Calculus

Problem 9001

Find the following limits as $x$ approaches infinity: (a) $\lim _{x \rightarrow \infty} \frac{7-2 x}{7 x^{3}-6}=$ (b) $\lim _{x \rightarrow \infty} \frac{7-2 x}{7 x-6}=$ (c) $\lim _{x \rightarrow \infty} \frac{7-2 x^{2}}{7 x-6}=$

See Solution

Problem 9002

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}-4}}{7 x-7}$ .

See Solution

Problem 9003

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin (3 x)}{x}$ .

See Solution

Problem 9004

Given the function $f(x)=(x+2)^{2}(x-1)$ , find:
(a) Critical numbers: $x=$
(b) Intervals where $f$ is increasing and decreasing: Increasing: $(-\infty,-2)$ , $(-2,0)$ , $(0, \infty)$ Decreasing: $(-\infty,-2)$ , $(-2,0)$ , $(0, \infty)$
(c) Use the First Derivative Test for relative extrema: Max $(x, y)=($ Min $(x, y)=($ )

See Solution

Problem 9005

Find the derivative of the composite function $\frac{d}{d x} f(g(x))$ with $f(x)=x^{5}$ and $g(x)=8 x-9$ .

See Solution

Problem 9006

Find the limits as $x$ approaches infinity for: (a) $h(x)=\frac{8x^2+2x+1}{x}$ , (b) $h(x)=\frac{8x^2+2x+1}{x^2}$ , (c) $h(x)=\frac{8x^2+2x+1}{x^3}$ .

See Solution

Problem 9007

Find the derivative of the composite function $f(g(x))$ with $f(x)=\frac{6}{x}$ and $g(x)=8-x^{2}$ .

See Solution

Problem 9008

Find the tangent line equation for $f(x)=x(4-x)^{2}$ at $x=2$ . What is $y=$ ?

See Solution

Problem 9009

Find the percentage of doctors prescribing medication after 10 months using $P(t)=100(1-e^{-0.42 t})$ and $P^{\prime}(10)$ . Interpret $P^{\prime}(10)$ .

See Solution

Problem 9010

A camera is 4,000 ft from a rocket pad. If the rocket rises at 700 ft/s from 3,000 ft, find: (a) rate of distance change (ft/s). (b) angle change rate (rad/s).

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Problem 9011

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{4} \sqrt{x+1}$ .

See Solution

Problem 9012

Find the limit as $x$ approaches negative infinity: $\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+2}$ .

See Solution

Problem 9013

A camera is $4,000 \mathrm{ft}$ from a rocket pad. If the rocket rises at $700 \mathrm{ft/s}$ when $3,000 \mathrm{ft}$ high, find:
(a) How fast is the distance to the rocket changing in $\mathrm{ft/s}$ ? (b) How fast is the angle of elevation changing in $\mathrm{rad/s}$ ?

See Solution

Problem 9014

Find $\frac{d y}{d x}$ for $y=x^{4} \sqrt{x+1}$ .

See Solution

Problem 9015

Calculate the total amount and interest from a \$3,000 deposit at 5.24% continuous compounding for 7 years.

See Solution

Problem 9016

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x \sin 4 x}{6 x^{2}}$ .

See Solution

Problem 9017

Find the value of $Q$ that maximizes total revenue for the demand equation $P=\sqrt{1000-4Q}$ .

See Solution

Problem 9018

Bestimme die Verdopplungszeit der Weltbevölkerung bei einer Wachstumsrate von 1,2\% jährlich.

See Solution

Problem 9019

Evaluate the integral: $\int 3 \cos \left(2 x-\frac{\pi}{2}\right) \cos \left(x+\frac{\pi}{4}\right) d x$

See Solution

Problem 9020

Calculate the integral: $\int 4 \sin 8 x \cos 3 x \, dx$

See Solution

Problem 9021

Find the derivative of $f(x) = (x-5.2)^{3} \cdot \sqrt{x}$ .

See Solution

Problem 9022

Calculate the integral: $\int \sin^{5} x \cos^{4} x \, dx$

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Problem 9023

Calculate the integral $\int \sin^{3} 2x \sqrt{\cos 2x} \, dx$ .

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Problem 9024

Calculate the integral of $\tan^{2}(2x) \sec^{4}(2x) \, dx$ .

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Problem 9025

Simplify $f(x)=\left(a \cdot x^{2}+44\right) \cdot \frac{1}{x}$ , find its derivative and roots.

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Problem 9026

Evaluate the integral: $\int \tan^{4}(3x) \, dx$

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Problem 9027

Find the integral of $\tan^{3} x \sec^{3/2} x \, dx$ .

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Problem 9028

Evaluate the integral: $\int 5 \sin \left(4 x+\frac{\pi}{3}\right) \sin \left(2 x-\frac{\pi}{6}\right) d x$

See Solution

Problem 9029

Find the derivative of $f(x)=(-x-a) \cdot \frac{10}{x^{2}}$ using the product rule.

See Solution

Problem 9030

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}$ . Untersuche Nullstellen, Extrempunkte, Wendepunkte und berechne Flächeninhalt und Volumen.

See Solution

Problem 9031

Which differentiation rule is incorrect? a. If $f(z)=z^{m}$ , then $f^{\prime}(z)=m z^{m-1}$ b. If $f(z)=a+b z^{m}$ , then $f^{\prime}(z)=a+b m z^{m-1}$ c. If $f(z)=a$ , then $f^{\prime}(z)=0$ d. If $f(z)=g(z)+h(z)$ , then $f^{\prime}(z)=g^{\prime}(z)+h^{\prime}(z)$

See Solution

Problem 9032

Find the value of $q$ that maximizes the profit given the marginal profit function $P^{\prime}(q)=4(3^{2}-q)+112$ . Select the correct option.

See Solution

Problem 9033

Bestimmen Sie die Extrem- und Wendepunkte der Funktionen: a) $f(x)=x \cdot(x+3)^{2}$ , b) $f(t)=(3 t-1)^{2}$ , c) $g(x)=(2 x-5)^{3}-150 x$ , d) $g(t)=t^{2} \cdot(2 t+5)$ , e) $h(x)=(3-4 x)^{2}+32 x$ , f) $h(t)=\left(\frac{1}{3} t+2\right)^{2} \cdot t$ .

See Solution

Problem 9034

Find the marginal revenue from renting the 20th apartment using the function $R(x)=6 x^{3}-8 x^{2}-\sqrt{x}$ .

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Problem 9035

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 3 \cos (t+2), 3 t e^{4 t},-3 t \ln (-4 t)\rangle$ .

See Solution

Problem 9036

Finde die Extrem- und Wendepunkte der Funktionen: a) $f(x)=x^{4}-4 x^{3}$ , b) $g(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{3}+3 x^{2}$ , c) $h(x)=\frac{2}{3} x^{3}-2 x$ .

See Solution

Problem 9037

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

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Problem 9038

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\left\langle 2 t^{2}+3, t^{2}+t^{4},-3 t^{3}\right\rangle$ at $(11,20,24)$ .

See Solution

Problem 9039

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 3 e^{3 t},-3 e^{3 t}, e^{3 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

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Problem 9040

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\left\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\right\rangle$ .

See Solution

Problem 9041

Find the integral of the vector function $\vec{r}(t)=\left\langle\sqrt{3 t+1},-\frac{4}{t+1},-4 \cos (-4 t)\right\rangle$ .

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Problem 9042

Berechne die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangenten bei $P(3, g(3))$ . Untersuche auch Asymptoten.

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Problem 9043

Bestimmen Sie die Intervalle für Linkskurve und Rechtskurve des Graphen von $f(x)=\frac{1}{3} x^{3}-x$ .

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Problem 9044

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 5 e^{2 t},-e^{2 t},-2 e^{2 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

See Solution

Problem 9045

Find the integral $\int \vec{r}(t) dt$ for $\vec{r}(t)=\left\langle\sqrt{3 t-1}, \frac{3}{t-1},-4 \cos (-4 t)\right\rangle$ .

See Solution

Problem 9046

Find the limit as $t$ approaches 0 for the expression: $\frac{5}{t^{2}-9} \vec{i}+e^{-t} \vec{j}+\frac{1-\cos (t)}{t} \vec{k}$ .

See Solution

Problem 9047

Evaluate the limit as $t$ approaches 0: $\ln(t+3) \vec{i} + \frac{\ln(t+3)}{t-5} \vec{j} + 5t^2 \vec{k}$ .

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Problem 9048

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 5 e^{2 t},-e^{2 t},-2 e^{2 t}\rangle$ for $0 \leq t \leq \ln(4)$ .

See Solution

Problem 9049

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\langle-4 \cos (-3 t+4),-5 t e^{-4 t},-t \ln (4 t)\rangle$ .

See Solution

Problem 9050

Find the 1st, 2nd, and 3rd derivatives of $f(x)=-\frac{1}{2} x^{8}-\frac{4}{3} x^{6}+7 x^{3}+3$ .

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Problem 9051

Find the integral of the vector function $\vec{r}(t)=\langle\sqrt{3 t-5,-\cos (-2 t), \frac{1}{-3 t+5}}\rangle$ .

See Solution

Problem 9052

Find $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

See Solution

Problem 9053

Find the tangent line $(L)$ to $\vec{r}(t)=\langle 4t^4+5, t^2+4t^4, 3t^3 \rangle$ at the point $(9,5,-3)$ .

See Solution

Problem 9054

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

See Solution

Problem 9055

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-4 \cos (-3 t+4),-5 t e^{-4 t},-t \ln (4 t)\rangle$ .

See Solution

Problem 9056

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at $(28,44,-20)$ .

See Solution

Problem 9057

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle-e^{3t},-5e^{3t},3e^{3t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

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Problem 9058

Find the limit as $t$ approaches 0 for $(t+3) \vec{i} + \frac{1-\cos(t)}{t} \vec{j} + e^{-t} \vec{k}$ .

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Problem 9059

Find the tangent line $(L)$ to $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at $(28,44,-20)$ .

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Problem 9060

Berechnen Sie die Werte der folgenden Integrale: a) $\int_{0}^{1} \frac{2 x}{x^{2}+1} dx$ b) $\int_{0}^{\frac{\pi}{2}} -\tan x \, dx$ c) $\int_{1}^{5} \frac{3 x^{2}}{x^{3}+8} dx$ d) $\int_{e}^{e} \frac{1}{x \ln x} dx$ e) $\int_{-2}^{0} \frac{e^{x}}{e^{x}+e} dx$ f) $\int_{\frac{\pi}{6}}^{\frac{\pi}{6}} \cot x \, dx$ g) $\int_{0}^{5} \frac{e^{x}}{e} dx$ h) $\int_{1}^{2} \frac{2x+4x^{3}}{x^{2}(1+x^{2})} dx$ i) $\int_{0}^{\frac{\pi}{2}} \sin(2x) \, dx$ j) $\int_{-2}^{6} e^{-x} dx$ k) $\int_{-1}^{1} e^{2x+3} dx$ l) $\int_{-\frac{1}{2}}^{0} \cos(\pi x) \, dx$

See Solution

Problem 9061

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at point $(28,44,-20)$ .

See Solution

Problem 9062

Bestimmen Sie die Ableitung der Funktion $f(x) = 2x - 1$ .

See Solution

Problem 9063

Bestimme die erste Ableitung von $f(x)=x^{2}-x-6$ .

See Solution

Problem 9064

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

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Problem 9065

Find the limit as $x$ approaches 2 from the left for the piecewise function $g(x)$ defined as: $g(x)=\begin{cases} \sqrt{5-x}, & x<-4 \\ x^{2}-5, & -4 \leq x<2 \\ x-3, & x \geq 2 \end{cases}$

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Problem 9066

Find positive $t$ such that vector $\overrightarrow{r(t)}=\langle 10t, 10t^2, 5t^2-10\rangle$ is perpendicular to $\overrightarrow{r'(t)}$ . $t=$

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Problem 9067

Find the unit tangent vector $\vec{T}(t)$ for $\vec{r}(t)=\langle 5t^4-1,-e^{4t},\sin(4t)\rangle$ at $t=0$ . Round to 4 decimal places.

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Problem 9068

Bestimmen Sie die Intervalle, in denen der Graph der Funktionen $f$ eine Links- oder Rechtskurve hat: a) $f(x)=\frac{3}{20} x^{5}-2 x^{3}+x$ b) $f(x)=\frac{1}{4} x^{4}+3 x^{3}-2$ c) $f(x)=x^{4}-2 x^{3}$

See Solution

Problem 9069

Bestimmen Sie die Ableitung von $f_{a}(x)=-x^{2}+a x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

See Solution

Problem 9070

Find the limit: $\lim _{x \rightarrow-4^{+}} g(x)$ for the piecewise function $g(x)=\left\{\begin{array}{cc}\sqrt{5-x}, & x<-4 \\ x^{2}-5, & -4 \leq x<2 \\ x-3, & x \geq 2\end{array}\right.$

See Solution

Problem 9071

Find the derivative of $y=\frac{6}{x}+8 \sec x$ . Show your work.

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Problem 9072

Find the unit tangent vector $\vec{T}(-1)$ for the curve $\vec{r}(t)=\langle 4t, -3t, \sqrt{25-t^2} \rangle$ .

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Problem 9073

Invest \$150000 at 4\% interest compounded continuously. (a) Find the account value after 9 years. (b) When will it reach \$240000?

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Problem 9074

Find the derivative of $y=\frac{\sec x+\csc x}{\csc x}$ . Show your work.

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Problem 9075

Find $\lim _{x \rightarrow-4^{-}} g(x)$ for the piecewise function $g(x)=\left\{\begin{array}{cc}\sqrt{5-x}, & x<-4 \\ x^{2}-5, & -4 \leq x<2 \\ x-3, & x \geq 2\end{array}\right.$

See Solution

Problem 9076

Find $y^{\prime \prime}$ for the function $y=3 \sin x$ . Show your calculations.

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Problem 9077

Find the unit tangent vector $\vec{T}\left(\frac{\pi}{4}\right)$ for the curve $\vec{r}(t)=\langle 4 \cos (t), 4 \sin (t), 3 \sin ^{2}(t)\rangle$ .

See Solution

Problem 9078

Does the function $y=3 x+6 \sin x$ have horizontal tangents in $0 \leq x \leq 2 \pi$ ? If yes, where?

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Problem 9079

Find the limits and value for the piecewise function:
$g(x)=\left\{\begin{array}{cc}\sqrt{5-x}, & x<-4 \\ x^{2}-5, & -4 \leq x<2 \\ x-3, & x \geq 2\end{array}\right.$
e. $\lim _{x \rightarrow 2^{+}} g(x)$ , f. $\lim _{x \rightarrow 2} g(x)$ , g. $\lim _{x \rightarrow-4} g(x)$ , h. $g(-4)$ .

See Solution

Problem 9080

Calculate the integral of the vector function $\vec{r}(t)=\langle 5 t^{4}+5, e^{5 t},-\sin (-t)\rangle$ .

See Solution

Problem 9081

Find the integral $\int \vec{r}(t) dt$ for $\vec{r}(t)=\langle 5t^4+5, e^{5t}, -\sin(-t)\rangle$ .

See Solution

Problem 9082

Calculate the integral of the vector function $\vec{r}(t)=\langle 3 \sin (-t),-\frac{1}{t-3}, \sqrt{5 t+3}\rangle$ . Find $\int \vec{r}(t) dt$ .

See Solution

Problem 9083

Evaluate the integral of $\vec{r}(t)=\left\langle\sqrt[3]{t-2}, \frac{4}{t-9}, 4 e^{t}\right\rangle$ from 10 to 29.

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Problem 9084

Find the derivative of $y=\sqrt{20 x-x^{3}}$ . Show your work.

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Problem 9085

Find $\frac{d y}{d x}$ for $y=\sin u$ and $u=9 x+18$ . Show your work using the chain rule.

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Problem 9086

Find the derivative of $(f+g)(x)=x^{3}+\sqrt[3]{x}+4$ and evaluate it at a specific point.

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Problem 9087

A 4 ft person walks from a 10 ft lamppost at 3 ft/s. Find shadow tip rates when the person is 8 ft from the pole.

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Problem 9088

Montrez que pour tout $x > 0$ , on a $x - \frac{x^3}{9} < G(x) < x$ avec $G$ la primitive de $y(x) = \frac{1}{x}\arctan(x)$ .

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Problem 9089

Bestimmen Sie den Differenzenquotienten für die Funktionen in den angegebenen Intervallen: a) $f(x)=(x-2)^{2}, I=[1 ; 6]$ b) $f(x)=\frac{9}{x^{2}}-3, I=[-3 ;-1]$ c) $f(x)=\sqrt{x+5}+x, I=[-4 ;-1]$ d) $f(x)=x^{3}+x^{2}, I=[-2 ; 4]$

See Solution

Problem 9090

Zeigen Sie, dass der Differenzenquotient von $f(x)=2^{x}$ im Intervall $[a ; a+2]$ gleich $\frac{3}{2} \cdot f(a)$ ist.

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Problem 9091

1. Blutzuckerwerte über Stunden: $f(x)=0,25 x^{4}+\frac{1}{3} x^{3}-8,5 x^{2}+15 x+100$ für $0 \leq x \leq 5$ . a. Glukosespiegel nach 2h. b. Zeitpunkt und Wert des tiefsten Glukosespiegels. c. Zeitpunkt, wenn Glukose > 120 mg/dl. d. Durchschnittlicher Anstieg in 5h.

See Solution

Problem 9092

Find the derivative $\frac{dy}{dx}$ for the equation $x y - y^{2} = 4$ .

See Solution

Problem 9093

Find the derivative of $h(x)=\left(\frac{\cos x}{1+\sin x}\right)^{5}$ . Show your work.

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Problem 9094

Given an invertible function $f(x)$ , use the table to find: a) $f^{-1}(3)$ , $f^{-1}(4)$ , b) $\left(f^{-1}\right)^{\prime}(4)$ , c) tangent line of $f^{-1}(x)$ at $x=4$ , d) tangent line of $f(x)$ at $x=1$ , e) relationship between the two tangent lines.

See Solution

Problem 9095

Find the derivative of $y=(1+7 x) e^{(-7 x)}$ . Show your work.

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Problem 9096

Bestimme die Ableitung und die Steigung von $f_{a}(x)$ an $x=0$ . Wann ist die Steigung 1? Funktionen: a) $-x^{2}+a x$ b) $a x^{3}-3 a x$ c) $a x^{4}-4 x^{3}+a^{2} x$ d) $a x^{4}+2 x^{2}+(a^{2}-3) x$ e) $\frac{1}{a} x^{2}+\frac{2}{a} x+a$ f) $x^{2}+(\frac{a}{2}-\frac{1}{3}) x+1$ .

See Solution

Problem 9097

Die Firma Meier verkauft Schokolade. Gegeben ist die Funktion $f(t)=-0,0001 t^{3}+0,15 t^{2}+15 t$ für $0 \leq t \leq 1500$ .
a. Wie viele Tafeln verkauft die Firma nach 700 Tagen? b. An welchem Tag werden die meisten Schokoladen verkauft? c. Wann ist die Zunahme der täglichen Verkaufszahlen am größten?

See Solution

Problem 9098

Given the function $F(x, y) = 7x^{2} + 6xy^{2} + 4y + 22^{2}$ , find the first derivative $F_{1}^{\prime}$ and coefficients $a, b, c, d, g, k$ .

See Solution

Problem 9099

Find $s^$ and $t^$ such that $v(s, t)=(7 s^{2}-2 s t) \cdot e^{\frac{1}{5} t}$ has one local extremum.

See Solution

Problem 9100

Rocket problem: Given exhaust velocity $20.4 \mathrm{~m/s}$ , initial mass $1.1 \mathrm{~kg}$ , and gravity $9.81 \mathrm{~ms}^{-2}$ , find:
(a) Fuel burn rate $\alpha$ in $\mathrm{kg/s}$ after $0.1$ seconds when velocity is $2.66 \mathrm{~m/s}$ .
(b) Time until all fuel is used up in seconds.
(c) Expected height when fuel is exhausted in metres (use $\lim_{x \to 0^+} x \ln x = 0$ ).

See Solution

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Calculus

Problem 9001

Problem 9002

Find the limit: lim⁡x→∞x2−47x−7\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}-4}}{7 x-7}limx→∞​7x−7x2−4​​.

Problem 9003

Find the limit: lim⁡x→∞sin⁡(3x)x\lim _{x \rightarrow \infty} \frac{\sin (3 x)}{x}limx→∞​xsin(3x)​.

Problem 9004

Problem 9005

Find the derivative of the composite function ddxf(g(x))\frac{d}{d x} f(g(x))dxd​f(g(x)) with f(x)=x5f(x)=x^{5}f(x)=x5 and g(x)=8x−9g(x)=8 x-9g(x)=8x−9.

Problem 9006

Find the limits as xxx approaches infinity for: (a) h(x)=8x2+2x+1xh(x)=\frac{8x^2+2x+1}{x}h(x)=x8x2+2x+1​, (b) h(x)=8x2+2x+1x2h(x)=\frac{8x^2+2x+1}{x^2}h(x)=x28x2+2x+1​, (c) h(x)=8x2+2x+1x3h(x)=\frac{8x^2+2x+1}{x^3}h(x)=x38x2+2x+1​.

Problem 9007

Find the derivative of the composite function f(g(x))f(g(x))f(g(x)) with f(x)=6xf(x)=\frac{6}{x}f(x)=x6​ and g(x)=8−x2g(x)=8-x^{2}g(x)=8−x2.

Problem 9008

Find the tangent line equation for f(x)=x(4−x)2f(x)=x(4-x)^{2}f(x)=x(4−x)2 at x=2x=2x=2. What is y=y=y=?

Problem 9009

Find the percentage of doctors prescribing medication after 10 months using P(t)=100(1−e−0.42t)P(t)=100(1-e^{-0.42 t})P(t)=100(1−e−0.42t) and P′(10)P^{\prime}(10)P′(10). Interpret P′(10)P^{\prime}(10)P′(10).

Problem 9010

A camera is 4,000 ft from a rocket pad. If the rocket rises at 700 ft/s from 3,000 ft, find: (a) rate of distance change (ft/s). (b) angle change rate (rad/s).

Problem 9011

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x4x+1y=x^{4} \sqrt{x+1}y=x4x+1​.

Problem 9012

Find the limit as xxx approaches negative infinity: lim⁡x→−∞9x6−xx3+2\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+2}limx→−∞​x3+29x6−x​​.

Problem 9013

Problem 9014

Find dydx\frac{d y}{d x}dxdy​ for y=x4x+1y=x^{4} \sqrt{x+1}y=x4x+1​.

Problem 9015

Calculate the total amount and interest from a \$3,000 deposit at 5.24% continuous compounding for 7 years.

Problem 9016

Find the limit: lim⁡x→0sin⁡3xsin⁡4x6x2\lim _{x \rightarrow 0} \frac{\sin 3 x \sin 4 x}{6 x^{2}}limx→0​6x2sin3xsin4x​.

Problem 9017

Find the value of QQQ that maximizes total revenue for the demand equation P=1000−4QP=\sqrt{1000-4Q}P=1000−4Q​.

Problem 9018

Bestimme die Verdopplungszeit der Weltbevölkerung bei einer Wachstumsrate von 1,2\% jährlich.

Problem 9019

Evaluate the integral: ∫3cos⁡(2x−π2)cos⁡(x+π4)dx\int 3 \cos \left(2 x-\frac{\pi}{2}\right) \cos \left(x+\frac{\pi}{4}\right) d x∫3cos(2x−2π​)cos(x+4π​)dx

Problem 9020

Calculate the integral: ∫4sin⁡8xcos⁡3x dx\int 4 \sin 8 x \cos 3 x \, dx∫4sin8xcos3xdx

Problem 9021

Find the derivative of f(x)=(x−5.2)3⋅xf(x) = (x-5.2)^{3} \cdot \sqrt{x}f(x)=(x−5.2)3⋅x​.

Problem 9022

Calculate the integral: ∫sin⁡5xcos⁡4x dx\int \sin^{5} x \cos^{4} x \, dx∫sin5xcos4xdx

Problem 9023

Calculate the integral ∫sin⁡32xcos⁡2x dx\int \sin^{3} 2x \sqrt{\cos 2x} \, dx∫sin32xcos2x​dx.

Problem 9024

Calculate the integral of tan⁡2(2x)sec⁡4(2x) dx\tan^{2}(2x) \sec^{4}(2x) \, dxtan2(2x)sec4(2x)dx.

Problem 9025

Simplify f(x)=(a⋅x2+44)⋅1xf(x)=\left(a \cdot x^{2}+44\right) \cdot \frac{1}{x}f(x)=(a⋅x2+44)⋅x1​, find its derivative and roots.

Problem 9026

Evaluate the integral: ∫tan⁡4(3x) dx\int \tan^{4}(3x) \, dx∫tan4(3x)dx

Problem 9027

Find the integral of tan⁡3xsec⁡3/2x dx\tan^{3} x \sec^{3/2} x \, dxtan3xsec3/2xdx.

Problem 9028

Evaluate the integral: ∫5sin⁡(4x+π3)sin⁡(2x−π6)dx\int 5 \sin \left(4 x+\frac{\pi}{3}\right) \sin \left(2 x-\frac{\pi}{6}\right) d x∫5sin(4x+3π​)sin(2x−6π​)dx

Problem 9029

Find the derivative of f(x)=(−x−a)⋅10x2f(x)=(-x-a) \cdot \frac{10}{x^{2}}f(x)=(−x−a)⋅x210​ using the product rule.

Problem 9030

Gegeben ist die Funktion y=f1(x)=(x+1t)⋅e−txy=f_{1}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}y=f1​(x)=(x+t1​)⋅e−tx. Untersuche Nullstellen, Extrempunkte, Wendepunkte und berechne Flächeninhalt und Volumen.

Problem 9031

Problem 9032

Find the value of qqq that maximizes the profit given the marginal profit function P′(q)=4(32−q)+112P^{\prime}(q)=4(3^{2}-q)+112P′(q)=4(32−q)+112. Select the correct option.

Problem 9033

Problem 9034

Find the marginal revenue from renting the 20th apartment using the function R(x)=6x3−8x2−xR(x)=6 x^{3}-8 x^{2}-\sqrt{x}R(x)=6x3−8x2−x​.

Problem 9035

Find the derivative r⃗′(t)\vec{r}^{\prime}(t)r′(t) for r⃗(t)=⟨3cos⁡(t+2),3te4t,−3tln⁡(−4t)⟩\vec{r}(t)=\langle 3 \cos (t+2), 3 t e^{4 t},-3 t \ln (-4 t)\rangler(t)=⟨3cos(t+2),3te4t,−3tln(−4t)⟩.

Problem 9036

Finde die Extrem- und Wendepunkte der Funktionen: a) f(x)=x4−4x3f(x)=x^{4}-4 x^{3}f(x)=x4−4x3, b) g(x)=14x4−32x3+3x2g(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{3}+3 x^{2}g(x)=41​x4−23​x3+3x2, c) h(x)=23x3−2xh(x)=\frac{2}{3} x^{3}-2 xh(x)=32​x3−2x.

Problem 9037

Find the derivative r⃗′(t)\vec{r}^{\prime}(t)r′(t) for r⃗(t)=⟨−3tln⁡(−2t),−4sin⁡(−t−3),5te−3t⟩\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangler(t)=⟨−3tln(−2t),−4sin(−t−3),5te−3t⟩.

Problem 9038

Find the tangent line (L)(L)(L) to the curve r⃗(t)=⟨2t2+3,t2+t4,−3t3⟩\vec{r}(t)=\left\langle 2 t^{2}+3, t^{2}+t^{4},-3 t^{3}\right\rangler(t)=⟨2t2+3,t2+t4,−3t3⟩ at (11,20,24)(11,20,24)(11,20,24).

Problem 9039

Find the arc length sss of the curve r⃗(t)=⟨3e3t,−3e3t,e3t⟩\vec{r}(t)=\langle 3 e^{3 t},-3 e^{3 t}, e^{3 t}\rangler(t)=⟨3e3t,−3e3t,e3t⟩ for 0≤t≤ln⁡(4)0 \leq t \leq \ln(4)0≤t≤ln(4). s=s= s=

Problem 9040

Find the derivative r⃗′(t)\vec{r}^{\prime}(t)r′(t) for r⃗(t)=⟨−3tln⁡(−2t),−4sin⁡(−t−3),5te−3t⟩\vec{r}(t)=\left\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\right\rangler(t)=⟨−3tln(−2t),−4sin(−t−3),5te−3t⟩.

Problem 9041

Find the integral of the vector function r⃗(t)=⟨3t+1,−4t+1,−4cos⁡(−4t)⟩\vec{r}(t)=\left\langle\sqrt{3 t+1},-\frac{4}{t+1},-4 \cos (-4 t)\right\rangler(t)=⟨3t+1​,−t+14​,−4cos(−4t)⟩.

Problem 9042

Berechne die Ableitung von g(x)=x2+2x3x−4g(x)=\frac{x^{2}+2x}{3x-4}g(x)=3x−4x2+2x​ und finde die Tangenten bei P(3,g(3))P(3, g(3))P(3,g(3)). Untersuche auch Asymptoten.

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}-4}}{7 x-7}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{\sin (3 x)}{x}$ .

Find the derivative of the composite function $\frac{d}{d x} f(g(x))$ with $f(x)=x^{5}$ and $g(x)=8 x-9$ .

Find the limits as $x$ approaches infinity for: (a) $h(x)=\frac{8x^2+2x+1}{x}$ , (b) $h(x)=\frac{8x^2+2x+1}{x^2}$ , (c) $h(x)=\frac{8x^2+2x+1}{x^3}$ .

Find the derivative of the composite function $f(g(x))$ with $f(x)=\frac{6}{x}$ and $g(x)=8-x^{2}$ .

Find the tangent line equation for $f(x)=x(4-x)^{2}$ at $x=2$ . What is $y=$ ?

Find the percentage of doctors prescribing medication after 10 months using $P(t)=100(1-e^{-0.42 t})$ and $P^{\prime}(10)$ . Interpret $P^{\prime}(10)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{4} \sqrt{x+1}$ .

Find the limit as $x$ approaches negative infinity: $\lim _{x \rightarrow-\infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+2}$ .

Find $\frac{d y}{d x}$ for $y=x^{4} \sqrt{x+1}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin 3 x \sin 4 x}{6 x^{2}}$ .

Find the value of $Q$ that maximizes total revenue for the demand equation $P=\sqrt{1000-4Q}$ .

Evaluate the integral: $\int 3 \cos \left(2 x-\frac{\pi}{2}\right) \cos \left(x+\frac{\pi}{4}\right) d x$

Calculate the integral: $\int 4 \sin 8 x \cos 3 x \, dx$

Find the derivative of $f(x) = (x-5.2)^{3} \cdot \sqrt{x}$ .

Calculate the integral: $\int \sin^{5} x \cos^{4} x \, dx$

Calculate the integral $\int \sin^{3} 2x \sqrt{\cos 2x} \, dx$ .

Calculate the integral of $\tan^{2}(2x) \sec^{4}(2x) \, dx$ .

Simplify $f(x)=\left(a \cdot x^{2}+44\right) \cdot \frac{1}{x}$ , find its derivative and roots.

Evaluate the integral: $\int \tan^{4}(3x) \, dx$

Find the integral of $\tan^{3} x \sec^{3/2} x \, dx$ .

Evaluate the integral: $\int 5 \sin \left(4 x+\frac{\pi}{3}\right) \sin \left(2 x-\frac{\pi}{6}\right) d x$

Find the derivative of $f(x)=(-x-a) \cdot \frac{10}{x^{2}}$ using the product rule.

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}$ . Untersuche Nullstellen, Extrempunkte, Wendepunkte und berechne Flächeninhalt und Volumen.

Find the value of $q$ that maximizes the profit given the marginal profit function $P^{\prime}(q)=4(3^{2}-q)+112$ . Select the correct option.

Find the marginal revenue from renting the 20th apartment using the function $R(x)=6 x^{3}-8 x^{2}-\sqrt{x}$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle 3 \cos (t+2), 3 t e^{4 t},-3 t \ln (-4 t)\rangle$ .

Finde die Extrem- und Wendepunkte der Funktionen: a) $f(x)=x^{4}-4 x^{3}$ , b) $g(x)=\frac{1}{4} x^{4}-\frac{3}{2} x^{3}+3 x^{2}$ , c) $h(x)=\frac{2}{3} x^{3}-2 x$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\left\langle 2 t^{2}+3, t^{2}+t^{4},-3 t^{3}\right\rangle$ at $(11,20,24)$ .

Find the arc length $s$ of the curve $\vec{r}(t)=\langle 3 e^{3 t},-3 e^{3 t}, e^{3 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\left\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\right\rangle$ .

Find the integral of the vector function $\vec{r}(t)=\left\langle\sqrt{3 t+1},-\frac{4}{t+1},-4 \cos (-4 t)\right\rangle$ .

Berechne die Ableitung von $g(x)=\frac{x^{2}+2x}{3x-4}$ und finde die Tangenten bei $P(3, g(3))$ . Untersuche auch Asymptoten.

Bestimmen Sie die Intervalle für Linkskurve und Rechtskurve des Graphen von $f(x)=\frac{1}{3} x^{3}-x$ .

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 5 e^{2 t},-e^{2 t},-2 e^{2 t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

Find the integral $\int \vec{r}(t) dt$ for $\vec{r}(t)=\left\langle\sqrt{3 t-1}, \frac{3}{t-1},-4 \cos (-4 t)\right\rangle$ .

Find the limit as $t$ approaches 0 for the expression: $\frac{5}{t^{2}-9} \vec{i}+e^{-t} \vec{j}+\frac{1-\cos (t)}{t} \vec{k}$ .

Evaluate the limit as $t$ approaches 0: $\ln(t+3) \vec{i} + \frac{\ln(t+3)}{t-5} \vec{j} + 5t^2 \vec{k}$ .

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle 5 e^{2 t},-e^{2 t},-2 e^{2 t}\rangle$ for $0 \leq t \leq \ln(4)$ .

Find the derivative $\vec{r}^{\prime}(t)$ of the vector function $\vec{r}(t)=\langle-4 \cos (-3 t+4),-5 t e^{-4 t},-t \ln (4 t)\rangle$ .

Find the 1st, 2nd, and 3rd derivatives of $f(x)=-\frac{1}{2} x^{8}-\frac{4}{3} x^{6}+7 x^{3}+3$ .

Find the integral of the vector function $\vec{r}(t)=\langle\sqrt{3 t-5,-\cos (-2 t), \frac{1}{-3 t+5}}\rangle$ .

Find $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

Find the tangent line $(L)$ to $\vec{r}(t)=\langle 4t^4+5, t^2+4t^4, 3t^3 \rangle$ at the point $(9,5,-3)$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-3 t \ln (-2 t),-4 \sin (-t-3), 5 t e^{-3 t}\rangle$ .

Find the derivative $\vec{r}^{\prime}(t)$ for $\vec{r}(t)=\langle-4 \cos (-3 t+4),-5 t e^{-4 t},-t \ln (4 t)\rangle$ .

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at $(28,44,-20)$ .

Calculate the arc length $s$ of the curve $\vec{r}(t)=\langle-e^{3t},-5e^{3t},3e^{3t}\rangle$ for $0 \leq t \leq \ln(4)$ . $s=$

Find the limit as $t$ approaches 0 for $(t+3) \vec{i} + \frac{1-\cos(t)}{t} \vec{j} + e^{-t} \vec{k}$ .

Find the tangent line $(L)$ to $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at $(28,44,-20)$ .

Find the tangent line $(L)$ to the curve $\vec{r}(t)=\langle-3 t^{3}+4, t^{2}-5 t^{3},-5 t^{2}\rangle$ at point $(28,44,-20)$ .

Bestimmen Sie die Ableitung der Funktion $f(x) = 2x - 1$ .

Bestimme die erste Ableitung von $f(x)=x^{2}-x-6$ .

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

Find positive $t$ such that vector $\overrightarrow{r(t)}=\langle 10t, 10t^2, 5t^2-10\rangle$ is perpendicular to $\overrightarrow{r'(t)}$ . $t=$

Find the unit tangent vector $\vec{T}(t)$ for $\vec{r}(t)=\langle 5t^4-1,-e^{4t},\sin(4t)\rangle$ at $t=0$ . Round to 4 decimal places.

Bestimmen Sie die Intervalle, in denen der Graph der Funktionen $f$ eine Links- oder Rechtskurve hat: a) $f(x)=\frac{3}{20} x^{5}-2 x^{3}+x$ b) $f(x)=\frac{1}{4} x^{4}+3 x^{3}-2$ c) $f(x)=x^{4}-2 x^{3}$

Bestimmen Sie die Ableitung von $f_{a}(x)=-x^{2}+a x$ und die Steigung bei $x=0$ . Für welches $a$ ist die Steigung 1?

Find the derivative of $y=\frac{6}{x}+8 \sec x$ . Show your work.

Find the unit tangent vector $\vec{T}(-1)$ for the curve $\vec{r}(t)=\langle 4t, -3t, \sqrt{25-t^2} \rangle$ .

Find the derivative of $y=\frac{\sec x+\csc x}{\csc x}$ . Show your work.