Calculus

Problem 27001

Leiten Sie die Funktionen ab und klammern Sie aus: a) $f(x)=2 x e^{3 x}$ , b) $g(x)=(x-3)e^{2 x}$ , c) $f(t)=t^{2} e^{\frac{1}{2} t}+5$ , d) $g(t)=(t^{3}-2 t^{2})e^{-2 t}$ , e) $f(x)=(2 x-4)e^{0,5 x}$ , f) $h(x)=(x^{2}-x)e^{-0,01 x}$ .

See Solution

Problem 27002

Find the tangent line equation to the curve $y=2^{-x^{2}}$ at $x=0$ and graph both the curve and tangent.

See Solution

Problem 27003

Analyze the model $P=\left(5 \times 10^{8}\right) e^{0.20015 t}$ for interest payments from 1967-1979. Calculate rates for 1968, 1978, 1988, and 1998, and verify with Statistics Canada.

See Solution

Problem 27004

Differentiate $y=\frac{6}{(x+2)}$ to find $\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}$ . What is 'a'?

See Solution

Problem 27005

Differentiate the following functions: a. $y=2^{3x}$ , b. $y=3.1^{x}+x^{3}$ , d. $w=10^{(5-6n+n^{2})}$ , e. $y=3^{x^{2}+2}$ .

See Solution

Problem 27006

Differentiate these functions: a. $y=2^{3x}$ , b. $y=3.1^{x}+x^{3}$ , d. $w=10^{(5-6n+n^{2})}$ , e. $y=3^{x^{2}+2}$ .

See Solution

Problem 27007

An epidemiologist starts with 2,000 bacteria decaying at $4.5\%$ per hour. How many will remain after 36 hours?

See Solution

Problem 27008

Find the derivatives of the following functions: a. $y=x^{5} \times(5)^{x}$ b. $y=x(3)^{x^{2}}$ c. $v=\frac{2^{t}}{t}$ d. $f(x)=\frac{\sqrt{3^{x}}}{2}$

See Solution

Problem 27009

Find the derivative of the following functions: a. $y=x^{5} \times(5)^{x}$ , c. $v=\frac{2^{t}}{t}$ .

See Solution

Problem 27010

Bestimme die Ableitung der Funktion $f(t) = \frac{1}{2} t^{-2}$ .

See Solution

Problem 27011

Graph the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<-1 \\ x+1 & \text { if } x \geq-1\end{array}\right.$ and find limits at $x=-1$ and $x=1$ .

See Solution

Problem 27012

Find the limit as $x$ approaches $-\infty$ for $\frac{2x - 4}{\sqrt{5x^2 + 3}}$ .

See Solution

Problem 27013

Find the derivatives: 1) $y = x(3)^{x^{2}}$ and 2) $f(x) = \frac{\sqrt{3^{x}}}{x^{2}}$ .

See Solution

Problem 27014

Find a function that behaves like $f(x)=\frac{6 x^{2}-4 x+18}{6 x+8}$ for large $|x|$ . $y=\square$

See Solution

Problem 27015

Find the derivative of the function $f(x) = \frac{\sqrt{3^{x}}}{x^{2}}$ .

See Solution

Problem 27016

Find the tangent line to the curve given by $x=t^{2}+2t$ and $y=t^{3}+t^{2}$ at $t=1$ .

See Solution

Problem 27017

Find the slope of the tangent line to the polar curve $r=\cos \theta$ at $\theta=\frac{\pi}{6}$ .

See Solution

Problem 27018

Find the half-life of a radioactive material with $P(t)=100(1.2)^{-t}$ and its decay rate at half-life.

See Solution

Problem 27019

Calculate the integral: $\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} d x$

See Solution

Problem 27020

Evaluate the integral: $\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} \, dx = ?$ Choose (A), (B), (C), or (D).

See Solution

Problem 27021

Determine the direction of the particle moving through the point $(3,-2)$ given its position $(x(t), y(t))=(5-2t, t^{2}-3)$ .

See Solution

Problem 27022

Find the integral: $\int \frac{6}{x^{2}+10 x+16} d x$ .

See Solution

Problem 27023

Evaluate the integral: $\int \frac{6}{x^{2}+10 x+16} d x$ . Choose the correct answer from the options.

See Solution

Problem 27024

Evaluate the integral: $\int \frac{8}{x^{2}-4} d x$ . Which is the correct answer? (A) (B) (C) (D) (E)

See Solution

Problem 27025

Find all values of $p$ for which the integral $\int_{1}^{\infty} \frac{1}{x^{3p+1}} dx$ converges.

See Solution

Problem 27026

Evaluate the integral $\int_{1}^{\infty} \frac{x}{(1+x^{2})^{2}} dx$ and choose the correct answer: (A) $-\frac{1}{2}$ , (B) $-\frac{1}{4}$ , (C) $\frac{1}{4}$ , (D) $\frac{1}{2}$ , (E) divergent.

See Solution

Problem 27027

Evaluate the integral $\int_{1}^{\infty} \frac{x^{2}}{(x^{2}+2)^{2}} dx$ . Options: (A) $-\frac{1}{8}$ , (B) हो, (C) $\frac{1}{8}$ , (b) 1, (1) divergent.

See Solution

Problem 27028

Find the area of the region $R$ between $y=e^{-2 x}$ and the $x$ -axis for $x \geq 3$ . Options: (A) $\frac{1}{2 e^{6}}$ , (B) $\frac{1}{e^{6}}$ , (C) $\frac{2}{e^{6}}$ , (D) $\frac{\pi}{2 e^{6}}$ , (E) infinite.

See Solution

Problem 27029

Montrer que le volume d'un cône de hauteur $h$ et rayon $r$ est $V=\frac{\pi r^{2} h}{3}$ en utilisant la méthode des disques.

See Solution

Problem 27030

Estimate the area change rate of a circle $A(r)=\pi r^{2}$ when the radius $r$ is 7.

See Solution

Problem 27031

Find the integral expression for the area of region $R$ between $f(\theta)=3+3\cos\theta$ and $g(\theta)=4+2\cos\theta$ .

See Solution

Problem 27032

Given $f(0)=2$ , $f(4)=-3$ , and $\int_{0}^{4} f(x) dx=8$ , find $\int_{0}^{4} x f'(x) dx$ . Choices: (A) -20 (B) -13 (C) -12 (D) -7 (E) 36.

See Solution

Problem 27033

Find the area between the polar curve $r=\sqrt{1+\frac{3}{\pi} \theta}$ and the $x$ -axis for $0 \leq \theta \leq \pi$ .

See Solution

Problem 27034

Find the area of the region $R$ in the first quadrant bounded by $r=\sqrt{6 \sin (3 \theta)}$ . Options: (A) 1 (B) 2 (C) 4 (D) 6

See Solution

Problem 27035

Find a differentiable function $f(x)$ such that $\int f(x) \sin x \, dx=-f(x) \cos x+\int 4 x^{3} \cos x \, dx$ . Which could be $f(x)$ ? (A) $\cos x$ (B) $\sin x$ (C) $4 x^{3}$ (D) $-x^{4}$ (E) $x^{4}$

See Solution

Problem 27036

Which integral finds the area bounded by $\theta=0$ , $\theta=\frac{\pi}{4}$ , and $r=\frac{2}{\cos \theta+\sin \theta}$ ?

See Solution

Problem 27037

Find the length of the path for $x=\sin(t^3)$ and $y=e^{5t}$ from $t=0$ to $t=\pi$ . Which integral is correct?

See Solution

Problem 27038

Which statement is true about the integral $\int_{0}^{\pi} \sec ^{2} x \, dx$ ? (A) 0, (B) $2/3$ , (C) diverges due to $\tan x$ , (D) diverges due to $\sec ^{2} x$ .

See Solution

Problem 27039

A particle moves on the parabola $y=x^{2}-x$ at speed $2\sqrt{10}$ . Find $\frac{d y}{d t}$ at $(2,2)$ .

See Solution

Problem 27040

Find the limit as $x$ approaches 5 for the function $\frac{1}{x-4}$ .

See Solution

Problem 27041

Which expression equals $\int_{0}^{2} \frac{17 x+4}{3 x^{2}-7 x-6} d x$ ? (A) $\int_{0}^{2} \frac{2}{x-3} d x+\int_{0}^{2} \frac{5}{3 x+2} d x$ (B) $\int_{0}^{2} \frac{5}{x-3} d x+\int_{0}^{2} \frac{2}{3 x+2} d x$ (C) $\int_{0}^{2} \frac{4}{x-3} d x+\int_{0}^{2} \frac{17 x}{3 x+2} d x$ (D) $\int_{0}^{2} \frac{17 x}{x-3} d x+\int_{0}^{2} \frac{4}{3 x+2} d x$

See Solution

Problem 27042

Find $\lim _{x \rightarrow-4} 3 x^{2}$ using the given values of $f(x)$ .

See Solution

Problem 27043

Find the limit: $\lim _{x \rightarrow 5} \frac{1}{x-4}=\square$ (Enter an integer or decimal.)

See Solution

Problem 27044

Find the limit: $\lim _{x \rightarrow 2} \frac{x+3}{x^{2}+1}=\square$ (Enter an integer or decimal.)

See Solution

Problem 27045

Find the slope of the tangent line to $f(x)=-4 x^{2}$ at $(3,-36)$ and its slope-intercept equation.

See Solution

Problem 27046

Find the slope of the tangent line to $f(x)=-4 x^{2}$ at $(3,-36)$ and its slope-intercept equation. What is $m_{\tan }=\square$ ?

See Solution

Problem 27047

A fan throws a puck at $15 \mathrm{~m/s}$ and 35° from $10 \mathrm{~m}$ . Find: a) time in air, b) max height, c) horizontal distance.

See Solution

Problem 27048

Determine the type of discontinuity for the graph of $-\frac{4}{x^{2}}$ : infinite, jump, point, or continuous.

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Problem 27049

Calculez l'augmentation du profit total lorsque $q$ passe de 5 à 10 avec $P_{m}(q)=46-4q$ .

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Problem 27050

Quel est le profit total pour 10 unités si le profit total pour 6 unités est de 230 \ $et$ P_{m}=\frac{80}{\sqrt{4 q+1}}$?

See Solution

Problem 27051

Determine the type of discontinuity for $f(x)=\frac{x^{3}+7 x^{2}+10 x}{x+5}$ : infinite, jump, point, or continuous?

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Problem 27052

Find when the particle with velocity $v(t)=-2t+6$ has increasing velocity and increasing speed.

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Problem 27053

Untersuchen Sie die Funktion $f(x)=2 x^{2}+3 x-5$ auf lokale Extremalpunkte und skizzieren Sie ihren Graphen.

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Problem 27054

Determine which integers $1, 2, 3, 4, 5$ could be $x$ -coordinates of relative minima or maxima based on $h'(x)$ and $h''(x)$ .

See Solution

Problem 27055

Determine the $x$ -coordinates of relative minima and maxima for $h(x)$ using $h'(x)$ and $h''(x)$ values at $1, 2, 3, 4, 5$ .

See Solution

Problem 27056

Find if the curve with slope $\frac{dy}{dx} = \frac{x-2}{y}$ has a local min, max, or neither at (2,3) using the second derivative test.

See Solution

Problem 27057

Ein radioaktives Material hat zu Beginn $0,5 \mathrm{~kg}$ . Nach einem Jahr sind es $0,4 \mathrm{~kg}$ . Bestimme die Exponentialfunktion und die Halbwertszeit. Wann sind $1 \mathrm{~g}$ Uran erreicht?

See Solution

Problem 27058

Wasserhöhe $h_{k}(t)=20 \cdot t \cdot e^{-k t}+10$ .
a) Höhe zu Beginn ( $t=0$ )? b) Finde $k$ , wenn $h_{k}(1)=12$ m. c) Änderungsrate bei $t=0$ ? d) Frage zu $f_{2}(t+1)-f_{2}(t)=0,5$ ?

See Solution

Problem 27059

Gegeben ist die Funktion $f(x)=-0,0002 x^{6}+0,0078 x^{5}-0,1098 x^{4}+0,6562 x^{3}-1,386 x^{2}+6$ für den Wasserverbrauch.
1. Berechne $f(7.5)$ .
2. Warum kann $f(x)$ nur im Intervall $[0; 14]$ den Verbrauch modellieren?
3. Finde die Hoch- und Tiefpunkte von $f(x)$ .
4. Analysiere das Monotonieverhalten von $f(x)$ und interpretiere es.
5. Bestimme die Zeiten, bei denen $f(x)=600$ .

See Solution

Problem 27060

Berechnen Sie die Fläche A zwischen den Graphen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ für $x_{1}=-1$ und $x_{2}=1,5$ .

See Solution

Problem 27061

Find the rate of change of the equation $y=3x-3$ .

See Solution

Problem 27062

Use the function $g(x)=x^{4}-3 x^{3}-3 x^{2}-3$ .
a. Calculate the average rate of change of $g(x)$ on $[0,3]$ .
b. Find all $x$ in $[0,3]$ where the instantaneous rate equals this average rate, accurate to three decimal places.

See Solution

Problem 27063

Find the $c$ values that satisfy the Mean Value Theorem for $f(x)=x+\sqrt{x}$ on the interval $[1,4]$ .

See Solution

Problem 27064

Approximate $f(3.1)$ using the tangent line at the point $(3,2)$ on $y(x)=x^{3}-3 x^{2}+2$ .

See Solution

Problem 27065

A kite string is held 5 feet high and let out at $2 \mathrm{ft/sec}$ . Find the kite's horizontal speed when the string is 125 feet.

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Problem 27066

1. Find $f^{\prime}(3)$ for the tangent line to $y=f(x)$ at $x=3$ .
2. Write the equation of the tangent line.
3. Use the tangent line to estimate $f(2.9)$ .

See Solution

Problem 27067

Calculate the integral: $\int\left(\frac{2}{x^{5}}+4 x^{7}\right) d x$ .

See Solution

Problem 27068

Find the integral of $x^{-7}$ with respect to $x$ .

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Problem 27069

Find the derivative of the function $f(x)=\frac{1}{3} x^{-9}$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 27070

Find the derivative of the function $f(x)=\sqrt[3]{x^{2}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 27071

Find the integral: $\int \frac{3 x}{2 x-3} d x$

See Solution

Problem 27072

Find the derivative of the function $f(x)=\frac{1}{x^{2}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 27073

Find the derivative of the function $f(x)=3 x^{4}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 27074

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x}{3 \sin \frac{3 x}{2}}$

See Solution

Problem 27075

Evaluate the integral $\int_{0}^{2}(x-3)^{-2} d x$ .

See Solution

Problem 27076

Evaluate the integral $\int_{-1}^{-0.5}\left(\frac{1}{x^{2}}+x^{3}\right) dx$ .

See Solution

Problem 27077

Calculate $10 \int_{1}^{4} \frac{x^{3}-x^{2}}{x^{5}} dx$ .

See Solution

Problem 27078

Gegeben ist die Funktion $f(x, y)=(y x^{y}+x y)^{x}$ . Welche Ableitungen sind korrekt?

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Problem 27079

Die Funktion $f(a, b, c)=4 a^{3} \sqrt[3]{b^{2}} c^{2}$ hat die dritte Ableitung 0. Nach welcher Variable wurde abgeleitet? $a$ , $b$ , $c$ oder keine?

See Solution

Problem 27080

Find the limit: $\lim _{x \rightarrow 0} \frac{x}{-2 \sin x}$ .

See Solution

Problem 27081

Calculate $10 \int_{1}^{4} \frac{x^{3}-x^{2}}{x^{5}} d x$ .

See Solution

Problem 27082

Find the limit as $x$ approaches infinity for $\frac{x-x^{4}}{(x^{3}-1)(2x+3)}$ .

See Solution

Problem 27083

Find where the function $f(x)=x e^{x}-3 e^{x}$ is decreasing, its relative extrema, concave up intervals, and inflection points.

See Solution

Problem 27084

Find the derivative $F'(3)$ for the function $F(x) = 4x - x^2$ using the limit definition.

See Solution

Problem 27085

Find the min and max values of $y=x^{3}-12 x$ on the interval $[0,3]$ .

See Solution

Problem 27086

Find the derivative $\frac{d y}{d x}$ for the equation $\sin (y+2)+\ln (\cot x)=0$ .

See Solution

Problem 27087

Find where the tangent to $f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}+\frac{1}{2}$ is parallel to $y=10$ .

See Solution

Problem 27088

Find the $x$ -value(s) that satisfy the Mean Value Theorem for $f(x)=\frac{1}{3} x^{3}-x$ on $[-3,0]$ .

See Solution

Problem 27089

Let $f$ be continuous with $f(b-x)=f(x)$ . Prove $\int_{0}^{b} x f(x) dx=\frac{b}{2} \int_{0}^{b} f(x) dx$ . Then evaluate $\int_{0}^{\pi} \frac{x \sin^{2022} x}{\sin^{2022} x+\cos^{2022} x} dx$ .

See Solution

Problem 27090

(a) For $n \geq 2$ , show that $f'(x)=n x^{n-2}(1-x)^{n-2}[G(n)+H(n)x(1-x)]$ and find $G(n)$ , $H(n)$ . (b) Compute $S_{n}=\int_{0}^{1} \frac{s^{n}(1-s)^{n}}{n !} e^{s} ds$ : (i) Find $S_{0}$ , $S_{1}$ . (ii) Show $S_{n}=S_{n-2}-(4n-2)S_{n-1}$ for $n \geq 2$ . (iii) Prove $0<n!S_{n}<e$ for all $n \geq 0$ .

See Solution

Problem 27091

Vrai ou faux : évaluez les limites suivantes et justifiez.
1. $\lim _{x \rightarrow 4}\left(\frac{2 x}{x-4}-\frac{8}{x-4}\right)=\lim _{x \rightarrow 4} \frac{2 x}{x-4}-\lim _{x \rightarrow 4} \frac{8}{x-4}$
2. $\lim _{x \rightarrow 1} \frac{x^{2}+6 x-7}{x^{2}+5 x-6}=\frac{\lim _{x \rightarrow 1}\left(x^{2}+6 x-7\right)}{\lim _{x \rightarrow 1}\left(x^{2}+5 x-6\right)}$

See Solution

Problem 27092

How long to charge a capacitor to 1.17% of max with a $17-\mathrm{V}$ battery and time constant $82.1 \mathrm{~s}$ ?

See Solution

Problem 27093

Calculate these limits: (a) $\lim _{x \rightarrow 0}\left[\frac{\cos (4x^{2})-1}{x^{2} \sin (2x^{2})}\right]$ , (b) $\lim _{x \rightarrow 0}\left[\frac{\ln (1+2x+x^{2})}{x}\right]$ , (c) $\lim _{x \rightarrow 0}\left[\frac{(1+2x) \sqrt{1-x^{2}}-1}{\arcsin (x)}\right]$ . Use limits: $\lim _{h \rightarrow 0} \frac{\sin (h)}{h}=1$ , $\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}=1$ , $\lim _{h \rightarrow 0} \frac{\cos (h)-1}{h}=0$ .

See Solution

Problem 27094

Calculez $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}+2 x-3}$ .

See Solution

Problem 27095

Find all maxima, minima, and x-intercepts of the function $y=3 \cos (2 x+\pi)$ .

See Solution

Problem 27096

Find the derivative of $(4x^3 + x)(6x^2 - 2x)$ with respect to $x$ .

See Solution

Problem 27097

Find the derivatives of: h(t) = e t^{2} + 3 e^{-1} and f \cdot g(t) = \frac{e^{2 t}}{1 + e^{2 t}}.

See Solution

Problem 27098

Trouvez la limite suivante : $\lim _{x \rightarrow 1^{+}} \frac{x^{2}-9}{x^{2}+2 x-3}$

See Solution

Problem 27099

Find the derivatives of: $h(t)=e t^{2}+3 e^{-t}$ and $g(t)=\frac{e^{2 t}}{1+e^{2 t}}$ .

See Solution

Problem 27100

Solve for $x$ in the equation $f'(x) = \frac{3}{2} x^2 - 3 x = 0$ .

See Solution

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Calculus

Problem 27001

Problem 27002

Find the tangent line equation to the curve y=2−x2y=2^{-x^{2}}y=2−x2 at x=0x=0x=0 and graph both the curve and tangent.

Problem 27003

Analyze the model P=(5×108)e0.20015tP=\left(5 \times 10^{8}\right) e^{0.20015 t}P=(5×108)e0.20015t for interest payments from 1967-1979. Calculate rates for 1968, 1978, 1988, and 1998, and verify with Statistics Canada.

Problem 27004

Differentiate y=6(x+2)y=\frac{6}{(x+2)}y=(x+2)6​ to find dydx=a(x2+bx+c)\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}dxdy​=(x2+bx+c)a​. What is 'a'?

Problem 27005

Differentiate the following functions: a. y=23xy=2^{3x}y=23x, b. y=3.1x+x3y=3.1^{x}+x^{3}y=3.1x+x3, d. w=10(5−6n+n2)w=10^{(5-6n+n^{2})}w=10(5−6n+n2), e. y=3x2+2y=3^{x^{2}+2}y=3x2+2.

Problem 27006

Differentiate these functions: a. y=23xy=2^{3x}y=23x, b. y=3.1x+x3y=3.1^{x}+x^{3}y=3.1x+x3, d. w=10(5−6n+n2)w=10^{(5-6n+n^{2})}w=10(5−6n+n2), e. y=3x2+2y=3^{x^{2}+2}y=3x2+2.

Problem 27007

An epidemiologist starts with 2,000 bacteria decaying at 4.5%4.5\%4.5% per hour. How many will remain after 36 hours?

Problem 27008

Find the derivatives of the following functions: a. y=x5×(5)xy=x^{5} \times(5)^{x}y=x5×(5)x b. y=x(3)x2y=x(3)^{x^{2}}y=x(3)x2 c. v=2ttv=\frac{2^{t}}{t}v=t2t​ d. f(x)=3x2f(x)=\frac{\sqrt{3^{x}}}{2}f(x)=23x​​

Problem 27009

Find the derivative of the following functions: a. y=x5×(5)xy=x^{5} \times(5)^{x}y=x5×(5)x, c. v=2ttv=\frac{2^{t}}{t}v=t2t​.

Problem 27010

Bestimme die Ableitung der Funktion f(t)=12t−2f(t) = \frac{1}{2} t^{-2}f(t)=21​t−2.

Problem 27011

Graph the piecewise function f(x)={x2 if x<−1x+1 if x≥−1f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<-1 \\ x+1 & \text { if } x \geq-1\end{array}\right.f(x)={x2x+1​ if x<−1 if x≥−1​ and find limits at x=−1x=-1x=−1 and x=1x=1x=1.

Problem 27012

Find the limit as xxx approaches −∞-\infty−∞ for 2x−45x2+3\frac{2x - 4}{\sqrt{5x^2 + 3}}5x2+3​2x−4​.

Problem 27013

Find the derivatives: 1) y=x(3)x2y = x(3)^{x^{2}}y=x(3)x2 and 2) f(x)=3xx2f(x) = \frac{\sqrt{3^{x}}}{x^{2}}f(x)=x23x​​.

Problem 27014

Find a function that behaves like f(x)=6x2−4x+186x+8f(x)=\frac{6 x^{2}-4 x+18}{6 x+8}f(x)=6x+86x2−4x+18​ for large ∣x∣|x|∣x∣. y=□y=\squarey=□

Problem 27015

Find the derivative of the function f(x)=3xx2f(x) = \frac{\sqrt{3^{x}}}{x^{2}}f(x)=x23x​​.

Problem 27016

Find the tangent line to the curve given by x=t2+2tx=t^{2}+2tx=t2+2t and y=t3+t2y=t^{3}+t^{2}y=t3+t2 at t=1t=1t=1.

Problem 27017

Find the slope of the tangent line to the polar curve r=cos⁡θr=\cos \thetar=cosθ at θ=π6\theta=\frac{\pi}{6}θ=6π​.

Problem 27018

Find the half-life of a radioactive material with P(t)=100(1.2)−tP(t)=100(1.2)^{-t}P(t)=100(1.2)−t and its decay rate at half-life.

Problem 27019

Calculate the integral: ∫−2x2+7x−8(x+2)(2x−1)(1−x)dx\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} d x∫(x+2)(2x−1)(1−x)−2x2+7x−8​dx

Problem 27020

Evaluate the integral: ∫−2x2+7x−8(x+2)(2x−1)(1−x) dx=?\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} \, dx = ?∫(x+2)(2x−1)(1−x)−2x2+7x−8​dx=? Choose (A), (B), (C), or (D).

Problem 27021

Determine the direction of the particle moving through the point (3,−2)(3,-2)(3,−2) given its position (x(t),y(t))=(5−2t,t2−3)(x(t), y(t))=(5-2t, t^{2}-3)(x(t),y(t))=(5−2t,t2−3).

Problem 27022

Find the integral: ∫6x2+10x+16dx\int \frac{6}{x^{2}+10 x+16} d x∫x2+10x+166​dx.

Problem 27023

Evaluate the integral: ∫6x2+10x+16dx\int \frac{6}{x^{2}+10 x+16} d x∫x2+10x+166​dx. Choose the correct answer from the options.

Problem 27024

Evaluate the integral: ∫8x2−4dx\int \frac{8}{x^{2}-4} d x∫x2−48​dx. Which is the correct answer? (A) (B) (C) (D) (E)

Problem 27025

Find all values of ppp for which the integral ∫1∞1x3p+1dx\int_{1}^{\infty} \frac{1}{x^{3p+1}} dx∫1∞​x3p+11​dx converges.

Problem 27026

Evaluate the integral ∫1∞x(1+x2)2dx\int_{1}^{\infty} \frac{x}{(1+x^{2})^{2}} dx∫1∞​(1+x2)2x​dx and choose the correct answer: (A) −12-\frac{1}{2}−21​, (B) −14-\frac{1}{4}−41​, (C) 14\frac{1}{4}41​, (D) 12\frac{1}{2}21​, (E) divergent.

Problem 27027

Evaluate the integral ∫1∞x2(x2+2)2dx\int_{1}^{\infty} \frac{x^{2}}{(x^{2}+2)^{2}} dx∫1∞​(x2+2)2x2​dx. Options: (A) −18-\frac{1}{8}−81​, (B) हो, (C) 18\frac{1}{8}81​, (b) 1, (1) divergent.

Problem 27028

Find the area of the region RRR between y=e−2xy=e^{-2 x}y=e−2x and the xxx-axis for x≥3x \geq 3x≥3. Options: (A) 12e6\frac{1}{2 e^{6}}2e61​, (B) 1e6\frac{1}{e^{6}}e61​, (C) 2e6\frac{2}{e^{6}}e62​, (D) π2e6\frac{\pi}{2 e^{6}}2e6π​, (E) infinite.

Problem 27029

Montrer que le volume d'un cône de hauteur hhh et rayon rrr est V=πr2h3V=\frac{\pi r^{2} h}{3}V=3πr2h​ en utilisant la méthode des disques.

Problem 27030

Estimate the area change rate of a circle A(r)=πr2A(r)=\pi r^{2}A(r)=πr2 when the radius rrr is 7.

Problem 27031

Find the integral expression for the area of region RRR between f(θ)=3+3cos⁡θf(\theta)=3+3\cos\thetaf(θ)=3+3cosθ and g(θ)=4+2cos⁡θg(\theta)=4+2\cos\thetag(θ)=4+2cosθ.

Problem 27032

Given f(0)=2f(0)=2f(0)=2, f(4)=−3f(4)=-3f(4)=−3, and ∫04f(x)dx=8\int_{0}^{4} f(x) dx=8∫04​f(x)dx=8, find ∫04xf′(x)dx\int_{0}^{4} x f'(x) dx∫04​xf′(x)dx. Choices: (A) -20 (B) -13 (C) -12 (D) -7 (E) 36.

Problem 27033

Find the area between the polar curve r=1+3πθr=\sqrt{1+\frac{3}{\pi} \theta}r=1+π3​θ​ and the xxx-axis for 0≤θ≤π0 \leq \theta \leq \pi0≤θ≤π.

Problem 27034

Find the area of the region RRR in the first quadrant bounded by r=6sin⁡(3θ)r=\sqrt{6 \sin (3 \theta)}r=6sin(3θ)​. Options: (A) 1 (B) 2 (C) 4 (D) 6

Problem 27035

Problem 27036

Which integral finds the area bounded by θ=0\theta=0θ=0, θ=π4\theta=\frac{\pi}{4}θ=4π​, and r=2cos⁡θ+sin⁡θr=\frac{2}{\cos \theta+\sin \theta}r=cosθ+sinθ2​?

Problem 27037

Find the length of the path for x=sin⁡(t3)x=\sin(t^3)x=sin(t3) and y=e5ty=e^{5t}y=e5t from t=0t=0t=0 to t=πt=\pit=π. Which integral is correct?

Problem 27038

Which statement is true about the integral ∫0πsec⁡2x dx\int_{0}^{\pi} \sec ^{2} x \, dx∫0π​sec2xdx? (A) 0, (B) 2/32/32/3, (C) diverges due to tan⁡x\tan xtanx, (D) diverges due to sec⁡2x\sec ^{2} xsec2x.

Problem 27039

A particle moves on the parabola y=x2−xy=x^{2}-xy=x2−x at speed 2102\sqrt{10}210​. Find dydt\frac{d y}{d t}dtdy​ at (2,2)(2,2)(2,2).

Problem 27040

Find the limit as xxx approaches 5 for the function 1x−4\frac{1}{x-4}x−41​.

Problem 27041

Find the tangent line equation to the curve $y=2^{-x^{2}}$ at $x=0$ and graph both the curve and tangent.

Analyze the model $P=\left(5 \times 10^{8}\right) e^{0.20015 t}$ for interest payments from 1967-1979. Calculate rates for 1968, 1978, 1988, and 1998, and verify with Statistics Canada.

Differentiate $y=\frac{6}{(x+2)}$ to find $\frac{d y}{d x}=\frac{a}{(x^{2}+b x+c)}$ . What is 'a'?

Differentiate the following functions: a. $y=2^{3x}$ , b. $y=3.1^{x}+x^{3}$ , d. $w=10^{(5-6n+n^{2})}$ , e. $y=3^{x^{2}+2}$ .

Differentiate these functions: a. $y=2^{3x}$ , b. $y=3.1^{x}+x^{3}$ , d. $w=10^{(5-6n+n^{2})}$ , e. $y=3^{x^{2}+2}$ .

An epidemiologist starts with 2,000 bacteria decaying at $4.5\%$ per hour. How many will remain after 36 hours?

Find the derivatives of the following functions: a. $y=x^{5} \times(5)^{x}$ b. $y=x(3)^{x^{2}}$ c. $v=\frac{2^{t}}{t}$ d. $f(x)=\frac{\sqrt{3^{x}}}{2}$

Find the derivative of the following functions: a. $y=x^{5} \times(5)^{x}$ , c. $v=\frac{2^{t}}{t}$ .

Bestimme die Ableitung der Funktion $f(t) = \frac{1}{2} t^{-2}$ .

Graph the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<-1 \\ x+1 & \text { if } x \geq-1\end{array}\right.$ and find limits at $x=-1$ and $x=1$ .

Find the limit as $x$ approaches $-\infty$ for $\frac{2x - 4}{\sqrt{5x^2 + 3}}$ .

Find the derivatives: 1) $y = x(3)^{x^{2}}$ and 2) $f(x) = \frac{\sqrt{3^{x}}}{x^{2}}$ .

Find a function that behaves like $f(x)=\frac{6 x^{2}-4 x+18}{6 x+8}$ for large $|x|$ . $y=\square$

Find the derivative of the function $f(x) = \frac{\sqrt{3^{x}}}{x^{2}}$ .

Find the tangent line to the curve given by $x=t^{2}+2t$ and $y=t^{3}+t^{2}$ at $t=1$ .

Find the slope of the tangent line to the polar curve $r=\cos \theta$ at $\theta=\frac{\pi}{6}$ .

Find the half-life of a radioactive material with $P(t)=100(1.2)^{-t}$ and its decay rate at half-life.

Calculate the integral: $\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} d x$

Evaluate the integral: $\int \frac{-2 x^{2}+7 x-8}{(x+2)(2 x-1)(1-x)} \, dx = ?$ Choose (A), (B), (C), or (D).

Determine the direction of the particle moving through the point $(3,-2)$ given its position $(x(t), y(t))=(5-2t, t^{2}-3)$ .

Find the integral: $\int \frac{6}{x^{2}+10 x+16} d x$ .

Evaluate the integral: $\int \frac{6}{x^{2}+10 x+16} d x$ . Choose the correct answer from the options.

Evaluate the integral: $\int \frac{8}{x^{2}-4} d x$ . Which is the correct answer? (A) (B) (C) (D) (E)

Find all values of $p$ for which the integral $\int_{1}^{\infty} \frac{1}{x^{3p+1}} dx$ converges.

Evaluate the integral $\int_{1}^{\infty} \frac{x}{(1+x^{2})^{2}} dx$ and choose the correct answer: (A) $-\frac{1}{2}$ , (B) $-\frac{1}{4}$ , (C) $\frac{1}{4}$ , (D) $\frac{1}{2}$ , (E) divergent.

Evaluate the integral $\int_{1}^{\infty} \frac{x^{2}}{(x^{2}+2)^{2}} dx$ . Options: (A) $-\frac{1}{8}$ , (B) हो, (C) $\frac{1}{8}$ , (b) 1, (1) divergent.

Find the area of the region $R$ between $y=e^{-2 x}$ and the $x$ -axis for $x \geq 3$ . Options: (A) $\frac{1}{2 e^{6}}$ , (B) $\frac{1}{e^{6}}$ , (C) $\frac{2}{e^{6}}$ , (D) $\frac{\pi}{2 e^{6}}$ , (E) infinite.

Montrer que le volume d'un cône de hauteur $h$ et rayon $r$ est $V=\frac{\pi r^{2} h}{3}$ en utilisant la méthode des disques.

Estimate the area change rate of a circle $A(r)=\pi r^{2}$ when the radius $r$ is 7.

Find the integral expression for the area of region $R$ between $f(\theta)=3+3\cos\theta$ and $g(\theta)=4+2\cos\theta$ .

Given $f(0)=2$ , $f(4)=-3$ , and $\int_{0}^{4} f(x) dx=8$ , find $\int_{0}^{4} x f'(x) dx$ . Choices: (A) -20 (B) -13 (C) -12 (D) -7 (E) 36.

Find the area between the polar curve $r=\sqrt{1+\frac{3}{\pi} \theta}$ and the $x$ -axis for $0 \leq \theta \leq \pi$ .

Find the area of the region $R$ in the first quadrant bounded by $r=\sqrt{6 \sin (3 \theta)}$ . Options: (A) 1 (B) 2 (C) 4 (D) 6

Which integral finds the area bounded by $\theta=0$ , $\theta=\frac{\pi}{4}$ , and $r=\frac{2}{\cos \theta+\sin \theta}$ ?

Find the length of the path for $x=\sin(t^3)$ and $y=e^{5t}$ from $t=0$ to $t=\pi$ . Which integral is correct?

Which statement is true about the integral $\int_{0}^{\pi} \sec ^{2} x \, dx$ ? (A) 0, (B) $2/3$ , (C) diverges due to $\tan x$ , (D) diverges due to $\sec ^{2} x$ .

A particle moves on the parabola $y=x^{2}-x$ at speed $2\sqrt{10}$ . Find $\frac{d y}{d t}$ at $(2,2)$ .

Find the limit as $x$ approaches 5 for the function $\frac{1}{x-4}$ .

Find $\lim _{x \rightarrow-4} 3 x^{2}$ using the given values of $f(x)$ .

Find the limit: $\lim _{x \rightarrow 5} \frac{1}{x-4}=\square$ (Enter an integer or decimal.)

Find the limit: $\lim _{x \rightarrow 2} \frac{x+3}{x^{2}+1}=\square$ (Enter an integer or decimal.)

Find the slope of the tangent line to $f(x)=-4 x^{2}$ at $(3,-36)$ and its slope-intercept equation.

Find the slope of the tangent line to $f(x)=-4 x^{2}$ at $(3,-36)$ and its slope-intercept equation. What is $m_{\tan }=\square$ ?

A fan throws a puck at $15 \mathrm{~m/s}$ and 35° from $10 \mathrm{~m}$ . Find: a) time in air, b) max height, c) horizontal distance.

Determine the type of discontinuity for the graph of $-\frac{4}{x^{2}}$ : infinite, jump, point, or continuous.

Calculez l'augmentation du profit total lorsque $q$ passe de 5 à 10 avec $P_{m}(q)=46-4q$ .

Quel est le profit total pour 10 unités si le profit total pour 6 unités est de 230 \ $et$ P_{m}=\frac{80}{\sqrt{4 q+1}}$?

Determine the type of discontinuity for $f(x)=\frac{x^{3}+7 x^{2}+10 x}{x+5}$ : infinite, jump, point, or continuous?

Find when the particle with velocity $v(t)=-2t+6$ has increasing velocity and increasing speed.

Untersuchen Sie die Funktion $f(x)=2 x^{2}+3 x-5$ auf lokale Extremalpunkte und skizzieren Sie ihren Graphen.

Determine which integers $1, 2, 3, 4, 5$ could be $x$ -coordinates of relative minima or maxima based on $h'(x)$ and $h''(x)$ .

Determine the $x$ -coordinates of relative minima and maxima for $h(x)$ using $h'(x)$ and $h''(x)$ values at $1, 2, 3, 4, 5$ .

Find if the curve with slope $\frac{dy}{dx} = \frac{x-2}{y}$ has a local min, max, or neither at (2,3) using the second derivative test.

Ein radioaktives Material hat zu Beginn $0,5 \mathrm{~kg}$ . Nach einem Jahr sind es $0,4 \mathrm{~kg}$ . Bestimme die Exponentialfunktion und die Halbwertszeit. Wann sind $1 \mathrm{~g}$ Uran erreicht?

Berechnen Sie die Fläche A zwischen den Graphen $f(x)=0,5 x^{2}+2$ und $g(x)=-0,5 x+1$ für $x_{1}=-1$ und $x_{2}=1,5$ .

Find the rate of change of the equation $y=3x-3$ .

Use the function $g(x)=x^{4}-3 x^{3}-3 x^{2}-3$ .
a. Calculate the average rate of change of $g(x)$ on $[0,3]$ .
b. Find all $x$ in $[0,3]$ where the instantaneous rate equals this average rate, accurate to three decimal places.

Find the $c$ values that satisfy the Mean Value Theorem for $f(x)=x+\sqrt{x}$ on the interval $[1,4]$ .

Approximate $f(3.1)$ using the tangent line at the point $(3,2)$ on $y(x)=x^{3}-3 x^{2}+2$ .

A kite string is held 5 feet high and let out at $2 \mathrm{ft/sec}$ . Find the kite's horizontal speed when the string is 125 feet.

1. Find $f^{\prime}(3)$ for the tangent line to $y=f(x)$ at $x=3$ .
2. Write the equation of the tangent line.
3. Use the tangent line to estimate $f(2.9)$ .

Calculate the integral: $\int\left(\frac{2}{x^{5}}+4 x^{7}\right) d x$ .

Find the integral of $x^{-7}$ with respect to $x$ .

Find the derivative of the function $f(x)=\frac{1}{3} x^{-9}$ , i.e., compute $f^{\prime}(x)$ .

Find the derivative of the function $f(x)=\sqrt[3]{x^{2}}$ . What is $f^{\prime}(x)$ ?

Find the integral: $\int \frac{3 x}{2 x-3} d x$

Find the derivative of the function $f(x)=\frac{1}{x^{2}}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $f(x)=3 x^{4}$ . What is $f^{\prime}(x)$ ?

Find the limit: $\lim _{x \rightarrow \pi} \frac{\cos x}{3 \sin \frac{3 x}{2}}$