Calculus

Problem 31001

Which option represents the acceleration of a particle: first or second time derivative of velocity?

See Solution

Problem 31002

Finde die Ableitung von $f(x) = x^2 - x$ .

See Solution

Problem 31003

What is the slope of the acceleration vs. time plot for a baseball tossed upward? Options: constant positive, constant negative, or zero slope?

See Solution

Problem 31004

A particle's velocity decreases at a constant rate. What does the acceleration vs. time plot look like?

See Solution

Problem 31005

Evaluate the integral $\int_{0}^{\pi} x \cdot \sin \left(x^{2}\right) d x$ .

See Solution

Problem 31006

A rumour spreads in a school of 1200 pupils. Given $\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x(1200-x)}{3600}$ with 300 pupils aware at $t=0$ , show $t=3 \ln \left(\frac{3 x}{1200-x}\right)$ .

See Solution

Problem 31007

Find the relative maximum point for the function $f(x)=\frac{x^{2}-3}{x-2}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31008

Integrate $\int \frac{1}{\sqrt{1-x^{2}}} dx$ using the substitution $x=\sin(z)$ and find $dx$ .

See Solution

Problem 31009

Find the integral of $\sqrt{1-x^{2}}$ with respect to $x$ .

See Solution

Problem 31010

Calculate the integral: $\int \frac{1}{\sqrt{1-x^{2}}} d x$

See Solution

Problem 31011

Evaluate the integral: $\int \frac{1}{\sqrt{4-4 x^{2}}} d x$

See Solution

Problem 31012

Find the price elasticity of demand when $\mathrm{P} = £10$ for the demand function $\mathrm{Qd} = 1000 - 20P$ .

See Solution

Problem 31013

How much work is done to stretch a spring from its natural length to $10 \mathrm{in}$ if a force of $10 \mathrm{lb}$ stretches it $4 \mathrm{in}$ ?

See Solution

Problem 31014

Find $x$ values for $y=\frac{x^{2}}{x^{2}-3 x+2}$ when $\frac{d y}{d x}=0$ .

See Solution

Problem 31015

Find the price elasticity of demand when $\mathrm{P} = £10$ for the demand function $\mathrm{Qd} = 1000 - 20\mathrm{P}$ .

See Solution

Problem 31016

Kaffee (85°C) und Milch (8°C) kühlen ab. a) Zeige $f(t)=20+65 e^{-0,1625 t}$ für Kaffee. b) Wer ist nach 5 Min. kälter? c) Wie lange bis Milchkaffee 40°C? d) Wann Milch für 40°C? e) Untersuche $f_a(x)=\ln((x-a)^{2}+1)$ .

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Problem 31017

Find the derivative of $\tan(x^2)$ with respect to $x$ .

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Problem 31018

Evaluate the integral $\int x e^{x^{2}} d x$ .

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Problem 31019

Berechnen Sie die Integrale und Stammfunktionen für gegebene Werte von $f(x)$ und $g(x)$ sowie bestimmte Integrale.

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Problem 31020

Calculate the integrals: a) $\int_{1}^{3}(6 x^{2}-5) dx$ and b) $\int_{-2}^{1}(-x) dx$ . What are the results?

See Solution

Problem 31021

Find the relative minimum point of the function $f(x) = \frac{x^{2}-3}{x-2}$ using $f'(x)$ and $f''(x)$ .

See Solution

Problem 31022

Find the price elasticity of demand when $\mathrm{Qd}=1000-20 \mathrm{P}$ and $P=£ 10$ . Options: 0.125, 0.025, 0.25, 0.5.

See Solution

Problem 31023

a) Evaluate $\int_{1}^{3}(6 x^{2}-5) dx$ and find the result. b) Calculate $\int_{-2}^{-1}(-x) dx$ and determine the value.

See Solution

Problem 31024

Find $\int_{4}^{8} f\left(\frac{x}{4}-1\right) d x$ given $\int_{0}^{1} f(x) d x=4$ . Options: 0, 10, 16, 20, 80.

See Solution

Problem 31025

Find the solution of the equation: $\frac{d y}{d x}=\frac{2 x y+y^{2}}{x^{2}+2 x y}$ .

See Solution

Problem 31026

Find the derivative of the function $f(x)=(x+1)^{3 x}$ . What is $f^{\prime}(x)$ ?

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Problem 31027

Find the integral $\int \sqrt{x+1} \, dx$ and choose the correct answer from the options provided.

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Problem 31028

Find local extrema of $f(x)=x^{4}-2$ . Where are the local maxima and minima?

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Problem 31029

Find values of $c$ for the Mean Value Theorem with $f(x)=x^{3}-4x$ , for $-2 \leq x \leq 2$ .

See Solution

Problem 31030

$y=x^{3}-5 x+3$ eğrisinin $O x$ -ekseniyle $45^{\circ}$ teğetinin denklemi hangisidir? A) B) C) D) E)

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Problem 31031

Find the limit: $\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / 2 x}$ . What is the result?

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Problem 31032

Eine Kugel wird aus $30 m$ Höhe auf der Erde ( $g_{\varepsilon}=9,81 \frac{m}{s^{2}}$ ) und Neptun ( $g_{N}=11 \frac{m}{s^{2}}$ ) geworfen. Berechne die Aufprallgeschwindigkeiten in $\frac{km}{h}$ und die Zeitdifferenz bis zum Aufprall.

See Solution

Problem 31033

Find the second derivative of the function $y=\frac{7}{8x}$ .

See Solution

Problem 31034

جد الفترات التي تزداد وتنقص فيها الدالة $f$ بناءً على الرسم البياني لـ $y=f^{\prime}(x)$ .

See Solution

Problem 31035

Evaluate the integral $\int \frac{d x}{x \ln x}$ . Choose the correct answer from the options.

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Problem 31036

$f(x)=\left\{\begin{array}{cc}2 \sin x & ; x<0 \\ 3 x^{2}+2 x & ; x \geq 0\end{array}\right.$ fonksiyonunun $(0,0)$ noktasındaki teğetinin denklemi nedir?

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Problem 31037

Determine where $f(x)=\cos x$ is concave up and down on $[0, 2\pi]$ .

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Problem 31038

Find the inflection points of $f$ given $f^{\prime \prime}(x)=(x-2)^{5}(x-1)^{3}(x+1)^{4}$ .

See Solution

Problem 31039

Find $\frac{d y}{d x}$ for $y=\int_{1}^{\cos x} \log _{7}(1+t^{3}) dt$ .

See Solution

Problem 31040

Find intervals where the function $f$ increases and decreases based on its derivative $y=f^{\prime}(x)$ .

See Solution

Problem 31041

Find the limit as $x$ approaches 0 for $\frac{2^{x}-1}{3^{x}-1}$ . What is the result?

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Problem 31042

احسب حجم الجسم الناتج عن دوران المنطقة بين $y=(x-1)^{2}$ و $y=\sqrt{x-1}$ حول $x=-2$ .

See Solution

Problem 31043

Find the length of the curve $y=\frac{1}{3}(x^{2}+2)^{3/2}$ for $0 \leq x \leq 12$ .

See Solution

Problem 31044

$y=x^{\sin x}$ fonksiyonunun türevini bulun. Aşağıdakilerden hangisi doğrudur? A) B) C) D) E)

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Problem 31045

Find when the particle at $x(t)=16 t-3.0 t^{3}$ is momentarily at rest for $t$ values: 0.75s, 1.3s, 1.8s, 5.3s, 7.3s.

See Solution

Problem 31046

Find the average velocity of an object with $x=7t-3t^{2}$ from $t=0$ to $t=2$ seconds. Choices: $5 \mathrm{~m/s}$ , $11 \mathrm{~m/s}$ , $1 \mathrm{~m/s}$ , $-5 \mathrm{~m/s}$ , $-11 \mathrm{~m/s}$ .

See Solution

Problem 31047

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ for a cylinder with constant height using $V=\pi r^{2} h$ .

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Problem 31048

Calculate the area $A$ using the integrals:
1. $A=\int_{0}^{1}(y^{3}-y^{2}) dy$
2. $A=\int_{0}^{1}(2 \sqrt{y}-y) dy$
3. $A=\int_{0}^{4}(2 \sqrt{x}-(2 x-4)) dx$
4. $A=\int_{0}^{2}\left(x-\frac{x^{2}}{4}\right) dx$
5. $A=\int_{-2}^{4}\left(\left(2+\frac{y}{2}\right)-\frac{y^{2}}{4}\right) dy$
6. $A=\int_{0}^{1}(\sqrt[3]{x}-\sqrt{x}) dx$

See Solution

Problem 31049

Find the intervals where the function $f$ increases and decreases based on the graph of its derivative $y=f'(x)$ .

See Solution

Problem 31050

Find the average velocity of an object with constant acceleration $4 \mathrm{~m/s}^2$ from $x=2 m$ to $x=8 m$ .

See Solution

Problem 31051

Find the acceleration of an automobile at $t=1.0$ s given $x(t)=27t-4.0t^{3}$ . Options: $23$ , $15$ , $-4$ , $-12$ , $-24$ m/s².

See Solution

Problem 31052

Find the acceleration of an object at the moment it stops, given $x(t)=75 t-1.0 t^{3}$ . Choices: $-9.8$ , $0$ , $9.2 \times 10^{3}$ , $-73$ , $-30$ m/s².

See Solution

Problem 31053

Evaluate the integral $\int \cos x 5^{\sin x} d x$ . Choose the correct answer from the options provided.

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Problem 31054

How far does an object fall during the second second of free fall? Options: $4.9 \mathrm{~m}$ , $9.8 \mathrm{~m}$ , $15 \mathrm{~m}$ , $20 \mathrm{~m}$ , $25 \mathrm{~m}$ .

See Solution

Problem 31055

Find critical values of $f(x)=\frac{x^{3}}{3}+\frac{x^{2}}{2}-2x$ for $x=\{-2, -1, 1, 2\}$ .

See Solution

Problem 31056

Find an antiderivative function $F(x)$ of $f(x)=|2 x|$ for $-2 \leq x \leq 2$ .

See Solution

Problem 31057

احسب نقطة الحد الأدنى النسبي لدالة $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ باستخدام $f^{\prime}(x)$ و $f^{\prime \prime}(x)$ .

See Solution

Problem 31058

Find the average velocity of an object with $v=4t-3t^{2}$ from $t=0$ to $t=2$ seconds. Options include 0, $-2 \mathrm{~m/s}$ , $2 \mathrm{~m/s}$ , $-4 \mathrm{~m/s}$ , or need initial position.

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Problem 31059

A plane at 40.0 m/s drops supplies from 200.0 m. When to release? Find the velocity just before hitting the ground. $t=\sqrt{\frac{2400}{-9.8}}$

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Problem 31060

Find $\int_{3}^{5} f(x) d x$ given $\int_{1}^{5} f(x) d x=5$ and $\int_{1}^{3} f(x) d x=3$ .

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Problem 31061

Ein Wassertank hat zu Beginn $2 \mathrm{~m}^{3}$ Wasser. Die Zuflussrate erreicht max. $3 \frac{\mathrm{m}^{3}}{\mathrm{h}}$ nach 3 Stunden. Berechne das Volumen nach 6 Stunden.

See Solution

Problem 31062

A stone is dropped from a 190 m roof. What is its speed just before hitting the ground? Options: $43 \mathrm{~m/s}$ , $61 \mathrm{~m/s}$ , $120 \mathrm{~m/s}$ , $190 \mathrm{~m/s}$ , $1400 \mathrm{~m/s}$ .

See Solution

Problem 31063

A car starts at $16 \mathrm{~m/s}$ and slows down with $a=-0.50 \mathrm{t}$ . When does it stop: $4s$ , $8s$ , $16s$ , $32s$ , or $64s$ ?

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Problem 31064

Bestimmen Sie die Stammfunktionen für die folgenden Funktionen: a) $f(x)=2 \sin \left(\frac{\pi}{2} x\right)$ , b) $f(x)=\frac{1}{3+x}$ , c) $f(x)=e^{4 x-1}$ , d) $f(x)=\frac{7}{3 x+1}$ , e) $f(x)=\frac{5}{2(1-x)^{2}}$ , f) $f(x)=\frac{2}{3}(5-4 x)^{4}$ , g) $f(x)=\sqrt{9-\frac{1}{3} x}$ , h) $f(x)=\frac{3}{(1-5 x)^{4}}$ .

See Solution

Problem 31065

Find the initial velocity and acceleration of an auto given $x(t)=27t-4.0t^{3}$ at $t=0$ .

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Problem 31066

A spot moves as $x=8.79 t-0.624 t^{3}$ . Find when it's at rest, its position, acceleration, and when it hits screen edges.

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Problem 31067

Analyse - Tangentengleichungen
A 9.1: Bestimme $f'(3)$ und die Tangentengleichung an $f(x)=\frac{1}{2}x^2$ bei $x_0=3$ .
A 9.2: Finde den Schnittpunkt $S$ der Tangente von $f(x)=\frac{1}{2}x^3$ durch $P(-2 \mid f(-2))$ mit der $x$ -Achse.
A 9.3: a) Bestimme die Tangentengleichung an $P(4 \mid f(4))$ für $f(x)=\frac{4}{x}$ . b) Wo hat $f$ die Steigung -1? c) Wo ist die Tangente parallel zu $y=-\frac{4}{9}x+5$ ?
A 9.4: Skizziere $f(x)=\frac{2}{3}x^3$ und $g(x)=\frac{1}{x^2}$ . a) Wo haben sie den gleichen Funktionswert? b) Wo haben die Tangenten die gleiche Steigung und wie groß ist der Steigungswinkel?
A 9.6: Bestimme die Tangentengleichung an $f(x)=\frac{1}{4}x^4-x^2+1$ bei $P(2 \mid f(2))$ .
A 9.7: Zeige, dass $f(x)=x^2$ und $g(x)=-x^2+4x-2$ sich berühren.

See Solution

Problem 31068

Finde a, sodass die Fläche zwischen der $x$ -Achse und $f(x)=\frac{1}{2} x^{2}$ im Intervall [a; 2] 12 beträgt.

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Problem 31069

Find $\frac{d y}{d t}$ for $y=\sin(2x)$ and $\frac{d x}{d t}=2$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 31070

Bestimmen Sie die Fläche zwischen dem Graphen $f(x)=-0,25 x^{4}+x^{3}$ und der x-Achse.

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Problem 31071

Find the limits or show divergence for these sequences: (i) $a_{n}=\sqrt{n}(\sqrt{n+1}-\sqrt{n-1})$ ; (ii) $b_{n}=(-1)^{n} 2^{1/n}$ .

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Problem 31072

Simplify the expression: $\frac{x^{2}}{x-1}-\frac{3}{x-1}-\frac{x-3}{x-1}$ .

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Problem 31073

A ball's velocity changes from $20 \mathrm{~cm/s}$ to $0 \mathrm{~cm/s}$ in 5 seconds. Find the average acceleration.

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Problem 31074

Find $\frac{d x}{d t}$ when $y=2 \sin (x)$ , $\frac{d y}{d t}=1$ , and $x=\frac{\pi}{3}$ .

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Problem 31075

Find the limit: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$ . What is it equal to?

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Problem 31076

Find the distance traveled in the first 3 seconds if the velocity is constant at $v_{s}=10 \mathrm{~m/s}$ .

See Solution

Problem 31077

Find the Linearization $L(x)$ of $f(x) = \left(1 - \frac{1}{2+x}\right)^{2/3}$ at $x=0$ .

See Solution

Problem 31078

Find the object's speed at $t=5$ seconds if its position is $x(t)=5+10 t^{2}$ . Options: $5 \mathrm{~m/s}$ , $50 \mathrm{~m/s}$ , $100 \mathrm{~m/s}$ , $15 \mathrm{~m/s}$ .

See Solution

Problem 31079

Find $\frac{dx}{dt}$ when $y = 2 \cot(x)$ and $\frac{dy}{dt} = 8$ unit/s at $x = \frac{\pi}{4}$ .

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Problem 31080

If \ $1,120 is invested at 15% interest compounded continuously, what is its value in 7 years using$ A=P e^{rt}$?

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Problem 31081

Determine if the speed of the particle at $t=4$ seconds is increasing, decreasing, or constant for $x(t)=10t-t^{2}$ .

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Problem 31082

Find $\frac{d x}{d t}$ when $y=2 \cot (x)$ , $\frac{d y}{d t}=8$ unit/s, and $x=\frac{\pi}{4}$ .

See Solution

Problem 31083

How fast is Butch's distance from the origin changing at $(7,24)$ if $\frac{d \tau}{d t}=-5 \mathrm{~m/s}$ and $\frac{d y}{d t}=-3 \mathrm{~m/s}$ ?

See Solution

Problem 31084

Find the inflection point of $f(x)=-0.1 x^{3}+3 x^{2}+1100$ . Choose from the options provided.

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Problem 31085

Find the linearization of $f(x)=\frac{1}{2 x}$ at $x=\frac{1}{10}$ . Provide exact coefficients: $L(x)=$

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Problem 31086

A radioactive substance has a half-life of 48 hours and starts with 20 grams. Find the function for remaining grams and when it reaches 11 grams.

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Problem 31087

Analyze a diver's ascent rate to see if they risk "the bends" when moving faster than $\pm 3 \frac{\mathrm{ft}}{10 \mathrm{ft}}$ .
1. Choose sections from three functions and check for "the bends."
2. Discuss the realism of your piecewise function.
3. Find the instantaneous rate of change at $1500 \mathrm{ft}$ .
The functions are:
1. $0.00005x(x-100)(x-206)$ for $0 \leq x \leq 235$
2. $\frac{0.6x+91}{x}-230$ for $235 < x \leq 800$
3. $\log(x-800)+1$ for $800 \leq x \leq 2000$

See Solution

Problem 31088

A cat jumps up $3.00 \mathrm{~m}$ . How long is it in the top $1.00 \mathrm{~m}$ (up and down)?

See Solution

Problem 31089

A radioactive substance has a half-life of 36 hours and starts with 13 grams. Find the function for remaining grams and when it reaches 5 grams.

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Problem 31090

Ein Land hat 20 Millionen Einwohner, die nach 5 Jahren auf 23 Millionen steigen. Bestimme die Wachstumsfunktion $N(t)$ und skizziere den Graphen. Berechne den Prozentsatz des Grenzbestandes nach 50 Jahren und die Zeit für 95\% des Grenzbestandes.

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Problem 31091

Bevölkerungswachstum: Ein Land hat 20 Mio. Einwohner, nach 5 Jahren 23 Mio. Maximal 100 Mio. Wachstumsfunktion: $N(t)=\frac{a}{1+b \cdot e^{-k t}}$ skizzieren für $0<t \leq 100$ .

See Solution

Problem 31092

Find the area of region $S$ bounded by $y=\sqrt{-x+4}$ , $y=2$ , and $x=4$ .
Also, write integrals for volumes with square cross sections (perpendicular to $y$ ) and rectangle cross sections (perpendicular to $x$ ) of height 3.

See Solution

Problem 31093

If $\frac{d f(x)}{d x}=x^{2}-3 x+2$ for $x>1$ , find $\frac{d f^{-1}}{d x}$ at $x=f(3)$ .

See Solution

Problem 31094

Find intervals where the function $f(x) = \frac{x^{2} - 2x + 1}{x + 1}$ is increasing or decreasing, and identify relative minima or maxima for integer $x$ .

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Problem 31095

A 250-kg car with a 75-kg passenger starts at rest 20.0 m high. Neglect friction. Find:
a) Gravitational potential energy at A. b) Speed at B. c) Speed at C (10.0 m high). d) Compare total mechanical energy at A, B, and C. (Ans: a) $63700 \mathrm{~J}$ ; b) $19.8 \mathrm{~m/s}$ ; c) $14 \mathrm{~m/s}$ ; d) TME is the same at all points)

See Solution

Problem 31096

Aufgabe: Schwimmbecken mit Leck. Gegeben ist die Funktion $f(x)=-\frac{1}{20} x^{3}+\frac{6}{5} x^{2}-\frac{21}{5} x+4$ .
a) Füllmenge zu Beginn. b) Durchschnittliche Zulaufgeschwindigkeit in 5 Stunden. c) Füllvorgang beschreiben (Zulauf, Abfluss, Füllstand). d) Momentane Füllgeschwindigkeit nach 10 Stunden. e) Warum ist die Füllgeschwindigkeit nach 8 Stunden am größten?

See Solution

Problem 31097

Find the relative minimum point of the function $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ .

See Solution

Problem 31098

Finde die Funktionen $f$ für die Ableitungen: a) $f^{\prime}(x)=x^{3}$ , b) $f^{\prime}(x)=10 x^{4}$ , c) $f^{\prime}(x)=6 x^{2}+8 x^{3}$ , d) $f^{\prime}(x)=x^{9}-5 x^{5}$ , e) $f^{\prime}(x)=\frac{2}{x^{2}}$ , f) $f^{\prime}(x)=\frac{10}{x^{3}}$ .

See Solution

Problem 31099

Schätzen und berechnen Sie die Fläche unter den Funktionen:
1. $f(x)=x^{2}-1$ , $f(x)=2 \cdot \sin (x)$ , $f(x)=-2 \cdot \sqrt{x}+2$
2. $f(x)=\frac{1}{(2 x-3)^{2}}$ , $f(x)=\frac{-1}{(x-1)^{2}}$ , $f(x)=-\frac{1}{3} \cdot\left[\frac{1}{(x+2)^{3}}-1\right]$

See Solution

Problem 31100

Bestimme den Funktionsterm $f$ aus der Ableitung $f^{\prime}$ und finde die Konstante $C$ für den Punkt $P$ . a) $f^{\prime}(x)=2 x, P(1 \mid 3)$ b) $f^{\prime}(x)=6 x^{2}, P(-1 \mid -1)$ c) $f^{\prime}(x)=\frac{5}{x^{2}}, P(0,5 \mid 5)$ d) $f^{\prime}(x)=\frac{3}{\sqrt{x}}, P(4 \mid 10)$ e) $f^{\prime}(x)=x^{2}+6 x+2, P(0 \mid 3)$ f) $f^{\prime}(x)=0, P(1 \mid 1)$

See Solution

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Calculus

Problem 31001

Which option represents the acceleration of a particle: first or second time derivative of velocity?

Problem 31002

Finde die Ableitung von f(x)=x2−xf(x) = x^2 - xf(x)=x2−x.

Problem 31003

What is the slope of the acceleration vs. time plot for a baseball tossed upward? Options: constant positive, constant negative, or zero slope?

Problem 31004

A particle's velocity decreases at a constant rate. What does the acceleration vs. time plot look like?

Problem 31005

Evaluate the integral ∫0πx⋅sin⁡(x2)dx\int_{0}^{\pi} x \cdot \sin \left(x^{2}\right) d x∫0π​x⋅sin(x2)dx.

Problem 31006

A rumour spreads in a school of 1200 pupils. Given dx dt=x(1200−x)3600\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x(1200-x)}{3600} dtdx​=3600x(1200−x)​ with 300 pupils aware at t=0t=0t=0, show t=3ln⁡(3x1200−x)t=3 \ln \left(\frac{3 x}{1200-x}\right)t=3ln(1200−x3x​).

Problem 31007

Find the relative maximum point for the function f(x)=x2−3x−2f(x)=\frac{x^{2}-3}{x-2}f(x)=x−2x2−3​ using f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x).

Problem 31008

Integrate ∫11−x2dx\int \frac{1}{\sqrt{1-x^{2}}} dx∫1−x2​1​dx using the substitution x=sin⁡(z)x=\sin(z)x=sin(z) and find dxdxdx.

Problem 31009

Find the integral of 1−x2\sqrt{1-x^{2}}1−x2​ with respect to xxx.

Problem 31010

Calculate the integral: ∫11−x2dx\int \frac{1}{\sqrt{1-x^{2}}} d x∫1−x2​1​dx

Problem 31011

Evaluate the integral: ∫14−4x2dx\int \frac{1}{\sqrt{4-4 x^{2}}} d x∫4−4x2​1​dx

Problem 31012

Find the price elasticity of demand when P=£10\mathrm{P} = £10P=£10 for the demand function Qd=1000−20P\mathrm{Qd} = 1000 - 20PQd=1000−20P.

Problem 31013

How much work is done to stretch a spring from its natural length to 10in10 \mathrm{in}10in if a force of 10lb10 \mathrm{lb}10lb stretches it 4in4 \mathrm{in}4in?

Problem 31014

Find xxx values for y=x2x2−3x+2y=\frac{x^{2}}{x^{2}-3 x+2}y=x2−3x+2x2​ when dydx=0\frac{d y}{d x}=0dxdy​=0.

Problem 31015

Find the price elasticity of demand when P=£10\mathrm{P} = £10P=£10 for the demand function Qd=1000−20P\mathrm{Qd} = 1000 - 20\mathrm{P}Qd=1000−20P.

Problem 31016

Problem 31017

Find the derivative of tan⁡(x2)\tan(x^2)tan(x2) with respect to xxx.

Problem 31018

Evaluate the integral ∫xex2dx\int x e^{x^{2}} d x∫xex2dx.

Problem 31019

Berechnen Sie die Integrale und Stammfunktionen für gegebene Werte von f(x)f(x)f(x) und g(x)g(x)g(x) sowie bestimmte Integrale.

Problem 31020

Calculate the integrals: a) ∫13(6x2−5)dx\int_{1}^{3}(6 x^{2}-5) dx∫13​(6x2−5)dx and b) ∫−21(−x)dx\int_{-2}^{1}(-x) dx∫−21​(−x)dx. What are the results?

Problem 31021

Find the relative minimum point of the function f(x)=x2−3x−2f(x) = \frac{x^{2}-3}{x-2}f(x)=x−2x2−3​ using f′(x)f'(x)f′(x) and f′′(x)f''(x)f′′(x).

Problem 31022

Find the price elasticity of demand when Qd=1000−20P\mathrm{Qd}=1000-20 \mathrm{P}Qd=1000−20P and P=£10P=£ 10P=£10. Options: 0.125, 0.025, 0.25, 0.5.

Problem 31023

a) Evaluate ∫13(6x2−5)dx\int_{1}^{3}(6 x^{2}-5) dx∫13​(6x2−5)dx and find the result. b) Calculate ∫−2−1(−x)dx\int_{-2}^{-1}(-x) dx∫−2−1​(−x)dx and determine the value.

Problem 31024

Find ∫48f(x4−1)dx\int_{4}^{8} f\left(\frac{x}{4}-1\right) d x∫48​f(4x​−1)dx given ∫01f(x)dx=4\int_{0}^{1} f(x) d x=4∫01​f(x)dx=4. Options: 0, 10, 16, 20, 80.

Problem 31025

Find the solution of the equation: dydx=2xy+y2x2+2xy \frac{d y}{d x}=\frac{2 x y+y^{2}}{x^{2}+2 x y} dxdy​=x2+2xy2xy+y2​.

Problem 31026

Find the derivative of the function f(x)=(x+1)3xf(x)=(x+1)^{3 x}f(x)=(x+1)3x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 31027

Find the integral ∫x+1 dx\int \sqrt{x+1} \, dx∫x+1​dx and choose the correct answer from the options provided.

Problem 31028

Find local extrema of f(x)=x4−2f(x)=x^{4}-2f(x)=x4−2. Where are the local maxima and minima?

Problem 31029

Find values of ccc for the Mean Value Theorem with f(x)=x3−4xf(x)=x^{3}-4xf(x)=x3−4x, for −2≤x≤2-2 \leq x \leq 2−2≤x≤2.

Problem 31030

y=x3−5x+3y=x^{3}-5 x+3y=x3−5x+3 eğrisinin OxO xOx-ekseniyle 45∘45^{\circ}45∘ teğetinin denklemi hangisidir? A) B) C) D) E)

Problem 31031

Find the limit: lim⁡x→0(ex+x)1/2x\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / 2 x}limx→0​(ex+x)1/2x. What is the result?

Problem 31032

Eine Kugel wird aus 30m30 m30m Höhe auf der Erde (gε=9,81ms2g_{\varepsilon}=9,81 \frac{m}{s^{2}}gε​=9,81s2m​) und Neptun (gN=11ms2g_{N}=11 \frac{m}{s^{2}}gN​=11s2m​) geworfen. Berechne die Aufprallgeschwindigkeiten in kmh\frac{km}{h}hkm​ und die Zeitdifferenz bis zum Aufprall.

Problem 31033

Find the second derivative of the function y=78xy=\frac{7}{8x}y=8x7​.

Problem 31034

جد الفترات التي تزداد وتنقص فيها الدالة fff بناءً على الرسم البياني لـ y=f′(x)y=f^{\prime}(x)y=f′(x).

Problem 31035

Evaluate the integral ∫dxxln⁡x\int \frac{d x}{x \ln x}∫xlnxdx​. Choose the correct answer from the options.

Problem 31036

f(x)={2sin⁡x;x<03x2+2x;x≥0f(x)=\left\{\begin{array}{cc}2 \sin x & ; x<0 \\ 3 x^{2}+2 x & ; x \geq 0\end{array}\right.f(x)={2sinx3x2+2x​;x<0;x≥0​ fonksiyonunun (0,0)(0,0)(0,0) noktasındaki teğetinin denklemi nedir?

Problem 31037

Determine where f(x)=cos⁡xf(x)=\cos xf(x)=cosx is concave up and down on [0,2π][0, 2\pi][0,2π].

Problem 31038

Find the inflection points of fff given f′′(x)=(x−2)5(x−1)3(x+1)4f^{\prime \prime}(x)=(x-2)^{5}(x-1)^{3}(x+1)^{4}f′′(x)=(x−2)5(x−1)3(x+1)4.

Problem 31039

Find dydx\frac{d y}{d x}dxdy​ for y=∫1cos⁡xlog⁡7(1+t3)dty=\int_{1}^{\cos x} \log _{7}(1+t^{3}) dty=∫1cosx​log7​(1+t3)dt.

Problem 31040

Find intervals where the function fff increases and decreases based on its derivative y=f′(x)y=f^{\prime}(x)y=f′(x).

Finde die Ableitung von $f(x) = x^2 - x$ .

Evaluate the integral $\int_{0}^{\pi} x \cdot \sin \left(x^{2}\right) d x$ .

A rumour spreads in a school of 1200 pupils. Given $\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x(1200-x)}{3600}$ with 300 pupils aware at $t=0$ , show $t=3 \ln \left(\frac{3 x}{1200-x}\right)$ .

Find the relative maximum point for the function $f(x)=\frac{x^{2}-3}{x-2}$ using $f'(x)$ and $f''(x)$ .

Integrate $\int \frac{1}{\sqrt{1-x^{2}}} dx$ using the substitution $x=\sin(z)$ and find $dx$ .

Find the integral of $\sqrt{1-x^{2}}$ with respect to $x$ .

Calculate the integral: $\int \frac{1}{\sqrt{1-x^{2}}} d x$

Evaluate the integral: $\int \frac{1}{\sqrt{4-4 x^{2}}} d x$

Find the price elasticity of demand when $\mathrm{P} = £10$ for the demand function $\mathrm{Qd} = 1000 - 20P$ .

How much work is done to stretch a spring from its natural length to $10 \mathrm{in}$ if a force of $10 \mathrm{lb}$ stretches it $4 \mathrm{in}$ ?

Find $x$ values for $y=\frac{x^{2}}{x^{2}-3 x+2}$ when $\frac{d y}{d x}=0$ .

Find the price elasticity of demand when $\mathrm{P} = £10$ for the demand function $\mathrm{Qd} = 1000 - 20\mathrm{P}$ .

Find the derivative of $\tan(x^2)$ with respect to $x$ .

Evaluate the integral $\int x e^{x^{2}} d x$ .

Berechnen Sie die Integrale und Stammfunktionen für gegebene Werte von $f(x)$ und $g(x)$ sowie bestimmte Integrale.

Calculate the integrals: a) $\int_{1}^{3}(6 x^{2}-5) dx$ and b) $\int_{-2}^{1}(-x) dx$ . What are the results?

Find the relative minimum point of the function $f(x) = \frac{x^{2}-3}{x-2}$ using $f'(x)$ and $f''(x)$ .

Find the price elasticity of demand when $\mathrm{Qd}=1000-20 \mathrm{P}$ and $P=£ 10$ . Options: 0.125, 0.025, 0.25, 0.5.

a) Evaluate $\int_{1}^{3}(6 x^{2}-5) dx$ and find the result. b) Calculate $\int_{-2}^{-1}(-x) dx$ and determine the value.

Find $\int_{4}^{8} f\left(\frac{x}{4}-1\right) d x$ given $\int_{0}^{1} f(x) d x=4$ . Options: 0, 10, 16, 20, 80.

Find the solution of the equation: $\frac{d y}{d x}=\frac{2 x y+y^{2}}{x^{2}+2 x y}$ .

Find the derivative of the function $f(x)=(x+1)^{3 x}$ . What is $f^{\prime}(x)$ ?

Find the integral $\int \sqrt{x+1} \, dx$ and choose the correct answer from the options provided.

Find local extrema of $f(x)=x^{4}-2$ . Where are the local maxima and minima?

Find values of $c$ for the Mean Value Theorem with $f(x)=x^{3}-4x$ , for $-2 \leq x \leq 2$ .

$y=x^{3}-5 x+3$ eğrisinin $O x$ -ekseniyle $45^{\circ}$ teğetinin denklemi hangisidir? A) B) C) D) E)

Find the limit: $\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / 2 x}$ . What is the result?

Eine Kugel wird aus $30 m$ Höhe auf der Erde ( $g_{\varepsilon}=9,81 \frac{m}{s^{2}}$ ) und Neptun ( $g_{N}=11 \frac{m}{s^{2}}$ ) geworfen. Berechne die Aufprallgeschwindigkeiten in $\frac{km}{h}$ und die Zeitdifferenz bis zum Aufprall.

Find the second derivative of the function $y=\frac{7}{8x}$ .

جد الفترات التي تزداد وتنقص فيها الدالة $f$ بناءً على الرسم البياني لـ $y=f^{\prime}(x)$ .

Evaluate the integral $\int \frac{d x}{x \ln x}$ . Choose the correct answer from the options.

$f(x)=\left\{\begin{array}{cc}2 \sin x & ; x<0 \\ 3 x^{2}+2 x & ; x \geq 0\end{array}\right.$ fonksiyonunun $(0,0)$ noktasındaki teğetinin denklemi nedir?

Determine where $f(x)=\cos x$ is concave up and down on $[0, 2\pi]$ .

Find the inflection points of $f$ given $f^{\prime \prime}(x)=(x-2)^{5}(x-1)^{3}(x+1)^{4}$ .

Find $\frac{d y}{d x}$ for $y=\int_{1}^{\cos x} \log _{7}(1+t^{3}) dt$ .

Find intervals where the function $f$ increases and decreases based on its derivative $y=f^{\prime}(x)$ .

Find the limit as $x$ approaches 0 for $\frac{2^{x}-1}{3^{x}-1}$ . What is the result?

احسب حجم الجسم الناتج عن دوران المنطقة بين $y=(x-1)^{2}$ و $y=\sqrt{x-1}$ حول $x=-2$ .

Find the length of the curve $y=\frac{1}{3}(x^{2}+2)^{3/2}$ for $0 \leq x \leq 12$ .

$y=x^{\sin x}$ fonksiyonunun türevini bulun. Aşağıdakilerden hangisi doğrudur? A) B) C) D) E)

Find when the particle at $x(t)=16 t-3.0 t^{3}$ is momentarily at rest for $t$ values: 0.75s, 1.3s, 1.8s, 5.3s, 7.3s.

Find the average velocity of an object with $x=7t-3t^{2}$ from $t=0$ to $t=2$ seconds. Choices: $5 \mathrm{~m/s}$ , $11 \mathrm{~m/s}$ , $1 \mathrm{~m/s}$ , $-5 \mathrm{~m/s}$ , $-11 \mathrm{~m/s}$ .

Find how $\frac{d V}{d t}$ relates to $\frac{d r}{d t}$ for a cylinder with constant height using $V=\pi r^{2} h$ .

Find the intervals where the function $f$ increases and decreases based on the graph of its derivative $y=f'(x)$ .

Find the average velocity of an object with constant acceleration $4 \mathrm{~m/s}^2$ from $x=2 m$ to $x=8 m$ .

Find the acceleration of an automobile at $t=1.0$ s given $x(t)=27t-4.0t^{3}$ . Options: $23$ , $15$ , $-4$ , $-12$ , $-24$ m/s².

Find the acceleration of an object at the moment it stops, given $x(t)=75 t-1.0 t^{3}$ . Choices: $-9.8$ , $0$ , $9.2 \times 10^{3}$ , $-73$ , $-30$ m/s².

Evaluate the integral $\int \cos x 5^{\sin x} d x$ . Choose the correct answer from the options provided.

How far does an object fall during the second second of free fall? Options: $4.9 \mathrm{~m}$ , $9.8 \mathrm{~m}$ , $15 \mathrm{~m}$ , $20 \mathrm{~m}$ , $25 \mathrm{~m}$ .

Find critical values of $f(x)=\frac{x^{3}}{3}+\frac{x^{2}}{2}-2x$ for $x=\{-2, -1, 1, 2\}$ .

Find an antiderivative function $F(x)$ of $f(x)=|2 x|$ for $-2 \leq x \leq 2$ .

احسب نقطة الحد الأدنى النسبي لدالة $f(x)=\frac{1}{3} x^{3}-2 x^{2}+3 x+1$ باستخدام $f^{\prime}(x)$ و $f^{\prime \prime}(x)$ .

Find the average velocity of an object with $v=4t-3t^{2}$ from $t=0$ to $t=2$ seconds. Options include 0, $-2 \mathrm{~m/s}$ , $2 \mathrm{~m/s}$ , $-4 \mathrm{~m/s}$ , or need initial position.

A plane at 40.0 m/s drops supplies from 200.0 m. When to release? Find the velocity just before hitting the ground. $t=\sqrt{\frac{2400}{-9.8}}$

Find $\int_{3}^{5} f(x) d x$ given $\int_{1}^{5} f(x) d x=5$ and $\int_{1}^{3} f(x) d x=3$ .

Ein Wassertank hat zu Beginn $2 \mathrm{~m}^{3}$ Wasser. Die Zuflussrate erreicht max. $3 \frac{\mathrm{m}^{3}}{\mathrm{h}}$ nach 3 Stunden. Berechne das Volumen nach 6 Stunden.

A stone is dropped from a 190 m roof. What is its speed just before hitting the ground? Options: $43 \mathrm{~m/s}$ , $61 \mathrm{~m/s}$ , $120 \mathrm{~m/s}$ , $190 \mathrm{~m/s}$ , $1400 \mathrm{~m/s}$ .

A car starts at $16 \mathrm{~m/s}$ and slows down with $a=-0.50 \mathrm{t}$ . When does it stop: $4s$ , $8s$ , $16s$ , $32s$ , or $64s$ ?

Find the initial velocity and acceleration of an auto given $x(t)=27t-4.0t^{3}$ at $t=0$ .

A spot moves as $x=8.79 t-0.624 t^{3}$ . Find when it's at rest, its position, acceleration, and when it hits screen edges.

Finde a, sodass die Fläche zwischen der $x$ -Achse und $f(x)=\frac{1}{2} x^{2}$ im Intervall [a; 2] 12 beträgt.

Find $\frac{d y}{d t}$ for $y=\sin(2x)$ and $\frac{d x}{d t}=2$ at $x=\frac{\pi}{2}$ .

Bestimmen Sie die Fläche zwischen dem Graphen $f(x)=-0,25 x^{4}+x^{3}$ und der x-Achse.

Find the limits or show divergence for these sequences: (i) $a_{n}=\sqrt{n}(\sqrt{n+1}-\sqrt{n-1})$ ; (ii) $b_{n}=(-1)^{n} 2^{1/n}$ .

Simplify the expression: $\frac{x^{2}}{x-1}-\frac{3}{x-1}-\frac{x-3}{x-1}$ .

A ball's velocity changes from $20 \mathrm{~cm/s}$ to $0 \mathrm{~cm/s}$ in 5 seconds. Find the average acceleration.

Find $\frac{d x}{d t}$ when $y=2 \sin (x)$ , $\frac{d y}{d t}=1$ , and $x=\frac{\pi}{3}$ .