e^{2 x+4} d x$ , c) $\int \frac{1}{(3 x+2)^{2}} d x$ , d) $\int(2-\cos (\pi x)) d x$ .

See Solution

Problem 5034

Find the second derivative of $V^{\prime}(h)=1000 h-130 h^{2}+4 h^{3}$ .

See Solution

Problem 5035

Find the second derivative of $v^{\prime}(h)=1000-260 h+12 h^{2}$ .

See Solution

Problem 5036

Find the integral of $(t-2)^{2}$ with respect to $t$ .

See Solution

Problem 5037

Find the derivative of $f(x) = 5x \cdot e^{-\frac{1}{2} x}$ and verify your steps.

See Solution

Problem 5038

Bestimme die unbestimmten Integrale: a) $\int\left(x^{2}-1\right) d x$ , b) $\int a t d t$ , c) $\int\left(x^{2}+3 x+1\right) d x$ , d) $\int(\sqrt{a}-x \sqrt{b}) d x$ , e) $\int\left(x+\frac{1}{x^{2}}+\frac{1}{x^{3}}\right) d x$ , f) $\int\left(a b+b^{4}\right) d b$ , g) $\int(t-2)^{2} d t$ .

See Solution

Problem 5039

Soit $f(x)=3x-3x\sqrt{x}$ . Calculez $f^{\prime \prime}(x)$ et étudiez son signe. Montrez que $f$ est concave et interprétez graphiquement.

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Problem 5040

Calculate the antiderivative of the function $f(x) = x^n - x$ .

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Problem 5041

Leite die Funktion $f$ einmal ab für: (a) $f(x)=\sin (3 x+3)$ , (b) $f(t)=2 \cdot \cos (\pi t-2)$ , (c) $f(t)=-\cos (5 t)$ , (d) $f(x)=-4 \cdot \sin (\pi-x)$ , (e) $f(t)=\sin (t-6)$ , (f) $f(x)=\cos (3 x+\pi)$ .

See Solution

Problem 5042

a) Zeigen Sie, dass der Hochpunkt $H\left(x_{0} \mid 0\right)$ von $f$ ein Tiefpunkt von $g(x)=-x^{2} \cdot f(x)$ ist. b) Untersuchen Sie, ob der Sattelpunkt $S\left(x_{0} \mid 0\right)$ von $f$ auch ein Sattelpunkt von $g(x)=x \cdot f(x)$ ist.

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Problem 5043

Wie verändert sich der Flächeninhalt unter $f$ im Intervall $[0 ; a]$ zu $[0 ; 2a]$ für $a>0$ ? a) $f(x)=x$ b) $f(x)=x^{2}$ c) $f(x)=2x$ d) $f(x)=x^{3}$

See Solution

Problem 5044

Berechnen Sie die Nullstellen der Funktionen und das Integral im Intervall der Nullstellen: a) $f(x)=\frac{1}{2} x^{2}-4 x+\frac{15}{2}$ , b) $f(x)=x^{3}+3 x^{2}$ , c) $f(x)=(x^{2}+4)(x^{2}-4)$ .

See Solution

Problem 5045

Finde $k \in \mathbb{R}$ für die folgenden Integrale: a) $\int_{0}^{k} x \, dx=8$ b) $\int_{0}^{k}(x+3) \, dx=8$ c) $\int_{0}^{k} \, dx=3 \int_{0}^{1} \, dx$ d) $\int_{-k}^{0} t^{2} \, dt=2$

See Solution

Problem 5046

Analyze the graph of $f(x)=-(x+7)^{2}+5$ . Is the rate of change of $f$ increasing or decreasing? Explain.

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Problem 5047

Crunchy Cookie Inc.'s profit model is $P(x)=5000+1000x-5x^{2}$ .
a. Describe the profit change per extra dollar spent. b. Calculate the average profit change from $x=50$ to $x=100$ (units?). c. Calculate the average profit change from $x=100$ to $x=150$ (units?).

See Solution

Problem 5048

Crunchy Cookie Inc. profit model: $P(x)=5000+1000 x-5 x^{2}$ (in thousands).
a. How does profit change with each extra dollar spent? b. Average profit change from $x=50$ to $x=100$ : \ $250,000. c. Average profit change from$ x=100 $to$ x=150 $: \$-250,000. d. Predict average change from$ x=150 $to$ x=200$. e. Is every advertising dollar equally effective?

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Problem 5049

Berechne die Integrale: a) $\int_{-4}^{2} \frac{1}{4} x d x$ und c) $\int_{1}^{-2}(1,5 t+0,5) d t$ .

See Solution

Problem 5050

Gegeben ist die Funktion $f$ , die zweimal differenzierbar ist und für $x>0$ gilt: $f(x)>0$ , $f'(x)>0$ , $f''(x)>0$ . Untersuchen Sie die Eigenschaften von $g(x)$ für A-D.

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Problem 5051

Prove that $\frac{\cos 2 x \sin 4 x-\sin 2 x \cos 4 x}{\sin ^{2} x-\cos ^{2} x}=-\tan 2 x$ and find $\int\left(\frac{\cos 2 x \sin 4 x-\sin 2 x \cos 4 x}{\sin ^{2} x-\cos ^{2} x}\right)^{2} d x$ .

See Solution

Problem 5052

Bestimme die Ableitung der Funktion $y=\varphi(x)=\frac{1}{\sqrt{2 \pi}} \cdot e^{-\frac{1}{2} x^{2}}$ in Teilschritten.

See Solution

Problem 5053

Die Geschwindigkeit eines Flugkörpers in den ersten 7 Sekunden ist $v(t)=0,1 t^{2}$ . Bestimme die Ableitungsfunktion und skizziere beide Graphen. Was bedeutet die momentane Änderungsrate?

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Problem 5054

Bestimme die Ableitung von $u(x) = \frac{3}{x}$ .

See Solution

Problem 5055

Bestimmen Sie die Ableitung von $u(x)=-3 \sin (x)$ .

See Solution

Problem 5056

Gegeben ist eine differenzierbare Funktion g. Finde die Ableitungen von $f_{1}(x)=g(3 x)$ , $f_{2}(x)=g(1-x)$ und $f_{3}(x)=g\left(\frac{1}{x}\right)$ .

See Solution

Problem 5057

Erlösfunktion $E(x)=-x^{2}+4x$ , $D_{\text {б̆k }}(E)=[0;4]$ . a) Finde $E^{\prime}$ . b) Zeichne $E$ und $E^{\prime}$ . c) Erlösentwicklung beschreiben. d) Bedeutung der Änderungsrate. e) Verlauf von $E^{\prime}$ interpretieren. f) Berechne und interpretiere $E^{\prime}(1)$ und $E^{\prime}(4)$ .

See Solution

Problem 5058

Evaluate the rate of change of $f(x) = 2x^{2} + 1$ over the interval $12 \leq x \leq 72$ .

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Problem 5059

Berechne das Integral $P(D) \approx \int_{-1,6}^{1,6} \varphi(x) dx$ mit acht Teilintervallen, wobei $\varphi(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}}$ .

See Solution

Problem 5060

Find the critical points of the function $F(t)=\frac{1}{3} t^{3}-4.5 t^{2}+18 t+27$ and characterize them.

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Problem 5061

Find the extremum of the function $g(z)=-28\left(e^{3 z}+5 e^{-5 z}\right)^{6}$ . Round answers to two digits.

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Problem 5062

Identify which scenario (1, 2, or 3) matches the optimization problem: $\max _{c} R \cdot \ln (c+1)-k \cdot c$ .

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Problem 5063

Find the derivative of the function $n(x)=\frac{2x+1}{x^2+1}$ .

See Solution

Problem 5064

Find the differential equation for $\tan (x-y)=4 x$ given $\frac{d y}{d x}=e^{y}$ .

See Solution

Problem 5065

Find the derivative of $g(t)=6 \cdot \sqrt{t}+\frac{2}{t^{2}}$ .

See Solution

Problem 5066

Find the derivative of $k(t)=(t^{2}-4t+5)^{3}$ .

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Problem 5067

A ball is dropped from $64 \mathrm{~m}$ and rebounds three-fourths of its previous fall. How far does it travel before stopping?

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Problem 5068

Find the derivative of $h(x)=(x^{4}+1)(x^{3}-2x)$ using the product rule. Let $f(x)=x^{4}+1$ and $g(x)=x^{3}-2x$ .

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Problem 5069

Zeichnen Sie die Graphen der Funktionen und prüfen Sie deren Stetigkeit: a) $H(x)=0$ für $x \leq 0$ , $H(x)=1$ für $x>0$ ; b) $\operatorname{sgn}(x)$ ; c) $f(x)=\frac{2}{(x+2)^{3}}$ ; d) $f(x)=\frac{x^{2}-1}{x+1}$ .

See Solution

Problem 5070

Find the tangent line equation to the curve $y=2 e^{x}+x$ at the point $(0,2)$ .

See Solution

Problem 5071

Bestimmen Sie den Grenzwert von $f(x)=-2 x^{3}+x^{2}$ für $x \rightarrow \pm \infty$ .

See Solution

Problem 5072

Differentiate the function without simplifying: $f(x)=x^{3/2}+x^{-3}$ .

See Solution

Problem 5073

Find the extreme values of $z=(x-17)^{2}+(y-14)^{2}+90$ with $x \geq 0$ , $y \geq 0$ , and $14x+17y \leq 238$ .
Complete: $f_{\min }=\square$ at $(x,y)=(\square, \square)$ ; $f_{\max }=\square$ at $(x,y)=(\square, \square)$ .

See Solution

Problem 5074

Berechne die Trapezstreifensumme $T_{4}$ für $f(x)=x^{2}$ im Intervall $[0; 1]$ und die Differenz zu $A$ . Vergleiche mit $U_{4}$ und $O_{4}$ .

See Solution

Problem 5075

Maximize $F(x) = 4.5 x^{2}-72 x+12$ for $-3 \leq x \leq 15$ . Find $x^{}$ and $F(x^{})$ .

See Solution

Problem 5076

Compute the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0}\left\langle e^{\theta+2}, \cos \theta, \frac{1}{\sqrt{\theta+16}}\right\rangle$

See Solution

Problem 5077

Find which pendulum configuration (A-D) has the maximum potential energy given by $U(x)=l(1-\cos x)$ .

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Problem 5078

Two isotopes decay with different half-lives: $X$ (0.25s) and $Y$ (0.5s). Will $X$ outlast $Y$ in quantity over time? Why?

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Problem 5079

Find the limit of the vector function as $t$ approaches -3: $\lim_{t \to -3} \ln(t+5) \mathbf{i} + \sin\left(\frac{\pi t}{2}\right) \mathbf{j} - 4 \mathbf{k}$ .

See Solution

Problem 5080

Finde die Stammfunktion für die folgenden Funktionen: a) $f(x)=(2 x+8)^{3}$ , b) $f(x)=5 \cdot(2-3 x)^{7}$ , c) $f(x)=\frac{3}{(2 x-1)^{2}}$ , d) $f(x)=(x-4)^{3}+\frac{1}{(x-4)^{3}}$ , e) $f(x)=-1,5 e^{2 x+3}$ , f) $f(x)=e^{m \cdot x+n}$ , g) $f(x)=\frac{1}{\sqrt{x+6}}$ , h) $f(t)=\sin (a \cdot t+b)$ , i) $f(t)=e^{4 t-3}+\cos (4 t-3)$ .

See Solution

Problem 5081

Find the $52^{\text{nd}}$ derivative of $\sin x$ .

See Solution

Problem 5082

How much will \$800 grow in 4 years at a continuous compounding rate of 3.15%?

See Solution

Problem 5083

Find the amount of radon-222 after 10 days using $A(t)=1.2(10)^{-0.0788 t}$ and time to reach $0.012 \mathrm{~g}$ .

See Solution

Problem 5084

Leiten Sie die Funktion $f(x)=-e^{-x}+e^{x}$ ab.

See Solution

Problem 5085

Ableiten von $f(x)=-e^{-x}+e^{x}$ .

See Solution

Problem 5086

Find the derivative of $f(x)=-5 x^{2} \ln(2 x^{2}-4 x^{-2})$ . What is $f'(x)$ ?

See Solution

Problem 5087

Rewrite $f(x)=9 \ln \left(\frac{15}{x}\right)$ using logarithm properties and find $f^{\prime}(x)$ .

See Solution

Problem 5088

Find the derivative of the function $f(x) = -14 \ln x + 3x^{2} - 10$ . What is $f'(x)$ ?

See Solution

Problem 5089

Find the derivative $f^{\prime}(x)$ for the function $f(x)=6 x-\ln x^{2}$ .

See Solution

Problem 5090

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

See Solution

Problem 5091

Find the derivative of $f(x)=x^{-1} \cdot(2x+3)^{2}$ .

See Solution

Problem 5092

Bestimme die Ableitung von $f(x)=(5-x^{4})^{5} \cdot(1-4 x)$ .

See Solution

Problem 5093

A potato is launched from a 50-ft building with an initial speed of $75 \, \mathrm{ft/s}$ . Find its velocity when it hits the ground.

See Solution

Problem 5094

Gegeben ist die Funktion $f: x \mapsto \frac{1}{x} \cdot \cos (x)$ . Bestimmen Sie die Definitionsmenge, Nullstellen und Grenzwert. Zeichnen Sie $f$ und $g: x \mapsto \frac{1}{x}$ . Untersuchen Sie die Funktion $h: x \mapsto \frac{1}{x}+\cos (x)$ .

See Solution

Problem 5095

Find the velocity of a potato launched from a 50-ft building with initial speed $75 \, \text{ft/s}$ using $s(t)=-16t^{2}+75t+50$ .

See Solution

Problem 5096

Find the derivative of the function $f(\underline{x})=(A B)^{n} \underline{x}$ for $m$ -by- $m$ matrices $A$ , $B$ and integer $n>0$ .

See Solution

Problem 5097

Find the rate of change of $y$ with respect to $x$ for the function $F(x, y)=0$ using implicit differentiation.

See Solution

Problem 5098

Berechnen Sie die Ableitung von $f(x)=3x \cdot \cos(2x^{2})$ .

See Solution

Problem 5099

Berechne die Ableitungen der Funktionen: b) $f(x)=(3 x+4)^{2} \cdot \sin (x)$ e) $f(x)=(5-4 x)^{3} \cdot x^{-2}$ h) $f(x)=(2 x-1) \cdot \sqrt{x^{3}-1}$

See Solution

Problem 5100

Find the partial derivative of $f=f(x, y)$ with respect to $v$ where $x=x(u, v)$ and $y=y(u, v)$ .

See Solution

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Calculus

Problem 5001

Find the derivative of the function y=3x2sin⁡xy=3 x^{2} \sin xy=3x2sinx, i.e., compute dydx\frac{d y}{d x}dxdy​.

Problem 5002

Find h′(2)h^{\prime}(2)h′(2) given that h(x)=f(x)⋅g(x)h(x)=f(x) \cdot g(x)h(x)=f(x)⋅g(x) and the values for fff and ggg at x=2x=2x=2.

Problem 5003

Find the derivative of y=f(g(2x8))y=f\left(g\left(2 x^{8}\right)\right)y=f(g(2x8)), where fff and ggg are differentiable functions.

Problem 5004

Find the derivative of y=f(x)y=\sqrt{f(x)}y=f(x)​, given that fff is differentiable and nonnegative at xxx.

Problem 5005

Approximate the root of 2x3+3x+3=02 x^{3}+3 x+3=02x3+3x+3=0 using Newton's method starting with x1=−1x_{1}=-1x1​=−1. Find x2x_{2}x2​ and x3x_{3}x3​.

Problem 5006

Approximate the root of 4x7+6x4+4=04 x^{7}+6 x^{4}+4=04x7+6x4+4=0 using Newton's method with initial guess x1=1x_{1}=1x1​=1. Find x2x_{2}x2​ and x3x_{3}x3​.

Problem 5007

For the function f(x)=x3−3x+4f(x)=x^{3}-3 x+4f(x)=x3−3x+4, write Newton's formula as x_n+1=x_n−[f(x_n)f′(x_n)]x \_ {n+1} = x \_ n - \left[\frac{f\left(x \_ n\right)}{f^{\prime}\left(x \_ n\right)}\right]x_n+1=x_n−[f′(x_n)f(x_n)​].

Problem 5008

Find the x\mathrm{x}x where the max of y=sin⁡x−3x2y=\sin x-3 x^{2}y=sinx−3x2 occurs. Use Newton's method, starting with x=0x=0x=0. Give the second guess as a fraction.

Problem 5009

Approximate a root of x3+x+3=0x^{3}+x+3=0x3+x+3=0 using Newton's method with x1=−1x_{1}=-1x1​=−1. Find x2x_{2}x2​ and x3x_{3}x3​.

Problem 5010

Find the object's height function s=f(t)s=f(t)s=f(t) given f(2)=90f(2)=90f(2)=90 and f′(2)=15f'(2)=15f′(2)=15. Fill in: (a) f(f(f( )=== (b) f′(f^{\prime}(f′( )===

Problem 5011

Tumor size S=2tS=2^{t}S=2t (mm³). (a) Find the change in size in 4 months. (b) Calculate the average rate of change in 4 months. (c) Estimate growth rate at t=4t=4t=4.

Problem 5012

Find approximate values for f′(x)f^{\prime}(x)f′(x) at x=0,4,8,12,16x = 0, 4, 8, 12, 16x=0,4,8,12,16 using right or left-hand approximations.

Problem 5013

Find the limit: lim⁡x→3excos⁡(πx3)\lim _{x \rightarrow 3} e^{x} \cos \left(\frac{\pi x}{3}\right)limx→3​excos(3πx​).

Problem 5014

Find approximate values for f′(x)f^{\prime}(x)f′(x) at x=0,2,4,6,8x = 0, 2, 4, 6, 8x=0,2,4,6,8 using right-hand or left-hand approximations.

Problem 5015

Find the limit: lim⁡x→−3+2x2−9x+9x2−9\lim _{x \rightarrow-3^{+}} \frac{2 x^{2}-9 x+9}{x^{2}-9}limx→−3+​x2−92x2−9x+9​.

Problem 5016

Find the limit: lim⁡x→−3+2x−3x+3\lim _{x \rightarrow-3^{+}} \frac{2 x-3}{x+3}limx→−3+​x+32x−3​.

Problem 5017

Find the limit as xxx approaches 0 for the expression 3sin⁡3x3x\frac{3 \sin 3 x}{3 x}3x3sin3x​.

Problem 5018

Find the limit: lim⁡θ→03−3cos⁡θθ\lim _{\theta \rightarrow 0} \frac{3-3 \cos \theta}{\theta}limθ→0​θ3−3cosθ​.

Problem 5019

Find the population size of a species modeled by P(t)=3401+9e−0.1tP(t)=\frac{340}{1+9 e^{-0.1 t}}P(t)=1+9e−0.1t340​ after 4 and 10 years.

Problem 5020

How long does it take for a bacteria population to double with a growth rate of 5.6%5.6\%5.6% per hour? Round to the nearest hundredth.

Problem 5021

A radioactive substance starts at 393 kg393 \mathrm{~kg}393 kg and decays at 17%17 \%17% daily. Find its mass after 4 days. Round to □kg\square \mathrm{kg}□kg.

Problem 5022

Find the hourly growth rate of a bacteria population that grows from 3000 to 3131 in 2 hours. Express as a percentage. ∣n%|n \%∣n%

Problem 5023

Find the derivative f′(x)f^{\prime}(x)f′(x) of the function f(x)=4x−4f(x)=4 \sqrt{x-4}f(x)=4x−4​ using the limit definition.

Problem 5024

Determine if the cereal consumption rate was higher in 2001 or 2009 using C(t)=−0.0036t3+0.096t2−0.349t+13C(t)=-0.0036 t^{3}+0.096 t^{2}-0.349 t+13C(t)=−0.0036t3+0.096t2−0.349t+13. Calculate C′(11)C'(11)C′(11) and C′(19)C'(19)C′(19).

Problem 5025

Find the first and second derivatives of the function f(x)=−3x3+5x−2f(x)=-3 x^{3}+5 x-2f(x)=−3x3+5x−2 using the limit definition.

Problem 5026

Given the position s=f(t)=6t3+4t+9s=f(t)=6 t^{3}+4 t+9s=f(t)=6t3+4t+9, find:(a) Velocity v(t)v(t)v(t), (b) Velocity at t=3t=3t=3 seconds, (c) Acceleration a(t)a(t)a(t), (d) Acceleration at t=3t=3t=3 seconds.

Problem 5027

Find the price elasticity of demand EEE for tissues at p=$24p = \$ 24p=$24 using q=9,409−194p+p2q=9,409-194 p+p^{2}q=9,409−194p+p2. Round to three decimals.

Problem 5028

Find the derivative of the function f(x)=4x7−5x6f(x)=4 x^{7}-5 x^{6}f(x)=4x7−5x6. What is f′(x)=?f^{\prime}(x)=?f′(x)=?

Problem 5029

A fried chicken franchise has a demand equation q=39p0.85q=\frac{39}{p^{0.85}}q=p0.8539​. Find the price elasticity E(p)E(p)E(p) at p=$4.00p=\$4.00p=$4.00. What is the % decrease in demand for a 1% price increase?

Problem 5030

Find the price elasticity of demand EEE for tissues at p=$24p=\$ 24p=$24 using q=9,409−194p+p2q=9,409-194 p+p^{2}q=9,409−194p+p2. Round to three decimals. Interpret EEE.

Problem 5031

A tank has 20 kg20 \mathrm{~kg}20 kg of salt in 5000 L5000 \mathrm{~L}5000 L of water. Brine with 0.03 kg/L0.03 \mathrm{~kg/L}0.03 kg/L enters at 25 L/min25 \mathrm{~L/min}25 L/min. How much salt after 30 minutes?

Problem 5032

Untersuchen Sie die Funktion f(x)=5x⋅e−12xf(x)=5x \cdot e^{-\frac{1}{2}x}f(x)=5x⋅e−21​x durch Differenzieren oder Integrieren.

Problem 5033

Berechnen Sie die Integrale: a) ∫(2x+1)2dx\int(2 x+1)^{2} d x∫(2x+1)2dx, b) ∫6e2x+4dx\int 6 e^{2 x+4} d x∫6e2x+4dx, c) ∫1(3x+2)2dx\int \frac{1}{(3 x+2)^{2}} d x∫(3x+2)21​dx, d) ∫(2−cos⁡(πx))dx\int(2-\cos (\pi x)) d x∫(2−cos(πx))dx.

Problem 5034

Find the second derivative of V′(h)=1000h−130h2+4h3V^{\prime}(h)=1000 h-130 h^{2}+4 h^{3}V′(h)=1000h−130h2+4h3.

Problem 5035

Find the second derivative of v′(h)=1000−260h+12h2v^{\prime}(h)=1000-260 h+12 h^{2}v′(h)=1000−260h+12h2.

Problem 5036

Find the integral of (t−2)2(t-2)^{2}(t−2)2 with respect to ttt.

Problem 5037

Find the derivative of f(x)=5x⋅e−12xf(x) = 5x \cdot e^{-\frac{1}{2} x}f(x)=5x⋅e−21​x and verify your steps.

Problem 5038

Problem 5039

Soit f(x)=3x−3xxf(x)=3x-3x\sqrt{x}f(x)=3x−3xx​. Calculez f′′(x)f^{\prime \prime}(x)f′′(x) et étudiez son signe. Montrez que fff est concave et interprétez graphiquement.

Problem 5040

Calculate the antiderivative of the function f(x)=xn−xf(x) = x^n - xf(x)=xn−x.

Find the derivative of the function $y=3 x^{2} \sin x$ , i.e., compute $\frac{d y}{d x}$ .

Find $h^{\prime}(2)$ given that $h(x)=f(x) \cdot g(x)$ and the values for $f$ and $g$ at $x=2$ .

Find the derivative of $y=f\left(g\left(2 x^{8}\right)\right)$ , where $f$ and $g$ are differentiable functions.

Find the derivative of $y=\sqrt{f(x)}$ , given that $f$ is differentiable and nonnegative at $x$ .

Approximate the root of $2 x^{3}+3 x+3=0$ using Newton's method starting with $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

Approximate the root of $4 x^{7}+6 x^{4}+4=0$ using Newton's method with initial guess $x_{1}=1$ . Find $x_{2}$ and $x_{3}$ .

For the function $f(x)=x^{3}-3 x+4$ , write Newton's formula as $x \_ {n+1} = x \_ n - \left[\frac{f\left(x \_ n\right)}{f^{\prime}\left(x \_ n\right)}\right]$ .

Find the $\mathrm{x}$ where the max of $y=\sin x-3 x^{2}$ occurs. Use Newton's method, starting with $x=0$ . Give the second guess as a fraction.

Approximate a root of $x^{3}+x+3=0$ using Newton's method with $x_{1}=-1$ . Find $x_{2}$ and $x_{3}$ .

Find the object's height function $s=f(t)$ given $f(2)=90$ and $f'(2)=15$ . Fill in: (a) $f($ ) $=$ (b) $f^{\prime}($ ) $=$

Tumor size $S=2^{t}$ (mm³).
(a) Find the change in size in 4 months.
(b) Calculate the average rate of change in 4 months.
(c) Estimate growth rate at $t=4$ .

Find approximate values for $f^{\prime}(x)$ at $x = 0, 4, 8, 12, 16$ using right or left-hand approximations.

Find the limit: $\lim _{x \rightarrow 3} e^{x} \cos \left(\frac{\pi x}{3}\right)$ .

Find approximate values for $f^{\prime}(x)$ at $x = 0, 2, 4, 6, 8$ using right-hand or left-hand approximations.

Find the limit: $\lim _{x \rightarrow-3^{+}} \frac{2 x^{2}-9 x+9}{x^{2}-9}$ .

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

Find the limit as $x$ approaches 0 for the expression $\frac{3 \sin 3 x}{3 x}$ .

Find the limit: $\lim _{\theta \rightarrow 0} \frac{3-3 \cos \theta}{\theta}$ .

Find the population size of a species modeled by $P(t)=\frac{340}{1+9 e^{-0.1 t}}$ after 4 and 10 years.

How long does it take for a bacteria population to double with a growth rate of $5.6\%$ per hour? Round to the nearest hundredth.

A radioactive substance starts at $393 \mathrm{~kg}$ and decays at $17 \%$ daily. Find its mass after 4 days. Round to $\square \mathrm{kg}$ .

Find the hourly growth rate of a bacteria population that grows from 3000 to 3131 in 2 hours. Express as a percentage. $|n \%$

Find the derivative $f^{\prime}(x)$ of the function $f(x)=4 \sqrt{x-4}$ using the limit definition.

Determine if the cereal consumption rate was higher in 2001 or 2009 using $C(t)=-0.0036 t^{3}+0.096 t^{2}-0.349 t+13$ . Calculate $C'(11)$ and $C'(19)$ .

Find the first and second derivatives of the function $f(x)=-3 x^{3}+5 x-2$ using the limit definition.

Given the position $s=f(t)=6 t^{3}+4 t+9$ , find:
(a) Velocity $v(t)$ ,
(b) Velocity at $t=3$ seconds,
(c) Acceleration $a(t)$ ,
(d) Acceleration at $t=3$ seconds.

Find the price elasticity of demand $E$ for tissues at $p = \$ 24$ using $q=9,409-194 p+p^{2}$ . Round to three decimals.

Find the derivative of the function $f(x)=4 x^{7}-5 x^{6}$ . What is $f^{\prime}(x)=?$

A fried chicken franchise has a demand equation $q=\frac{39}{p^{0.85}}$ . Find the price elasticity $E(p)$ at $p=\$4.00$ . What is the % decrease in demand for a 1% price increase?

Find the price elasticity of demand $E$ for tissues at $p=\$ 24$ using $q=9,409-194 p+p^{2}$ . Round to three decimals. Interpret $E$ .

A tank has $20 \mathrm{~kg}$ of salt in $5000 \mathrm{~L}$ of water. Brine with $0.03 \mathrm{~kg/L}$ enters at $25 \mathrm{~L/min}$ . How much salt after 30 minutes?

Untersuchen Sie die Funktion $f(x)=5x \cdot e^{-\frac{1}{2}x}$ durch Differenzieren oder Integrieren.

Berechnen Sie die Integrale: a) $\int(2 x+1)^{2} d x$ , b) $\int 6 e^{2 x+4} d x$ , c) $\int \frac{1}{(3 x+2)^{2}} d x$ , d) $\int(2-\cos (\pi x)) d x$ .

Find the second derivative of $V^{\prime}(h)=1000 h-130 h^{2}+4 h^{3}$ .

Find the second derivative of $v^{\prime}(h)=1000-260 h+12 h^{2}$ .

Find the integral of $(t-2)^{2}$ with respect to $t$ .

Find the derivative of $f(x) = 5x \cdot e^{-\frac{1}{2} x}$ and verify your steps.

Soit $f(x)=3x-3x\sqrt{x}$ . Calculez $f^{\prime \prime}(x)$ et étudiez son signe. Montrez que $f$ est concave et interprétez graphiquement.

Calculate the antiderivative of the function $f(x) = x^n - x$ .

Wie verändert sich der Flächeninhalt unter $f$ im Intervall $[0 ; a]$ zu $[0 ; 2a]$ für $a>0$ ? a) $f(x)=x$ b) $f(x)=x^{2}$ c) $f(x)=2x$ d) $f(x)=x^{3}$

Berechnen Sie die Nullstellen der Funktionen und das Integral im Intervall der Nullstellen: a) $f(x)=\frac{1}{2} x^{2}-4 x+\frac{15}{2}$ , b) $f(x)=x^{3}+3 x^{2}$ , c) $f(x)=(x^{2}+4)(x^{2}-4)$ .

Analyze the graph of $f(x)=-(x+7)^{2}+5$ . Is the rate of change of $f$ increasing or decreasing? Explain.

Berechne die Integrale: a) $\int_{-4}^{2} \frac{1}{4} x d x$ und c) $\int_{1}^{-2}(1,5 t+0,5) d t$ .

Gegeben ist die Funktion $f$ , die zweimal differenzierbar ist und für $x>0$ gilt: $f(x)>0$ , $f'(x)>0$ , $f''(x)>0$ . Untersuchen Sie die Eigenschaften von $g(x)$ für A-D.

Bestimme die Ableitung der Funktion $y=\varphi(x)=\frac{1}{\sqrt{2 \pi}} \cdot e^{-\frac{1}{2} x^{2}}$ in Teilschritten.

Die Geschwindigkeit eines Flugkörpers in den ersten 7 Sekunden ist $v(t)=0,1 t^{2}$ . Bestimme die Ableitungsfunktion und skizziere beide Graphen. Was bedeutet die momentane Änderungsrate?

Bestimme die Ableitung von $u(x) = \frac{3}{x}$ .

Bestimmen Sie die Ableitung von $u(x)=-3 \sin (x)$ .

Gegeben ist eine differenzierbare Funktion g. Finde die Ableitungen von $f_{1}(x)=g(3 x)$ , $f_{2}(x)=g(1-x)$ und $f_{3}(x)=g\left(\frac{1}{x}\right)$ .

Evaluate the rate of change of $f(x) = 2x^{2} + 1$ over the interval $12 \leq x \leq 72$ .

Berechne das Integral $P(D) \approx \int_{-1,6}^{1,6} \varphi(x) dx$ mit acht Teilintervallen, wobei $\varphi(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}}$ .

Find the critical points of the function $F(t)=\frac{1}{3} t^{3}-4.5 t^{2}+18 t+27$ and characterize them.

Find the extremum of the function $g(z)=-28\left(e^{3 z}+5 e^{-5 z}\right)^{6}$ . Round answers to two digits.

Identify which scenario (1, 2, or 3) matches the optimization problem: $\max _{c} R \cdot \ln (c+1)-k \cdot c$ .

Find the derivative of the function $n(x)=\frac{2x+1}{x^2+1}$ .

Find the differential equation for $\tan (x-y)=4 x$ given $\frac{d y}{d x}=e^{y}$ .

Find the derivative of $g(t)=6 \cdot \sqrt{t}+\frac{2}{t^{2}}$ .

Find the derivative of $k(t)=(t^{2}-4t+5)^{3}$ .

A ball is dropped from $64 \mathrm{~m}$ and rebounds three-fourths of its previous fall. How far does it travel before stopping?

Find the derivative of $h(x)=(x^{4}+1)(x^{3}-2x)$ using the product rule. Let $f(x)=x^{4}+1$ and $g(x)=x^{3}-2x$ .

Find the tangent line equation to the curve $y=2 e^{x}+x$ at the point $(0,2)$ .

Bestimmen Sie den Grenzwert von $f(x)=-2 x^{3}+x^{2}$ für $x \rightarrow \pm \infty$ .

Differentiate the function without simplifying: $f(x)=x^{3/2}+x^{-3}$ .

Berechne die Trapezstreifensumme $T_{4}$ für $f(x)=x^{2}$ im Intervall $[0; 1]$ und die Differenz zu $A$ . Vergleiche mit $U_{4}$ und $O_{4}$ .