Calculus

Problem 33001

Find the velocity function $v(t)$ from $s(t)=6 t^{3}-72 t^{2}+270 t$ . Where is $v(t)=0$ ? Factor out GCF. Also find $a(t)$ .

See Solution

Problem 33002

Given $s(t)=2 t^{3}+7 t+9$ , find: (a) $v(t)$ , (b) $v(3)$ , (c) $a(t)$ , (d) $a(3)$ .

See Solution

Problem 33003

Find $M_{6}$ for $f(x)=\sqrt{6 x}$ on the interval $[3,6]$ . Round your answer to two decimal places.

See Solution

Problem 33004

Estimate the mosquito population after 14 days if $P(0)=589$ and $P^{\prime}(0)=95$ . Round to the nearest whole number.

See Solution

Problem 33005

Find the derivative of $\frac{2 \sin (x)-5}{3 x^{8}+7}$ using the quotient rule. No need to expand your answer.

See Solution

Problem 33006

Find the derivative of $f(x)=4 x(\sin x+\cos x)$ and evaluate it at $x=1$ .

See Solution

Problem 33007

Find the derivative of $g(x)=\sin (x)$ and evaluate it at $x=2$ . What is $g^{\prime}(2)=?$

See Solution

Problem 33008

Find the derivative of $g(x)=\cos(x)$ and evaluate it at $x=2$ . What is $g^{\prime}(2)$ ?

See Solution

Problem 33009

Find the derivative $f'(x)$ of $f(x)=2 \sin x + 2 \cos x$ and evaluate $f'(4)$ .

See Solution

Problem 33010

Find the derivative of $f(x)=\frac{3 x^{2} \tan x}{\sec x}$ and evaluate $f^{\prime}(4)$ .

See Solution

Problem 33011

Find the average velocity of the object described by $s(t)=-4.9 t^{2}+34 t+25$ over intervals: a. $[0,1]$ b. $[0,2]$ c. $[0,4]$ d. $[0,h]$ , where $h>0$ .

See Solution

Problem 33012

Find $f(x)=\frac{x^{2}-16}{x+4}$ for $x=-3.9, -3.99, -3.999, -4.1, -4.01, -4.001$ . Conjecture $\lim_{x \to -4} f(x)$ .

See Solution

Problem 33013

Find the limit as $x$ approaches -6 for $\frac{5x+30}{x^2+2x-24}$ using values: -5.99, -5.999, -6.01, -6.001.

See Solution

Problem 33014

Find the limit as $x$ approaches -4 for $2x^3 - 3x^2 + 5x + 8$ or state if it doesn't exist.

See Solution

Problem 33015

Calculate the right Riemann sum $R_4$ for $\frac{1}{x^2 + 1}$ on the interval $[-2, 2]$ .

See Solution

Problem 33016

Find the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$ . Simplify and evaluate if possible. A. Exact answer, B. Limit does not exist.

See Solution

Problem 33017

Find the limit: $\lim _{x \rightarrow 4} \frac{5 x^{2}-80}{x-4}$ . Simplify if possible and evaluate the limit.

See Solution

Problem 33018

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{2 \sin x}{1+\cos x}$ and calculate $f^{\prime}(3)$ .

See Solution

Problem 33019

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4+9 x+2 x^{2}}{x^{2}}$ .

See Solution

Problem 33020

Find the inverse Laplace transform of $\frac{1}{s^{3}+4 s^{2}+3 s}$ .

See Solution

Problem 33021

Find the derivative $f'(x)$ of $f(x)=\frac{5 \sin x}{2+\cos x}$ and evaluate $f'(3)$ .

See Solution

Problem 33022

Find the limits: $\lim _{x \rightarrow \infty} e^{x}$ , $\lim _{x \rightarrow-\infty} e^{x}$ , and $\lim _{x \rightarrow \infty} e^{-x}$ .

See Solution

Problem 33023

Find $f^{\prime}(x)$ for $f(x)=\cos x-7 \tan x$ and calculate $f^{\prime}(3)$ .

See Solution

Problem 33024

Curve $C$ : $(x^{2}+y^{2})^{2}=6xy$ for $x>0, y>0$ .
(a) Show $r^{2}=3\sin 2\theta$ for $0<\theta<\frac{\pi}{2}$ .
(b) Find the area of shaded region $R$ using calculus.

See Solution

Problem 33025

Which investment yields more in 1 year: $12\%$ monthly or $11.86\%$ continuously on \$13,000?

See Solution

Problem 33026

Find the derivative of $f(x)=5 x \sin x \cos x$ and evaluate $f^{\prime}(1)$ .

See Solution

Problem 33027

Find the derivative $f'(x)$ of the function $f(x) = -7 \cos x - 2 \tan x$ and evaluate $f'(\frac{\pi}{6})$ .

See Solution

Problem 33028

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x+3$ , where $h \neq 0$ .

See Solution

Problem 33029

Express the limit as an integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{}\right)^{2}-3\left(x_{i}^{}\right)^{3}\right) \Delta x$ over $[0,2]$ .

See Solution

Problem 33030

List the three conditions for a function to be continuous at a point. Select all that apply: A. $\lim _{x \rightarrow a} f(x)$ exists. B. $\lim _{x \rightarrow a} f(x)=f(a)$ C. $\lim _{x \rightarrow a} f(x) \neq f(a)$ D. $f(a)$ is defined.

See Solution

Problem 33031

Find the formula for $R_{N}$ for $f(x)=9x+7$ on $[3,6]$ and compute the limit as $N$ approaches infinity: $\lim _{N \rightarrow \infty} R_{N}.$

See Solution

Problem 33032

Find the derivative $f'(x)$ of the function $f(x)=\frac{\tan x-5}{\sec x}$ and evaluate $f'(5)$ .

See Solution

Problem 33033

Determine if the function $F(r)$ is continuous at $r=R$ using the definition of continuity.

See Solution

Problem 33034

Find $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ for $f(x)=\frac{5x^3-9}{x^4+7x^2}$ . What are the horizontal asymptotes?

See Solution

Problem 33035

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=\frac{5 x^{3}-9}{x^{4}+7 x^{2}}$ .

See Solution

Problem 33036

Calculate the average rate of change of $f(x)=x^{2}+9x$ between $x=1$ and $x=2$ .

See Solution

Problem 33037

Find the limit as $x$ approaches 0 for $\sin \left(\frac{200 \pi}{x}\right)$ .

See Solution

Problem 33038

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=9$ to $x=49$ .

See Solution

Problem 33039

Find the tangent line equation for $y=6 \sin x$ at $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .

See Solution

Problem 33040

Find the limit: $\lim _{x \rightarrow 1} \frac{5-5 x}{1-\sqrt{x}}$ .

See Solution

Problem 33041

Find the 28th derivative of $f(x)=\cos(x)$ . What is the result?

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Problem 33042

Find the oscillation rate of the spring at $t=2$ s for $s(t)=3 \cos t$ . Round to four decimal places.

See Solution

Problem 33043

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{39 x^{7}+2 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

See Solution

Problem 33044

Is the function $f(x)=\frac{2 x^{2}+3 x+1}{x^{2}-7 x}$ continuous at $a=7$ ? Justify using the checklist. Select all that apply.

See Solution

Problem 33045

Calculate the integral $I=\int_{0}^{\pi / 2}(2 \sin (\theta)+\sin ^{3}(\theta)) d \theta$ .

See Solution

Problem 33046

Is the function $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text{if } x \neq 2 \\ 10 & \text{if } x=2\end{array}\right.$ continuous at $a=2$ ? Select true statements.

See Solution

Problem 33047

Calculate the derivative of $f(x)=3 x^{2}-2 x$ at the point $(-1,5)$ . Find $f'(-1)=\square$ .

See Solution

Problem 33048

Redefine the function $f$ for continuity at $x=0$ : $f(x) = \begin{cases} \frac{9}{x} + \frac{-8x + 36}{x(x-4)} & x \neq 0, 4 \\ 4 & x = 0, 4 \end{cases}$

See Solution

Problem 33049

Determine where the function $p(x)=4 x^{6}-5 x^{3}+1$ is continuous. Provide your answer in interval notation.

See Solution

Problem 33050

Find the slope of the tangent line to $f(x)=x^{2}+5$ at point $P(-5,30)$ and its equation.

See Solution

Problem 33051

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{39 x^{7}+2 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

See Solution

Problem 33052

1. Find the domain of these functions in interval notation: (a) $f(x)=\frac{3x+7}{x^{2}-2x-8}$ (b) $g(x)=\sqrt{3-x}+\sqrt{x+5}$
2. Graph the piecewise function: $f(x)=\begin{cases} x+1 & \text{if } x<-1 \\ 2-x^{2} & \text{if } -1 \leq x<1 \\ 2x-1 & \text{if } x \geq 1 \end{cases}$
Find limits and continuity at $x=-1$ and $x=1$ .

See Solution

Problem 33053

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=\frac{39 x^{7}+4 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

See Solution

Problem 33054

Find the slope of the tangent line to $f(x)=x^{2}-1$ at $P(2,3)$ using $m_{\tan }=\lim _{x \rightarrow 2} \frac{f(x)-3}{x-2}$ . Then, find the tangent line equation and plot both.

See Solution

Problem 33055

Classify these first order equations as linear homogeneous, linear inhomogeneous, or nonlinear:
1) $u_{t}+x u_{x}=0$ 2) $u_{t}+u u_{x}=0$ 3) $u_{t}+x u_{x}-u=0$ 4) $u_{t}+u u_{x}+x=0$ 5) $u_{t}+u_{x}-u^{2}=0$ 6) $u_{t}^{2}-u_{x}^{2}-1=0$ 7) $u_{x}^{2}+u_{y}^{2}-1=0$ 8) $x u_{x}+y u_{y}+z u_{z}=0$ 9) $u_{x}^{2}+u_{y}^{2}+u_{z}^{2}-1=0$ 10) $u_{t}+u_{x}^{2}+u_{y}^{2}=0$

See Solution

Problem 33056

Find the slope of the tangent line to $f(x)=x^{2}-2$ at $P(4,14)$ using $m_{\tan }=\lim _{x \rightarrow 4} \frac{f(x)-14}{x-4}$ .

See Solution

Problem 33057

Find the average rate of change of $h(x)=x^{2}-9x$ from 2 to 3 and the secant line equation in slope-intercept form.

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Problem 33058

Classify these equations by order and type: linear homogeneous, linear inhomogeneous, or non-linear (where $u$ is unknown):
1. $u_{t}+(1+x^{2}) u_{x x}=0$
2. $u_{t}-(1+u^{2}) u_{x x}=0$
3. $u_{t}+u_{x x x}=0$
4. $u_{t}+u u_{x}+u_{x x x}=0$
5. $u_{t t}+u_{x x x x}=0$
6. $u_{t t}+u_{x x x x}+u=0$
7. $u_{t t}+u_{x x x x}+\sin (x)=0$
8. $u_{t t}+u_{x x x x}+\sin (x) \sin (u)=0$

See Solution

Problem 33059

Find the slope of the tangent line to $f(x)=\frac{1}{3+2x}$ at $P=\left(2, \frac{1}{7}\right)$ using $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$ . Then, write the tangent line equation.

See Solution

Problem 33060

Find the point of discontinuity of $F(x)=\frac{5}{x-8}$ and evaluate the one-sided limits at that point.

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Problem 33061

a) Find the IROC of $f(x)=0.2 x^{4}-3 x^{2}-5 x+6$ on $[-1,4]$ . b) Find the IROC of $f(x)=-\sqrt{3 x^{3}-5 x}$ on $[0,3]$ . c) Find the IROC of $h(x)=\frac{3 x}{x+6}$ at $x=-5$ and $x=15$ .

See Solution

Problem 33062

Express the Riemann sum $R_{100}$ for $\ln x$ on $[1, e]$ in sigma notation without evaluating it.

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Problem 33063

Find the absolute extreme values of $f(x)=2x^{2}-6x$ on $[0,5]$ . What are the global max and min?

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Problem 33064

Find the derivative $f'$ of $f(x)=\frac{2}{5x+3}$ and the tangent line at $(a, f(a))$ where $a=-1$ .

See Solution

Problem 33065

Find if the limit exists: $\lim _{x \rightarrow 5}(-7 x+11)$ . Choose A or B.

See Solution

Problem 33066

Find if the limit exists: $\lim _{x \rightarrow 11} \frac{\sqrt{x^{2}-3 x-54}}{10-4 x}$ . What is the limit or does it not exist?

See Solution

Problem 33067

Find if the limit exists: $\lim _{x \rightarrow 0} \frac{x^{5}+3 x}{x}$ . Choose A or B and simplify if A.

See Solution

Problem 33068

Find if the limit exists: $\lim _{x \rightarrow-13} \frac{x^{2}-169}{x+13}$ . What is the limit or does it not exist?

See Solution

Problem 33069

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{2x^{2}+2x}{-5x^{2}+7500}$ .

See Solution

Problem 33070

Find the derivative $f^{\prime}$ of $f(x)=\sqrt{3x+7}$ and the tangent line at $(3, f(3))$ .

See Solution

Problem 33071

Find the limit: $\lim _{t \rightarrow 5} \frac{t^{2}+3 t-40}{t^{2}-25}$ . Does it exist? If so, compute it.

See Solution

Problem 33072

Find the derivative $f^{\prime}(9)$ for $f(x)=x^{2}+5$ using limits. $f^{\prime}(9)=\square$ .

See Solution

Problem 33073

Given $\lim _{x \rightarrow 8} f(x)=3$ and $\lim _{x \rightarrow 8} g(x)=-4$ , find these limits:
a. $\lim _{x \rightarrow 8}[f(x)+7 g(x)]$ b. $\lim _{x \rightarrow 8}[f(x) g(x)]$ c. $\lim _{x \rightarrow 8}[8 f(x) g(x)]$ d. $\lim _{x \rightarrow 8}\left[\frac{f(x)}{f(x)-g(x)}\right]$

See Solution

Problem 33074

For the function $F(x)=-x^{4}+8 x^{2}+9$ , determine if it's even/odd, find a second local max, and calculate area from $x=-3$ to $x=0$ .

See Solution

Problem 33075

Find the derivative $f^{\prime}$ of $f(x)=2 x^{2}-7 x+5$ and the tangent line at $(1, f(1))$ .

See Solution

Problem 33076

Find the derivative $f^{\prime}$ of $f(s)=5 s^{3}+2 s$ and evaluate $f^{\prime}(-2)$ and $f^{\prime}(-1)$ .

See Solution

Problem 33077

Find the derivative using limits: $f^{\prime}(4)$ for $f(x)=9 x^{3}$ . Calculate $f^{\prime}(4)=$ (simplify).

See Solution

Problem 33078

Evaluate the series $\sum_{n=3}^{+\infty} n \sin \left(\frac{6}{n}\right)$ .

See Solution

Problem 33079

Find $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ for $f(x) = \frac{6x^3 - 9}{x^4 + 7x^2}$ . Determine horizontal asymptotes.

See Solution

Problem 33080

Find the derivative $f^{\prime}(x)$ using limits for the function $f(x)=\sqrt{9-x}$ .

See Solution

Problem 33081

Find the derivative $f^{\prime}(x)$ using limits for the function $f(x)=\frac{5 x}{x-1}$ .

See Solution

Problem 33082

Find the function $f(x)$ and the value of $a$ from the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{32+h}-0.03125}{h}$ .

See Solution

Problem 33083

Find the function $f(x)$ and the value of $a$ from the limit definition: $\lim _{h \rightarrow 0} \frac{(8+h)^{6}-8^{6}}{h}$ . The function is $f(x)=\square$ .

See Solution

Problem 33084

Calculate the limit: $\lim _{x \rightarrow \infty} \left(\sqrt{144 x^{2}+x}-12 x\right)$ .

See Solution

Problem 33085

Calculate the sum: $\sum_{n=6}^{\infty} n^{3} e^{-n^{4}}$ .

See Solution

Problem 33086

Find the limit: $\lim _{x \rightarrow \infty} \frac{6 x+5}{2 x+9}$ . Choose A or B.

See Solution

Problem 33087

Evaluate the series $\sum_{n=11}^{+\infty} \frac{\sin ^{2}(4 n)}{n^{16}}$ .

See Solution

Problem 33088

Find $\lim _{x \rightarrow \infty} \frac{10 x+100}{x^{2}-20}$ . Is it A: $\square$ or B: limit does not exist?

See Solution

Problem 33089

Find all numbers $x$ in $(-1,3)$ where $f'(x)=3x^2+6x-1$ equals the average rate of change of $f(x)=x^3+3x^2-x+6$ .

See Solution

Problem 33090

Find the limit: $\lim _{x \rightarrow \infty} \frac{5}{x^{3}}$ . Choose A or B and simplify your answer.

See Solution

Problem 33091

Find the sum: $\sum_{n=3}^{+\infty}\left(\ln (n)+\frac{1}{n^{4}}\right)^{2 n}$ .

See Solution

Problem 33092

Find the limit as $x$ approaches infinity for $\frac{\sqrt{5 x^{2}+25}}{9 x+6}$ .

See Solution

Problem 33093

Find the limit: $\lim _{x \rightarrow+\infty}\left(5-4 x+3 x^{3}\right)$ .

See Solution

Problem 33094

Calculate the limit: $\lim _{x \rightarrow \infty} \frac{\sqrt{2 x^{2}+49}}{11 x+1}$ .

See Solution

Problem 33095

Find the limit as $x$ approaches infinity for the expression $\frac{3 x+1}{2 x-5}$ .

See Solution

Problem 33096

Find the real number $a$ where the squeeze theorem applies to $f(x)$ , and determine the limit $L = \lim_{x \to a} f(x)$ .

See Solution

Problem 33097

Find the limit as $x$ approaches infinity for the expression $\sqrt[4]{x}$ .

See Solution

Problem 33098

Find the limit as $y$ approaches $-\infty$ for the expression $\frac{3}{y+4}$ .

See Solution

Problem 33099

What does the concentration $C(t)=\frac{44 t}{2+11 t}$ approach as $t \rightarrow \infty$ ?

See Solution

Problem 33100

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x-2}{x^{2}+2 x+1}$ .

See Solution

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Calculus

Problem 33001

Find the velocity function v(t)v(t)v(t) from s(t)=6t3−72t2+270ts(t)=6 t^{3}-72 t^{2}+270 ts(t)=6t3−72t2+270t. Where is v(t)=0v(t)=0v(t)=0? Factor out GCF. Also find a(t)a(t)a(t).

Problem 33002

Given s(t)=2t3+7t+9s(t)=2 t^{3}+7 t+9s(t)=2t3+7t+9, find: (a) v(t)v(t)v(t), (b) v(3)v(3)v(3), (c) a(t)a(t)a(t), (d) a(3)a(3)a(3).

Problem 33003

Find M6M_{6}M6​ for f(x)=6xf(x)=\sqrt{6 x}f(x)=6x​ on the interval [3,6][3,6][3,6]. Round your answer to two decimal places.

Problem 33004

Estimate the mosquito population after 14 days if P(0)=589P(0)=589P(0)=589 and P′(0)=95P^{\prime}(0)=95P′(0)=95. Round to the nearest whole number.

Problem 33005

Find the derivative of 2sin⁡(x)−53x8+7\frac{2 \sin (x)-5}{3 x^{8}+7}3x8+72sin(x)−5​ using the quotient rule. No need to expand your answer.

Problem 33006

Find the derivative of f(x)=4x(sin⁡x+cos⁡x)f(x)=4 x(\sin x+\cos x)f(x)=4x(sinx+cosx) and evaluate it at x=1x=1x=1.

Problem 33007

Find the derivative of g(x)=sin⁡(x)g(x)=\sin (x)g(x)=sin(x) and evaluate it at x=2x=2x=2. What is g′(2)=?g^{\prime}(2)=?g′(2)=?

Problem 33008

Find the derivative of g(x)=cos⁡(x)g(x)=\cos(x)g(x)=cos(x) and evaluate it at x=2x=2x=2. What is g′(2)g^{\prime}(2)g′(2)?

Problem 33009

Find the derivative f′(x)f'(x)f′(x) of f(x)=2sin⁡x+2cos⁡xf(x)=2 \sin x + 2 \cos xf(x)=2sinx+2cosx and evaluate f′(4)f'(4)f′(4).

Problem 33010

Find the derivative of f(x)=3x2tan⁡xsec⁡xf(x)=\frac{3 x^{2} \tan x}{\sec x}f(x)=secx3x2tanx​ and evaluate f′(4)f^{\prime}(4)f′(4).

Problem 33011

Find the average velocity of the object described by s(t)=−4.9t2+34t+25s(t)=-4.9 t^{2}+34 t+25s(t)=−4.9t2+34t+25 over intervals: a. [0,1][0,1][0,1] b. [0,2][0,2][0,2] c. [0,4][0,4][0,4] d. [0,h][0,h][0,h], where h>0h>0h>0.

Problem 33012

Find f(x)=x2−16x+4f(x)=\frac{x^{2}-16}{x+4}f(x)=x+4x2−16​ for x=−3.9,−3.99,−3.999,−4.1,−4.01,−4.001x=-3.9, -3.99, -3.999, -4.1, -4.01, -4.001x=−3.9,−3.99,−3.999,−4.1,−4.01,−4.001. Conjecture lim⁡x→−4f(x)\lim_{x \to -4} f(x)limx→−4​f(x).

Problem 33013

Find the limit as xxx approaches -6 for 5x+30x2+2x−24\frac{5x+30}{x^2+2x-24}x2+2x−245x+30​ using values: -5.99, -5.999, -6.01, -6.001.

Problem 33014

Find the limit as xxx approaches -4 for 2x3−3x2+5x+82x^3 - 3x^2 + 5x + 82x3−3x2+5x+8 or state if it doesn't exist.

Problem 33015

Calculate the right Riemann sum R4R_4R4​ for 1x2+1\frac{1}{x^2 + 1}x2+11​ on the interval [−2,2][-2, 2][−2,2].

Problem 33016

Find the limit: lim⁡x→−5x2−25x+5\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}x→−5lim​x+5x2−25​. Simplify and evaluate if possible. A. Exact answer, B. Limit does not exist.

Problem 33017

Find the limit: lim⁡x→45x2−80x−4\lim _{x \rightarrow 4} \frac{5 x^{2}-80}{x-4}limx→4​x−45x2−80​. Simplify if possible and evaluate the limit.

Problem 33018

Find the derivative f′(x)f^{\prime}(x)f′(x) of f(x)=2sin⁡x1+cos⁡xf(x)=\frac{2 \sin x}{1+\cos x}f(x)=1+cosx2sinx​ and calculate f′(3)f^{\prime}(3)f′(3).

Problem 33019

Find the limit as xxx approaches infinity: lim⁡x→∞4+9x+2x2x2\lim _{x \rightarrow \infty} \frac{4+9 x+2 x^{2}}{x^{2}}x→∞lim​x24+9x+2x2​.

Problem 33020

Find the inverse Laplace transform of 1s3+4s2+3s\frac{1}{s^{3}+4 s^{2}+3 s}s3+4s2+3s1​.

Problem 33021

Find the derivative f′(x)f'(x)f′(x) of f(x)=5sin⁡x2+cos⁡xf(x)=\frac{5 \sin x}{2+\cos x}f(x)=2+cosx5sinx​ and evaluate f′(3)f'(3)f′(3).

Problem 33022

Find the limits: lim⁡x→∞ex\lim _{x \rightarrow \infty} e^{x}limx→∞​ex, lim⁡x→−∞ex\lim _{x \rightarrow-\infty} e^{x}limx→−∞​ex, and lim⁡x→∞e−x\lim _{x \rightarrow \infty} e^{-x}limx→∞​e−x.

Problem 33023

Find f′(x)f^{\prime}(x)f′(x) for f(x)=cos⁡x−7tan⁡xf(x)=\cos x-7 \tan xf(x)=cosx−7tanx and calculate f′(3)f^{\prime}(3)f′(3).

Problem 33024

Curve CCC: (x2+y2)2=6xy(x^{2}+y^{2})^{2}=6xy(x2+y2)2=6xy for x>0,y>0x>0, y>0x>0,y>0. (a) Show r2=3sin⁡2θr^{2}=3\sin 2\thetar2=3sin2θ for 0<θ<π20<\theta<\frac{\pi}{2}0<θ<2π​. (b) Find the area of shaded region RRR using calculus.

Problem 33025

Which investment yields more in 1 year: 12%12\%12% monthly or 11.86%11.86\%11.86% continuously on \$13,000?

Problem 33026

Find the derivative of f(x)=5xsin⁡xcos⁡xf(x)=5 x \sin x \cos xf(x)=5xsinxcosx and evaluate f′(1)f^{\prime}(1)f′(1).

Problem 33027

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=−7cos⁡x−2tan⁡xf(x) = -7 \cos x - 2 \tan xf(x)=−7cosx−2tanx and evaluate f′(π6)f'(\frac{\pi}{6})f′(6π​).

Problem 33028

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the function f(x)=9x+3f(x)=9x+3f(x)=9x+3, where h≠0h \neq 0h=0.

Problem 33029

Express the limit as an integral: lim⁡n→∞∑i=1n(5(xi∗)2−3(xi∗)3)Δx\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{*}\right)^{2}-3\left(x_{i}^{*}\right)^{3}\right) \Delta xlimn→∞​∑i=1n​(5(xi∗​)2−3(xi∗​)3)Δx over [0,2][0,2][0,2].

Problem 33030

Problem 33031

Find the formula for RNR_{N}RN​ for f(x)=9x+7f(x)=9x+7f(x)=9x+7 on [3,6][3,6][3,6] and compute the limit as NNN approaches infinity: lim⁡N→∞RN.\lim _{N \rightarrow \infty} R_{N}.N→∞lim​RN​.

Problem 33032

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=tan⁡x−5sec⁡xf(x)=\frac{\tan x-5}{\sec x}f(x)=secxtanx−5​ and evaluate f′(5)f'(5)f′(5).

Problem 33033

Determine if the function F(r)F(r)F(r) is continuous at r=Rr=Rr=R using the definition of continuity.

Problem 33034

Find lim⁡x→∞f(x)\lim_{x \to \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x)limx→−∞​f(x) for f(x)=5x3−9x4+7x2f(x)=\frac{5x^3-9}{x^4+7x^2}f(x)=x4+7x25x3−9​. What are the horizontal asymptotes?

Problem 33035

Find lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x)limx→∞​f(x) and lim⁡x→−∞f(x)\lim _{x \rightarrow -\infty} f(x)limx→−∞​f(x) for f(x)=5x3−9x4+7x2f(x)=\frac{5 x^{3}-9}{x^{4}+7 x^{2}}f(x)=x4+7x25x3−9​.

Problem 33036

Calculate the average rate of change of f(x)=x2+9xf(x)=x^{2}+9xf(x)=x2+9x between x=1x=1x=1 and x=2x=2x=2.

Problem 33037

Find the limit as xxx approaches 0 for sin⁡(200πx)\sin \left(\frac{200 \pi}{x}\right)sin(x200π​).

Problem 33038

Calculate the average rate of change of f(x)=xf(x)=\sqrt{x}f(x)=x​ from x=9x=9x=9 to x=49x=49x=49.

Problem 33039

Find the tangent line equation for y=6sin⁡xy=6 \sin xy=6sinx at (π6,3)\left(\frac{\pi}{6}, 3\right)(6π​,3) in the form y=mx+by=m x+by=mx+b.

Problem 33040

Find the limit: lim⁡x→15−5x1−x\lim _{x \rightarrow 1} \frac{5-5 x}{1-\sqrt{x}}limx→1​1−x​5−5x​.

Find the velocity function $v(t)$ from $s(t)=6 t^{3}-72 t^{2}+270 t$ . Where is $v(t)=0$ ? Factor out GCF. Also find $a(t)$ .

Given $s(t)=2 t^{3}+7 t+9$ , find: (a) $v(t)$ , (b) $v(3)$ , (c) $a(t)$ , (d) $a(3)$ .

Find $M_{6}$ for $f(x)=\sqrt{6 x}$ on the interval $[3,6]$ . Round your answer to two decimal places.

Estimate the mosquito population after 14 days if $P(0)=589$ and $P^{\prime}(0)=95$ . Round to the nearest whole number.

Find the derivative of $\frac{2 \sin (x)-5}{3 x^{8}+7}$ using the quotient rule. No need to expand your answer.

Find the derivative of $f(x)=4 x(\sin x+\cos x)$ and evaluate it at $x=1$ .

Find the derivative of $g(x)=\sin (x)$ and evaluate it at $x=2$ . What is $g^{\prime}(2)=?$

Find the derivative of $g(x)=\cos(x)$ and evaluate it at $x=2$ . What is $g^{\prime}(2)$ ?

Find the derivative $f'(x)$ of $f(x)=2 \sin x + 2 \cos x$ and evaluate $f'(4)$ .

Find the derivative of $f(x)=\frac{3 x^{2} \tan x}{\sec x}$ and evaluate $f^{\prime}(4)$ .

Find the average velocity of the object described by $s(t)=-4.9 t^{2}+34 t+25$ over intervals: a. $[0,1]$ b. $[0,2]$ c. $[0,4]$ d. $[0,h]$ , where $h>0$ .

Find $f(x)=\frac{x^{2}-16}{x+4}$ for $x=-3.9, -3.99, -3.999, -4.1, -4.01, -4.001$ . Conjecture $\lim_{x \to -4} f(x)$ .

Find the limit as $x$ approaches -6 for $\frac{5x+30}{x^2+2x-24}$ using values: -5.99, -5.999, -6.01, -6.001.

Find the limit as $x$ approaches -4 for $2x^3 - 3x^2 + 5x + 8$ or state if it doesn't exist.

Calculate the right Riemann sum $R_4$ for $\frac{1}{x^2 + 1}$ on the interval $[-2, 2]$ .

Find the limit: $\lim _{x \rightarrow-5} \frac{x^{2}-25}{x+5}$ . Simplify and evaluate if possible. A. Exact answer, B. Limit does not exist.

Find the limit: $\lim _{x \rightarrow 4} \frac{5 x^{2}-80}{x-4}$ . Simplify if possible and evaluate the limit.

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{2 \sin x}{1+\cos x}$ and calculate $f^{\prime}(3)$ .

Find the limit as $x$ approaches infinity: $\lim _{x \rightarrow \infty} \frac{4+9 x+2 x^{2}}{x^{2}}$ .

Find the inverse Laplace transform of $\frac{1}{s^{3}+4 s^{2}+3 s}$ .

Find the derivative $f'(x)$ of $f(x)=\frac{5 \sin x}{2+\cos x}$ and evaluate $f'(3)$ .

Find the limits: $\lim _{x \rightarrow \infty} e^{x}$ , $\lim _{x \rightarrow-\infty} e^{x}$ , and $\lim _{x \rightarrow \infty} e^{-x}$ .

Find $f^{\prime}(x)$ for $f(x)=\cos x-7 \tan x$ and calculate $f^{\prime}(3)$ .

Curve $C$ : $(x^{2}+y^{2})^{2}=6xy$ for $x>0, y>0$ .
(a) Show $r^{2}=3\sin 2\theta$ for $0<\theta<\frac{\pi}{2}$ .
(b) Find the area of shaded region $R$ using calculus.

Which investment yields more in 1 year: $12\%$ monthly or $11.86\%$ continuously on \$13,000?

Find the derivative of $f(x)=5 x \sin x \cos x$ and evaluate $f^{\prime}(1)$ .

Find the derivative $f'(x)$ of the function $f(x) = -7 \cos x - 2 \tan x$ and evaluate $f'(\frac{\pi}{6})$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=9x+3$ , where $h \neq 0$ .

Express the limit as an integral: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(5\left(x_{i}^{}\right)^{2}-3\left(x_{i}^{}\right)^{3}\right) \Delta x$ over $[0,2]$ .

Find the formula for $R_{N}$ for $f(x)=9x+7$ on $[3,6]$ and compute the limit as $N$ approaches infinity: $\lim _{N \rightarrow \infty} R_{N}.$

Find the derivative $f'(x)$ of the function $f(x)=\frac{\tan x-5}{\sec x}$ and evaluate $f'(5)$ .

Determine if the function $F(r)$ is continuous at $r=R$ using the definition of continuity.

Find $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ for $f(x)=\frac{5x^3-9}{x^4+7x^2}$ . What are the horizontal asymptotes?

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=\frac{5 x^{3}-9}{x^{4}+7 x^{2}}$ .

Calculate the average rate of change of $f(x)=x^{2}+9x$ between $x=1$ and $x=2$ .

Find the limit as $x$ approaches 0 for $\sin \left(\frac{200 \pi}{x}\right)$ .

Calculate the average rate of change of $f(x)=\sqrt{x}$ from $x=9$ to $x=49$ .

Find the tangent line equation for $y=6 \sin x$ at $\left(\frac{\pi}{6}, 3\right)$ in the form $y=m x+b$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{5-5 x}{1-\sqrt{x}}$ .

Find the 28th derivative of $f(x)=\cos(x)$ . What is the result?

Find the oscillation rate of the spring at $t=2$ s for $s(t)=3 \cos t$ . Round to four decimal places.

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{39 x^{7}+2 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

Is the function $f(x)=\frac{2 x^{2}+3 x+1}{x^{2}-7 x}$ continuous at $a=7$ ? Justify using the checklist. Select all that apply.

Calculate the integral $I=\int_{0}^{\pi / 2}(2 \sin (\theta)+\sin ^{3}(\theta)) d \theta$ .

Is the function $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2} & \text{if } x \neq 2 \\ 10 & \text{if } x=2\end{array}\right.$ continuous at $a=2$ ? Select true statements.

Calculate the derivative of $f(x)=3 x^{2}-2 x$ at the point $(-1,5)$ . Find $f'(-1)=\square$ .

Redefine the function $f$ for continuity at $x=0$ : $f(x) = \begin{cases} \frac{9}{x} + \frac{-8x + 36}{x(x-4)} & x \neq 0, 4 \\ 4 & x = 0, 4 \end{cases}$

Determine where the function $p(x)=4 x^{6}-5 x^{3}+1$ is continuous. Provide your answer in interval notation.

Find the slope of the tangent line to $f(x)=x^{2}+5$ at point $P(-5,30)$ and its equation.

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ for $f(x)=\frac{39 x^{7}+2 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

Find $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow -\infty} f(x)$ for $f(x)=\frac{39 x^{7}+4 x^{2}}{19 x^{6}-3 x}$ . What are the horizontal asymptotes?

Find the slope of the tangent line to $f(x)=x^{2}-1$ at $P(2,3)$ using $m_{\tan }=\lim _{x \rightarrow 2} \frac{f(x)-3}{x-2}$ . Then, find the tangent line equation and plot both.

Find the slope of the tangent line to $f(x)=x^{2}-2$ at $P(4,14)$ using $m_{\tan }=\lim _{x \rightarrow 4} \frac{f(x)-14}{x-4}$ .

Find the average rate of change of $h(x)=x^{2}-9x$ from 2 to 3 and the secant line equation in slope-intercept form.

Find the point of discontinuity of $F(x)=\frac{5}{x-8}$ and evaluate the one-sided limits at that point.

Express the Riemann sum $R_{100}$ for $\ln x$ on $[1, e]$ in sigma notation without evaluating it.

Find the absolute extreme values of $f(x)=2x^{2}-6x$ on $[0,5]$ . What are the global max and min?

Find the derivative $f'$ of $f(x)=\frac{2}{5x+3}$ and the tangent line at $(a, f(a))$ where $a=-1$ .

Find if the limit exists: $\lim _{x \rightarrow 5}(-7 x+11)$ . Choose A or B.

Find if the limit exists: $\lim _{x \rightarrow 11} \frac{\sqrt{x^{2}-3 x-54}}{10-4 x}$ . What is the limit or does it not exist?

Find if the limit exists: $\lim _{x \rightarrow 0} \frac{x^{5}+3 x}{x}$ . Choose A or B and simplify if A.

Find if the limit exists: $\lim _{x \rightarrow-13} \frac{x^{2}-169}{x+13}$ . What is the limit or does it not exist?

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{2x^{2}+2x}{-5x^{2}+7500}$ .

Find the derivative $f^{\prime}$ of $f(x)=\sqrt{3x+7}$ and the tangent line at $(3, f(3))$ .

Find the limit: $\lim _{t \rightarrow 5} \frac{t^{2}+3 t-40}{t^{2}-25}$ . Does it exist? If so, compute it.

Find the derivative $f^{\prime}(9)$ for $f(x)=x^{2}+5$ using limits. $f^{\prime}(9)=\square$ .

For the function $F(x)=-x^{4}+8 x^{2}+9$ , determine if it's even/odd, find a second local max, and calculate area from $x=-3$ to $x=0$ .