Browse Calculus problems Studdy Solved!

Problem 1201

2 3- جد مركبات شعاع الوحدة المماسي

See Solution

Problem 1202

5. Найти наибольшее и наименьшее значения функции $z=-10 x y^{2}+x^{2}+10 x+1$ на замкнутом множестве $D: \frac{|x|}{7}+\frac{|y|}{2} \leq 1$ . (4 балла)

See Solution

Problem 1203

${ }^{3}$ For the following exercises, graph the equations and shade the area of the region between the curves on over specified interval. Determine its area. (a) $y=x^{2}$ and $y=-x^{2}+18 x$ over $[0,9]$ . (b) $y=\frac{1}{x}$ and $y=1 / x^{2}$ over $[1,3]$ . (c) $y=\cos (x)$ and $y=\cos (x)^{2}$ over $[-\pi, \pi]$ . (d) $y=e^{x}$ and $y=e^{2 x-1}$ over $[0,1]$ . (e) $y=e^{x}$ and $y=e^{-x}$ over $[-1,1]$ .

See Solution

Problem 1204

Use the Ratio Test to determine whether the series is convergent or divergent. $\sum_{n=1}^{\infty} \frac{n \pi^{n}}{(-5)^{n-1}}$
Identify $a_{n}$ . $\square$
Evaluate the following limit. $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|$ $\square$
Since $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|$ ? 1, ---Select--- $\square$

See Solution

Problem 1205

Use the Root Test to determine whether the series convergent or divergent. $\sum_{n=2}^{\infty}\left(\frac{-7 n}{n+1}\right)^{3 n}$
Identify $a_{n}$ . $\square$
Evaluate the following limit. $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$ $\square$
Since $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$ $\square$ ? 1, $\square$ -Select--

See Solution

Problem 1206

Question: Compute the directional derivatives of the following functions along unit vectors at the indicated points in directions parallel to the given vector. (a) $f(x, y)=x^{y},\left(x_{0}, y_{0}\right)=(e, e), d=20 i+21 j$ (b) $f(x, y, z)=e x+y z,\left(x_{0}, y_{0}, z_{0}\right)=(1,1,1), d$

See Solution

Problem 1207

1. $\lim _{x \rightarrow+\infty} \frac{\frac{1}{x}}{\tan \left(\frac{2}{x}\right)}$

See Solution

Problem 1208

3. $\lim _{x \rightarrow 0^{+}} x^{x^{x}}$

See Solution

Problem 1209

Вычислить неопределенные интегралы, сводя их к табличным:
1. $\int\left(2 x^{5}-3 x^{2}\right) d x=\frac{x^{6}}{3}-x^{3}+C$

See Solution

Problem 1210

44. The discriminant $f_{x x} f_{y y}-f_{x y}{ }^{2}$ is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface $z=f(x, y)$ looks like. Describe your reasoning in each case. a. $f(x, y)=x^{2} y^{2}$ b. $f(x, y)=1-x^{2} y^{2}$ c. $f(x, y)=x y^{2}$ d. $f(x, y)=x^{3} y^{2}$ e. $f(x, y)=x^{3} y^{3}$ f. $f(x, y)=x^{4} y^{4}$

See Solution

Problem 1211

13. Consider the curve defined by $x^{2}+x y+y^{2}=27$ . a) Show that $\frac{d y}{d x}=\frac{-2 x-y}{x+2 y} \cdot(2 \mathrm{pts})$ b) Write an equation for the line tangent to the curve at the point $(3,-6)$ . (3 pts) c) Where is the curve not differentiable? Justify your answer. (3 pts) $x+2 y=0$ d) Find $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ . (2 pts)

See Solution

Problem 1212

14. Use the table below to find the following values. \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $f(x)$ & $g(x)$ & $f^{\prime}(x)$ & $g^{\prime}(x)$ \\ \hline \hline-1 & -5 & 1 & 3 & 0 \\ \hline 0 & -2 & 0 & 1 & 1 \\ \hline 1 & 0 & -3 & 0 & 0.5 \\ \hline 2 & 5 & -1 & 5 & 2 \\ \hline \end{tabular} (a) Find $h^{\prime}(2)$ whenh $(x)=-3 g(x)$ (2 pts) (b) Find $h^{\prime}(-1)$ when $h(x)=f(-1-3 x)-e^{3 x}$ (2 pts) (c) Find $h^{\prime}(2)$ when $h(x)=x^{3} g(x)(2 \mathrm{pts})$ (d) Find $h^{\prime}(0)$ whenh $(x)=g\left(-\frac{1}{2} x\right) \cdot f(x)(2 \mathrm{pts})$ (e) Find $h^{\prime}(1)$ when $h(x)=\frac{g(x)}{\left(x^{2}-4\right)}(2 \mathrm{pts})$ (f) Find $\left(f^{-1}\right)^{\prime}(-5)$ (2 pts)

See Solution

Problem 1213

oncepts Araph of a function h mathematics, the graph of a function is the set of ordered pairs, where in the common ce find the end behavior in limit notation $f(x)=\frac{x-1}{x^{2}-4}$

See Solution

Problem 1214

$\int \sin ^{-1}(5 x) d x$

See Solution

Problem 1215

If $e^{x y}-y^{2}=e-4$ , then at $x=\frac{1}{2}$ and $y=2, \frac{d y}{d x}=$

See Solution

Problem 1216

(a) $\lim _{x \rightarrow 3} \frac{\sqrt{x-1}-\sqrt{2}}{9-x^{2}}$ .

See Solution

Problem 1217

10. Do the following series converge or diverge? Be sure to state which test you are using show that that test is able to be applied. All work must be shown. (a) $\sum_{n=1}^{\infty} \frac{1}{(2 n+7)^{3}}$ (b) $\sum_{n=1}^{\infty} \frac{(n-1)!}{5^{n}}$ (c) $\sum_{n=1}^{\infty} \frac{1}{n \sin ^{2}(n)}$ (d) $\sum_{n=1}^{\infty} \frac{n^{3}-2 n^{2}+n-1}{n^{5}-2}$ (e) $\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{\sqrt{n+1}}$ (f) $\sum_{n=1}^{\infty} \frac{n^{2 n}}{\left(1+2 n^{2}\right)^{n}}$ (g) $\sum_{n=1}^{\infty} \frac{(-2)^{1+3 n}(n+1)}{n^{2} 5^{1+n}}$ (h) $\sum_{n=1}^{\infty} \ln \left(\frac{n}{3 n+1}\right)$ (i) $\sum_{n=1}^{\infty} \frac{e^{1 / n}}{n^{2}}$ (j) $\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+1}$

See Solution

Problem 1218

Let $H(t)=\frac{a}{t^{40}}-b \cdot e^{t}$ where $a$ and $b$ are both fixed constants. What is $H^{\prime}(t)$ ?

See Solution

Problem 1219

PREVIOUS ANSWERS PRACTICE ANOTHER
Find the equation for the plane tangent to each surface $z=f(x, y)$ at the indicated point. (a) $z=x^{3}+y^{3}-5 x y$ , at the point $(1,2,-1)$

See Solution

Problem 1220

14. Given $h(x)=f(g(x))$ , use the graph to the right to find $h^{\prime}(4)$ . (A) $\frac{1}{2}$ (B) $\frac{5}{4}$ $f^{\prime}(g(x)) g^{\prime}(x)$ (C) $\frac{5}{8}$ (D) 5 $\begin{array}{l} f^{\prime}(g(4)) g^{\prime}(4) \\ f^{\prime}(5)\left(\frac{1}{2}\right)=\frac{5}{4}\left(\frac{1}{2}\right) \end{array}$

See Solution

Problem 1221

Для вычисления этого интеграла вы сделали замену. Чему равен dt ? $\int e^{5 x+1} d x$
Ответ записать в строку без пробелов и скобок. Знак умножения не пишем. Знак деления:/

See Solution

Problem 1222

$\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{\infty} e^{t x} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \, dx$

See Solution

Problem 1223

Для вычисления этого интеграла вы сделали замену. Чему равен dx? $\int \cos (3 x+5) d x$
Ответ записать в строку без пробелов и скобок. Знак умножения не пишем. Знак деления: /

See Solution

Problem 1224

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) $\lim _{x \rightarrow 0} x^{4} \cos \left(\frac{8}{x}\right)$

See Solution

Problem 1225

If the figure below is the graph of the derivative $f^{\prime}$ , answer the following:
Where do the points of inflection of $f$ occur? $\square$
On which interval(s) is $f$ concave down? $\square$
Note: You can earn partial credit on this problem.

See Solution

Problem 1226

$\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$

See Solution

Problem 1227

26. What is wrong with the following argument? Suppose $w=f(x, y, z)$ and $z=g(x, y)$ . By the chain rule, $\frac{\partial w}{\partial x}=\frac{\partial w}{\partial x} \frac{\partial x}{\partial x}+\frac{\partial w}{\partial y} \frac{\partial y}{\partial x}+\frac{\partial w}{\partial z} \frac{\partial z}{\partial x}=\frac{\partial w}{\partial x}+\frac{\partial w}{\partial z} \frac{\partial z}{\partial x}$
Hence, $0=(\partial w / \partial z)(\partial z / \partial x)$ , and so $\partial w / \partial z=0$ or $\partial z / \partial x=0$ , which is, in general, absurd.

See Solution

Problem 1228

Suppose $f\left(\frac{\pi}{3}\right)=4$ and $f^{\prime}\left(\frac{\pi}{3}\right)=-5$ . Let $g(x)=f(x) \sin (x)$ and $h(x)=\frac{\cos (x)}{f(x)}$ . Find the following. (a) $g^{\prime}\left(\frac{\pi}{3}\right)$ $\square$ (b) $h^{\prime}\left(\frac{\pi}{3}\right)$ $\square$

See Solution

Problem 1229

Find the limit, if it exists. (If an answer does not exist, enter DNE.) $\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{8}}{x^{3}+y^{12}}$

See Solution

Problem 1230

Determine whether the statement is true or false. The derivative of a polynomial is a polynomial. True False Submit Answer

See Solution

Problem 1231

Suppose $f\left(\frac{\pi}{3}\right)=2$ and $f^{\prime}\left(\frac{\pi}{3}\right)=-3$ . Let $g(x)=f(x) \sin (x)$ and $h(x)=\frac{\cos (x)}{f(x)}$ . Find the following. (a) $g^{\prime}\left(\frac{\pi}{3}\right)$ $\square$ (b) $h^{\prime}\left(\frac{\pi}{3}\right)$ $\square$ Submit Answer

See Solution

Problem 1232

2. Найдите пределы следующих последовательностей: 2.1. $x_{n}=\left(\frac{5 n^{2}-3}{5 n^{2}+1}\right)^{n}$ 2.2. $x_{n}=\frac{4 n^{4}+2 n-3}{n(3 n-2)\left(n+1-2 n^{2}\right)}$ 2.3. $x_{n}=\left(\frac{1-3 n}{20}\right)^{1-5 n}$ 2.4. $x_{n}=\sqrt{4 n^{2}+3 n+7}-2 n$

See Solution

Problem 1233

5. Determine whether the series $\sum_{k=1}^{\infty} \frac{7^{k-2}}{8^{k+1}}$ converges or diverges. If it converges, find its sum. (A) converges and equals $\frac{1}{64}$ (B) converges and equals $\frac{1}{56}$ (C) converges and equals $\frac{1}{8}$ (D) The series diverges.

See Solution

Problem 1234

A tank contains 80 gallons ( gal ) of pure water. A brine (salt) solution with $2 \mathrm{lb} / \mathrm{gal}$ of salt enters at $2 \mathrm{gal} / \mathrm{min}$ , and the well stirred mixtures leaves at the same rate. Find (a) the amount of salt in the tank at any time, and (b) the time at which the brine leaving will contain $1 \mathrm{lb} / \mathrm{gal}$ of salt.

See Solution

Problem 1235

8. $\lim _{X \rightarrow 0} \frac{\sqrt{1-x}-\sqrt{1+X}}{X}=$

See Solution

Problem 1236

4) The horizontal asymptote for $f(x)=\frac{5 x^{2}-1}{(x-1)(x-3)}$ is

See Solution

Problem 1237

13. $\lim _{x \rightarrow 0} \frac{4 X^{3}+6 x}{3 x^{3}+2 x}=$

See Solution

Problem 1238

Use the graph to answer the question. Describe the continuity of the graphed function. (1 point) The function is continuous. The function has a removable discontinuity at $x=3$ . The function has a jump discontinuity at $x=3$ . The function has an infinite discontinuity at $x=3$ .

See Solution

Problem 1239

reach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

See Solution

Problem 1240

Let $f(x)=\frac{1}{x^{2}-4}$ Where does $f$ have critical points? Choose all answers that apply:
A $x=-2$ B $x=0$ c. $x=2$
D $f$ has no critical points.

See Solution

Problem 1241

Let $g(x)=\sin (2 x)$ , for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ . Where does $g$ have critical points?

See Solution

Problem 1242

Let $h(x)=\frac{e^{2 x}}{x-3}$ Where does $h$ have critical points?

See Solution

Problem 1243

Determine whether the series $\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+3}}$ converges.

See Solution

Problem 1244

Problem 2. Use geometry to calculate each of the following integrals exactly. The graph of $f(x)$ is shown below. a) $\int_{0}^{2} f(x) d x$ c) $\int_{0}^{4} f(x) d x$ e) $\int_{7}^{9} f(x) d x$ b) $\int_{2}^{4} f(x) d x$ d) $\int_{4}^{7} f(x) d x$ f) $\int_{0}^{9} f(x) d x$ 160

See Solution

Problem 1245

For $-14 \leq x \leq 13$ the function $f$ is defined by $f(x)=x^{7}(x+8)^{2}$ On which two intervals is the function increasing (enter intervals in ascending order)? $x=$ $\square$ $\square$ and $\square$ to $x=$ $x=$ to $x=$ $\square$ Find the interval on which the function is positive: $x$ - $\square$ to $x=$ $\square$ Where does the function achieve its minimum? $x=$ $\square$

See Solution

Problem 1246

Taylor's Remainder Theorem (see CLP-1 section 3.4.9) Suppose that $f$ is an $n+1$ -times differentiable function on an interval $I$ containing the point $x=a$ . There exists a point $c$ strictly between $x$ and $a$ so that $f(x)=T_{n}(x)+\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ where $T_{n}(x)$ is the $n$ -th degree Taylor polynomial of $f$ centred at $x=a$ . Moreover, if $M$ is a constant and $\left|f^{(n+1)}(c)\right| \leq M$ , then we can estimate the error in the approximation of $f$ by $T_{n}$ as follows: $\left|f(x)-T_{n}(x)\right| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$
In practice, we take $M$ to be the maximum of $\left|f^{(n+1)}(x)\right|$ on the interval $I$ . Question: (a) Approximate $\sin (1)$ using the 5 -th degree Taylor polynomial of $f(x)=\sin (x)$ centred at $x=0$ . (b) Use Taylor's Remainder Theorem to approximate the error between $\sin (1)$ and the 5 -th degree Taylor polynomial approximation on the interval $I=\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ . Hint: You can use the closed interval method to find " $M$ " in this context. (c) Let $T_{n}(x)$ be the $n$ -th degree Taylor polynomial for $f(x)=\sin (x)$ centred at $x=0$ . Using Taylor's Remainder Theorem, find the smallest integer $n$ so that $\left|f(x)-T_{n}(x)\right| \leq 0.0001$ for all $x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ . Hint: See the Fall 2021 Written Question Exam Solution / Key and the Written Lab Task \#1 solutions posted to D2L for a sample of what good mathematical writing looks like. Reminder: Follow the checklist on the Written Lab Task Rubric. (C) University of Calgary

See Solution

Problem 1247

Let $h$ be a polynomial function and let $h^{\prime}$ , its derivative, be defined as $h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}$ .
At how many points does the graph of $h$ have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

See Solution

Problem 1248

7. Consider the curve given by the equation $y^{3}-3 x y=2$ . a. Find $\frac{d y}{d x}$ .

See Solution

Problem 1249

Find a general formula for the average rate of change on the interval $[4,4+h]$ . Given $f(x)=\frac{1}{x+11}$ , use the Difference Quotient on $[4,4+h]$ . (Note: Your answer will be an expression involving $h$ and must be simplified.) $\square$ The interval $[4,4+h]$ gets smaller when $h \rightarrow 0$ . If we substitute $h=0$ into our formula, the A.R.O.C simplies to what? $\square$ This will be our estimate for the instanteous rate of change for $x=4$ .

See Solution

Problem 1250

Question 7
Let $f(x)=\sqrt{3 x+4}$ . Calculate the difference quotient: $\frac{f(x+h)-f(x)}{h}$

See Solution

Problem 1251

27-28 Explain why the Integral Test can't be used to determine whether the series is convergent.
27. $\sum_{n=1}^{\infty} \frac{\cos \pi n}{\sqrt{n}}$
28. $\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}$

See Solution

Problem 1252

2. (a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

See Solution

Problem 1253

Use the substitution $u=7 x^{2}+2 x$ to evaluate the following indefinite integral. $\int(14 x+2) \sqrt{7 x^{2}+2 x} d x$

See Solution

Problem 1254

Evaluate the given indefinite integrals. a) $\int \frac{6}{y^{5}} d y=\square+C$ , b) $\int\left(\frac{1}{u^{1 / 4}}+5 \sqrt{u}\right) d u=\square+C$ . c) $\int \frac{1}{6 x^{5}} d x=\square+C$ .

See Solution

Problem 1255

3. Use Green's Theorem to evaluate $\oint_{C} 2 x y d x+(x+y) d y$ . $C$ is the boundary of the region lying between the graphs of $y=0$ and $y=1-x^{2}$ , oriented counterclockwise.

See Solution

Problem 1256

Prove that $\sin ^{2} x+\cos ^{2} x=1$ using Calculus by: (1) Find the derivative of $f(x)=\sin ^{2} x+\cos ^{2} x$ (2) Use corollay to show that $f(x)=1$ on all of $[0,2 \pi]$ .

See Solution

Problem 1257

Let $f(x)=x^{2} \tan ^{-1}(8 x)$ $f^{\prime}(x)=$

See Solution

Problem 1258

af caffeine. If the body filters the $28 \%$ per hour, how much caffeine is left after 6 hours assuming no more caffeine is taken into the body during that time? What is the half-life of caffeine in the body? $50(1,0+6)^{\frac{6}{2}}$ 9) $\$ 17000$ invested at $\frac{6 \%}{}$ compounded semi-annually will be worth how much after 8 years? $17000(1.03)^{16}$ $=27250$ . 10) If the population of the earth is increasing at $1.27 \%$ per year and reached 7 billion in the year 2011. When will the population of the earth reach 9 billion? 11) If a population of rabbits doubles every 5 weeks, how long would it take a population of 100 rabbits to turn into 900 ? 12) What is the annual interest rate of an investment if it takes 8 years for the money to double in value? 13) A pair of jeans fades by $1.3 \%$ every time they are washed. How many washes will it take for the jeans to fade to half of their original colour? 14) The half life of iodine-131 is 8.2 days. How long would it take a sample of iodine-131 to be reduced to $2 \%$ of it's original amount?

See Solution

Problem 1259

8) An energy drink contains 80 mg of caffeine. If the body filters the caffeine out at a rate of $28 \%$ per hour, how much caffeine is left after 6 hours assuming no more caffeine is taken into the body during that time? What is the half-life of caffeine in the body? $00(80(4) 2+6)^{\frac{5}{2}}$ 9) $\$ 17000$ invested at $\frac{6 \%}{}$ compounded semi-annually will be worth how much after 8 years? $\begin{array}{l} =17000(1.03)^{16} \\ =27250 . \end{array}$ 10) If the population of the earth is increasing at $1.27 \%$ per year and reached 7 billion in the year 2011. When will the population of the earth reach 9 billion?
9billan = 7 bition $(1,0 \mid P$

See Solution

Problem 1260

Find the general antiderivative, $F(x)$ , of the function $f(x)=-\frac{2}{11} x^{5}+\frac{9}{7} x^{11}-\frac{6}{11} \frac{1}{x^{7}}$ $F(x)=$

See Solution

Problem 1261

Solve the initial value problem (IVP) $\frac{d y}{d x}=9 \sqrt{x}-4 x$ where $y(1)=9$ $y(x)=6 x^{3 / 2}-2 x^{2}-13$ Nothing in this list is correct. $y(x)=6 x^{3 / 2}-2 x^{2}-2$ $y(x)=6 x^{3 / 2}-2 x^{2}+1$ $y(x)=4 x^{3 / 2}+x^{2}-8$ $y(x)=4 x^{3 / 2}+x^{2}-41$ $y(x)=4 x^{3 / 2}+x^{2}-36$ $y(x)=4 x^{3 / 2}+x^{2}+5$ $y(x)=6 x^{3 / 2}-2 x^{2}+5$

See Solution

Problem 1262

Find any horizontal or vertical asymptotes. $f(x)=\frac{x^{4}+1}{x^{2}+3 x-18}$
Select the correct choice below and, if necessary, fill in the answer box to complete A. The horizontal asymptotes are $y=$ $\square$ (Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

See Solution

Problem 1263

Find the indefinite integral. (Use C for the constant of integration.) $\int(x-2)(x+4) d x$

See Solution

Problem 1264

Current Attempt in Progress Let $C(q)$ represent the cost, $R(q)$ the revenue, and $\pi(q)$ the total profit, in dollars, of producing $q$ items. (a) If $C^{\prime}(50)=77$ and $R^{\prime}(50)=83$ , approximately how much profit is earned by the $51^{\text {st }}$ item?
The profit earned from the $51^{\text {st }}$ item will be approximately $\$$ i $\square$ (b) If $C^{\prime}(90)=73$ and $R^{\prime}(90)=69$ , approximately how much profit is earned by the $91^{\text {st }}$ item?
The profit earned from the $91^{\text {st }}$ item will be approximately \\square $i . (c) If$ \pi(q) $is a maximum when$ q=78 $, how do you think$ C^{\prime}(78) $and$ R^{\prime}(78) $compare?$ C^{\prime}(78)\squareR^{\prime}(78)$ eTextbook and Media

See Solution

Problem 1265

$\lim _{x \rightarrow 4} \frac{x^{2}-16}{\sqrt{x}-2}$

See Solution

Problem 1266

9. Use an appropriate substitution to evaluate the definite integral $\int_{2}^{6} \frac{x}{\sqrt{2 x-3}} d x$ .

See Solution

Problem 1267

What is $\int_{0}^{1}(2 t+2) \, dx ?$

See Solution

Problem 1268

6) An observer stands 700 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of $900 \mathrm{ft} / \mathrm{sec}$ . Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 2400 ft from the ground?

See Solution

Problem 1269

(1 point) Let $f(x)=(\ln (x))^{\operatorname{sed}(x)}$ . Find $f^{\prime}(x)$ . $f^{\prime}(x)=$

See Solution

Problem 1270

$\lim _{x \rightarrow \infty} \frac{\ln \left(x^{2}+1\right)}{\ln \left(3 x^{2}+7\right)}$

See Solution

Problem 1271

Crowdmark linear approximation of sc test-3-07073 $-x^{1}-1$ $-\cot (x)-0.5$ $\ln |x|-0.5$
Q2 X18 (20 pts) Sketch the graph of a function $f(x)$ satisfying the following: $\lim _{x \rightarrow \infty} f(x)=3 \quad \lim _{x \rightarrow-\infty} f(x)=2$
Increasing on $(-\infty,-4) \cup(1, \infty)$
Concave up on $(-\infty, 1)$ Concave down on $(1, \infty)$ minimum at $(1,0)$
Vertical Asymptote at $x=-4$ Dccrcasing on $(-4,1)$
1 Concavity -2

See Solution

Problem 1272

نعثر الدالة $f$ المعر فلة على

See Solution

Problem 1273

1. If $f(x)=2 x^{2}+4$ , which of the following will calculate the derivative of $f(x)$ ? (a) $\frac{\left[2(x+\Delta x)^{2}+4\right]-\left(2 x^{2}+4\right)}{\Delta x}$ (b) $\lim _{\Delta x \rightarrow 0} \frac{\left(2 x^{2}+4+\Delta x\right)-\left(2 x^{2}+4\right)}{\Delta x}$ (c) $\lim _{\Delta x \rightarrow 0} \frac{\left[2(x+\Delta x)^{2}+4\right]-\left(2 x^{2}+4\right)}{\Delta x}$ (d) $\frac{\left(2 x^{2}+4+\Delta x\right)-\left(2 x^{2}+4\right)}{\Delta x}$ (e) None of these

See Solution

Problem 1274

For problems 11-14, use proper notation throughout. 11.) Consider the function $f(t)$ Int a.) Calculate the instantancous rate of change of the function at $t=\frac{1}{2}$ , b.) Find the equation of the tangent line at the point where $t=3$ . Leave your answer in terms of the natural logarithm.

See Solution

Problem 1275

[-/1 Points] DETAILS MY NOTES TANAPCALC10 6.5.028.EP.
Consider the following. $\int_{1}^{9} \frac{9 \ln (x)}{x} d x$
Let $u=\ln (x)$ . Find $d u$ . $d u=$ $\square$ $d x$
Indicate how the limits of integration should be adjusted in order to perform the integration with respect to $u$ . (Enter your answer using interval nota $[1,9]$ $\square$ Evaluate the definite integral. $\square$

See Solution

Problem 1276

12.) Find the following using the Limit Definition of Derivative. You should be able to do this with very little computation. \begin{tabular}{|l|l|} \hline a.) $\lim _{\Delta x \rightarrow 0} \frac{\sin (x+\Delta x)-\sin (x)}{\Delta x}$ & b.) $\lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h}$ \\ c.) $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\sin \frac{\pi}{4}}{x-\frac{\pi}{4}}$ & d.) $\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\frac{1}{2}}{h}$ \\ \hline e.) $\lim _{\Delta x \rightarrow 0} \frac{(2+\Delta x)^{3}-8}{\Delta x}$ & f.) $\lim _{x \rightarrow \pi} \frac{\cos x+1}{x-\pi}$ \\ \hline & \\ \hline \end{tabular}

See Solution

Problem 1277

5. Find $\frac{d^{2} y}{d x^{2}}$ for $y=\frac{x+3}{x-1}$ (a) 0 (b) $\frac{-8}{(x-1)^{3}}$ (c) $\frac{-4}{(x-1)^{3}}$ (d) $\frac{8}{(x-1)^{3}}$ (e) None of these

See Solution

Problem 1278

3. Find the a bsolute max and min $f(x)=x^{\frac{3}{2}}-3 \sqrt{x}, 0 \leq x \leq 4$ $\begin{array}{l} f(x)=x^{\frac{3}{2}}-\frac{3 x^{\frac{1}{2}}}{} \\ f^{\prime}(x)=\frac{3}{2} x^{\frac{1}{2}}-\frac{3}{2} x^{-\frac{1}{2}} \end{array}$

See Solution

Problem 1279

7. Find $\frac{d y}{d x}$ if $y^{2}-3 x y+x^{2}=7$ . (a) $\frac{2 x+y}{3 x-2 y}$ (b) $\frac{3 y-2 x}{2 y-3 x}$ (c) $\frac{2 x}{3-2 y}$ (d) $\frac{2 x}{y}$ (e) None of these

See Solution

Problem 1280

8. Find $y^{\prime}$ if $y=\sin (x+y)$ . (a) 0 (b) $\frac{\cos (x+y)}{1-\cos (x+y)}$ (c) $\cos (x+y)$ (d) 1 (c) None of these

See Solution

Problem 1281

57-63 Calcule los valores de $x$ para los cuales la serie converge. Determine la suma de la serie para dichos valores de $x$ .
57. $\sum_{n=1}^{\infty}(-5)^{n} x^{n}$
58. $\sum_{n=1}^{\infty}(x+2)^{n}$

See Solution

Problem 1282

12.) Find the following using the Limit Definition of Derivative. You should be able to do this with very little computation. g.) $\lim _{x \rightarrow 2} \frac{\ln x-\ln 2}{x-2}$ h.) $\lim _{\Delta x \rightarrow 0} \frac{(3+\Delta x)^{2}+(3+\Delta x)-12}{\Delta x}$

See Solution

Problem 1283

```latex A manufacturer of tennis rackets finds that the total cost $C(x)$ (in dollars) of manufacturing $x$ rackets/day is given by $C(x)=400+4x+0.0001x^{2}$ . Each racket can be sold at a price of $p$ dollars, where $p$ is related to $x$ by the demand equation $p=10-0.0004x$ . If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer. ```

See Solution

Problem 1284

Submit Answer
8. [-/1 Points]
DETAILS MY NOTES The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure.
Let $x=$ the distance from the left end of the pool. Determine $\Delta x$ if the midpoint rule with $n=4$ will be used to estimate the area (in $\mathrm{m}^{2}$ ) of the pool. $\Delta x=$ $\square$ Use the midpoint rule with $n=4$ to estimate the area (in $\mathrm{m}^{2}$ ) of the pool. $\square$ $m^{2}$ Submit Answer Home My Assignments Request Exten

See Solution

Problem 1285

Compute the derivatives of the given functions. a) $f(r)=10^{r} . \quad f^{\prime}(r)=$ $\square$ . b) $g(s)=17^{9} . \quad g^{\prime}(s)=$ $\square$ . b) $h(t)=\frac{5^{t}}{6^{t}} \quad h^{\prime}(t)=$ $\square$ .

See Solution

Problem 1286

ค. $\int_{0}^{6} \sin 2 x \cos 4 x d x 9$

See Solution

Problem 1287

Problem 13. *Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 1800 bacteria selected from this population reached the size of 2115 bacteria in five hours. Find the hourly growth rate parameter. Note: This is a continuous exponential growth model. Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.

See Solution

Problem 1288

10. [0/1 Points] DETAILS
MYNOTES
Find the absolute maximum and absolute minimum values of $f$ on the given interval. $f(x)=4 x^{3}-6 x^{2}-144 x+1, \quad[-4,5]$ absolute minimum $\square$ $\square$ absolute maximum Need Help? Readit $\square$ Watchlt $\square$ $\square$ Masterlit

See Solution

Problem 1289

points) $f(x)=\sin (\sin (x))$ then $f^{\prime}(x)=\square$ Preview My Answers Submit Answers

See Solution

Problem 1290

2-30 Determine si la serie es absolutamente convergente, condicionalmente convergente o divergente.
3. $\sum_{n=1}^{\infty} \frac{n}{5^{n}}$
4. $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n^{2}+4}$

See Solution

Problem 1291

(1 point) Please answer the following questions about the function $f(x)=\frac{2 x^{2}}{x^{2}-25}$
Instructions: - If you are asked for a function, enter a function. - If you are asked to find $x$ - or $y$ -values, enter either a number or a list of numbers separated by commas. If there are no solutions, enter None. - If you are asked to find an interval or union of intervals, use interval notation. Enter \{ \} if an interval is empty. - If you are asked to find a limit, enter either a number, I for $\infty,-I$ for $-\infty$ , or $D N E$ if the limit does not exist. (a) Calculate the first derivative of $f$ . Find the critical numbers of $f$ , where it is increasing and decreasing, and its local extrema. $f^{\prime}(x)=$ $\square$ Critical numbers $x=$ $\square$ Union of the intervals where $f(x)$ is increasing $\square$ Union of the intervals where $f(x)$ is decreasing $\square$ Local maxima $x=$ $\square$ Local minima $x=$ $\square$ (b) Find the following left- and right-hand limits at the vertical asymptote $x=-5$ . $\lim _{x \rightarrow-5^{-}} \frac{2 x^{2}}{x^{2}-25}=$ $\square$ $\lim _{x \rightarrow-5^{+}} \frac{2 x^{2}}{x^{2}-25}=?$ $\square$ Find the following left-and right-hand limits at the vertical asymptote $x=5$ . $\lim _{x \rightarrow 5^{-}} \frac{2 x^{2}}{x^{2}-25}=?$ $\square$ $\lim _{x \rightarrow 5^{+}} \frac{2 x^{2}}{x^{2}-25}=$ $\square$ Find the following limits at infinity to determine any horizontal asymptotes. $\lim _{x \rightarrow-\infty} \frac{2 x^{2}}{x^{2}-25}=? \quad \vee \quad \lim _{x \rightarrow+\infty} \frac{2 x^{2}}{x^{2}-25}=?$ $\square$ $\square$ (c) Calculate the second derivative of $f$ . Find where $f$ is concave up, concave down, and has inflection points. $f^{\prime \prime}(x)=\square$
Union of the intervals where $f(x)$ is concave up $\square$ Union of the intervals where $f(x)$ is concave down $\square$ $\square$ Inflection points $x=$

See Solution

Problem 1292

Suppose that $f(x)=(5-\ln (x))^{3}$ . Find $f^{\prime}(1)$ . $f^{\prime}(1)=$

See Solution

Problem 1293

Suppose that $f(x)=\frac{3}{\ln \left(x^{2}+2\right)}$
Find $f^{\prime}(1)$ . $f^{\prime}(1)=$

See Solution

Problem 1294

37-43 La región delimitada por las curvas dadas gira alrededor del eje especificado. Determine el volumen del sólido resultante, por medio de cualquier método.
37. $y=-x^{2}+6 x-8, y=0$ ; alrededor del eje $y$
38. $y=-x^{2}+6 x-8, y=0$ ; alrededor del eje $x$

See Solution

Problem 1295

If $f(x)=(5 x+4)^{-1}$
Find $f^{\prime}(x)$ . Then $f^{\prime}(x)=$ $\square$ Find $f^{\prime}(3)$ . Then $f^{\prime}(3)=$ $\square$

See Solution

Problem 1296

(1 point) Let $f(x)=e^{-7 x^{2}}$ . Then $f(x)$ has a relative minimum at $x=$ a relative maximum at $x=\square$ and inflection points at $x=\square$ and at $x=\square$
Write DNE if any of the above do not exist. Write the inflection points (if ar

See Solution

Problem 1297

Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. (If an answer is undefined, enter UNDEFINED.) $y=\frac{25}{(x+5)^{2}},(0,1)$ $\square$ Need Help? Read It Submit Answer

See Solution

Problem 1298

3. Determine la derivada direccional de $f(x, y)=2 x^{2} y^{3}+6 x y$ en $(1,1)$ en la dirección del vector unitario cuyo ángulo con el eje x positivo es $\frac{\pi}{6}$ . Además, determine el gradiente de fen ( 1,1 ) Recuerde: $u=\cos \theta i+\operatorname{sen} \theta j$

See Solution

Problem 1299

$\lim _{x \rightarrow 0} \frac{5 x^{3}+8 x^{2}}{3 x^{4}-16 x^{2}}$

See Solution

Problem 1300

Substitution in the Definite Integral
Suppose we want to evaluate the definite integral, $\int_{0}^{\sqrt{60}} t\left(4+t^{2}\right)^{\frac{1}{3}} d t$ using the substitution, $u=4+t^{2}$ .
Part 1.
Re-write the definite integral in terms of the variable $u$ and remember to use the limits of integration for the function $u=f(t)$ . Then, input antiderivative of the integrand and the limits of integration you found. $\square$ $=\square$ $=[\square]$

See Solution

Calculus

Problem 1201

2 3- جد مركبات شعاع الوحدة المماسي

Problem 1202

Problem 1203

Problem 1204

Problem 1205

Problem 1206

Problem 1207

1. lim⁡x→+∞1xtan⁡(2x)\lim _{x \rightarrow+\infty} \frac{\frac{1}{x}}{\tan \left(\frac{2}{x}\right)}limx→+∞​tan(x2​)x1​​

Problem 1208

3. lim⁡x→0+xxx\lim _{x \rightarrow 0^{+}} x^{x^{x}}limx→0+​xxx

Problem 1209

Вычислить неопределенные интегралы, сводя их к табличным:1. ∫(2x5−3x2)dx=x63−x3+C\int\left(2 x^{5}-3 x^{2}\right) d x=\frac{x^{6}}{3}-x^{3}+C∫(2x5−3x2)dx=3x6​−x3+C

Problem 1210

Problem 1211

Problem 1212

Problem 1213

oncepts Araph of a function h mathematics, the graph of a function is the set of ordered pairs, where in the common ce find the end behavior in limit notation f(x)=x−1x2−4f(x)=\frac{x-1}{x^{2}-4}f(x)=x2−4x−1​

Problem 1214

∫sin⁡−1(5x)dx\int \sin ^{-1}(5 x) d x∫sin−1(5x)dx

Problem 1215

If exy−y2=e−4e^{x y}-y^{2}=e-4exy−y2=e−4, then at x=12x=\frac{1}{2}x=21​ and y=2,dydx=y=2, \frac{d y}{d x}=y=2,dxdy​=

Problem 1216

(a) lim⁡x→3x−1−29−x2\lim _{x \rightarrow 3} \frac{\sqrt{x-1}-\sqrt{2}}{9-x^{2}}limx→3​9−x2x−1​−2​​.

Problem 1217

Problem 1218

Let H(t)=at40−b⋅etH(t)=\frac{a}{t^{40}}-b \cdot e^{t}H(t)=t40a​−b⋅et where aaa and bbb are both fixed constants. What is H′(t)H^{\prime}(t)H′(t) ?

Problem 1219

PREVIOUS ANSWERS PRACTICE ANOTHERFind the equation for the plane tangent to each surface z=f(x,y)z=f(x, y)z=f(x,y) at the indicated point. (a) z=x3+y3−5xyz=x^{3}+y^{3}-5 x yz=x3+y3−5xy, at the point (1,2,−1)(1,2,-1)(1,2,−1)

Problem 1220

Problem 1221

Problem 1222

12πσ∫−∞∞etxe−(x−μ)22σ2 dx\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{\infty} e^{t x} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \, dx2π​σ1​∫−∞∞​etxe−2σ2(x−μ)2​dx

Problem 1223

Problem 1224

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim⁡x→0x4cos⁡(8x)\lim _{x \rightarrow 0} x^{4} \cos \left(\frac{8}{x}\right)x→0lim​x4cos(x8​)

Problem 1225

If the figure below is the graph of the derivative f′ f^{\prime} f′, answer the following:Where do the points of inflection of f f f occur? □\square□On which interval(s) is f f f concave down? □\square□Note: You can earn partial credit on this problem.

Problem 1226

∑n=1∞2n+2\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}∑n=1∞​n​+22​

Problem 1227

Problem 1228

Problem 1229

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim⁡(x,y)→(0,0)xy8x3+y12\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{8}}{x^{3}+y^{12}}(x,y)→(0,0)lim​x3+y12xy8​

Problem 1230

Determine whether the statement is true or false. The derivative of a polynomial is a polynomial. True False Submit Answer

Problem 1231

Problem 1232

Problem 1233

Problem 1234

Problem 1235

8. lim⁡X→01−x−1+XX=\lim _{X \rightarrow 0} \frac{\sqrt{1-x}-\sqrt{1+X}}{X}=limX→0​X1−x​−1+X​​=

Problem 1236

4) The horizontal asymptote for f(x)=5x2−1(x−1)(x−3)f(x)=\frac{5 x^{2}-1}{(x-1)(x-3)}f(x)=(x−1)(x−3)5x2−1​ is

Problem 1237

13. lim⁡x→04X3+6x3x3+2x=\lim _{x \rightarrow 0} \frac{4 X^{3}+6 x}{3 x^{3}+2 x}=limx→0​3x3+2x4X3+6x​=

Problem 1238

Use the graph to answer the question. Describe the continuity of the graphed function. (1 point) The function is continuous. The function has a removable discontinuity at x=3x=3x=3. The function has a jump discontinuity at x=3x=3x=3. The function has an infinite discontinuity at x=3x=3x=3.

Problem 1239

reach series will converge or diverge. Then sible. 7) 8.1+3.24+1.296+0.5184…8.1+3.24+1.296+0.5184 \ldots8.1+3.24+1.296+0.5184…

Problem 1240

Let f(x)=1x2−4f(x)=\frac{1}{x^{2}-4}f(x)=x2−41​ Where does fff have critical points? Choose all answers that apply:A x=−2x=-2x=−2 B x=0x=0x=0 c. x=2x=2x=2D fff has no critical points.

Problem 1241

Let g(x)=sin⁡(2x)g(x)=\sin (2 x)g(x)=sin(2x), for −π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}−2π​≤x≤2π​. Where does ggg have critical points?

Problem 1242

Let h(x)=e2xx−3h(x)=\frac{e^{2 x}}{x-3}h(x)=x−3e2x​ Where does hhh have critical points?

Problem 1243

Determine whether the series ∑k=0∞kk2+3\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+3}}∑k=0∞​k2+3​k​ converges.

Problem 1244

Problem 1245

Problem 1246

Problem 1247

Let hhh be a polynomial function and let h′h^{\prime}h′, its derivative, be defined as h′(x)=x2(x−2)2(x−1)2h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}h′(x)=x2(x−2)2(x−1)2.At how many points does the graph of hhh have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

Problem 1248

7. Consider the curve given by the equation y3−3xy=2y^{3}-3 x y=2y3−3xy=2. a. Find dydx\frac{d y}{d x}dxdy​.

Problem 1249

Problem 1250

Question 7Let f(x)=3x+4f(x)=\sqrt{3 x+4}f(x)=3x+4​. Calculate the difference quotient: f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​

Problem 1251

1. $\lim _{x \rightarrow+\infty} \frac{\frac{1}{x}}{\tan \left(\frac{2}{x}\right)}$

3. $\lim _{x \rightarrow 0^{+}} x^{x^{x}}$

Вычислить неопределенные интегралы, сводя их к табличным:
1. $\int\left(2 x^{5}-3 x^{2}\right) d x=\frac{x^{6}}{3}-x^{3}+C$

oncepts Araph of a function h mathematics, the graph of a function is the set of ordered pairs, where in the common ce find the end behavior in limit notation $f(x)=\frac{x-1}{x^{2}-4}$

$\int \sin ^{-1}(5 x) d x$

If $e^{x y}-y^{2}=e-4$ , then at $x=\frac{1}{2}$ and $y=2, \frac{d y}{d x}=$

(a) $\lim _{x \rightarrow 3} \frac{\sqrt{x-1}-\sqrt{2}}{9-x^{2}}$ .

Let $H(t)=\frac{a}{t^{40}}-b \cdot e^{t}$ where $a$ and $b$ are both fixed constants. What is $H^{\prime}(t)$ ?

PREVIOUS ANSWERS PRACTICE ANOTHER
Find the equation for the plane tangent to each surface $z=f(x, y)$ at the indicated point. (a) $z=x^{3}+y^{3}-5 x y$ , at the point $(1,2,-1)$

$\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{\infty} e^{t x} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \, dx$

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) $\lim _{x \rightarrow 0} x^{4} \cos \left(\frac{8}{x}\right)$

If the figure below is the graph of the derivative $f^{\prime}$ , answer the following:
Where do the points of inflection of $f$ occur? $\square$
On which interval(s) is $f$ concave down? $\square$
Note: You can earn partial credit on this problem.

$\sum_{n=1}^{\infty} \frac{2}{\sqrt{n}+2}$

Find the limit, if it exists. (If an answer does not exist, enter DNE.) $\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{8}}{x^{3}+y^{12}}$

8. $\lim _{X \rightarrow 0} \frac{\sqrt{1-x}-\sqrt{1+X}}{X}=$

4) The horizontal asymptote for $f(x)=\frac{5 x^{2}-1}{(x-1)(x-3)}$ is

13. $\lim _{x \rightarrow 0} \frac{4 X^{3}+6 x}{3 x^{3}+2 x}=$

Use the graph to answer the question. Describe the continuity of the graphed function. (1 point) The function is continuous. The function has a removable discontinuity at $x=3$ . The function has a jump discontinuity at $x=3$ . The function has an infinite discontinuity at $x=3$ .

reach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

Let $f(x)=\frac{1}{x^{2}-4}$ Where does $f$ have critical points? Choose all answers that apply:
A $x=-2$ B $x=0$ c. $x=2$
D $f$ has no critical points.

Let $g(x)=\sin (2 x)$ , for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ . Where does $g$ have critical points?

Let $h(x)=\frac{e^{2 x}}{x-3}$ Where does $h$ have critical points?

Determine whether the series $\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+3}}$ converges.

Let $h$ be a polynomial function and let $h^{\prime}$ , its derivative, be defined as $h^{\prime}(x)=x^{2}(x-2)^{2}(x-1)^{2}$ .
At how many points does the graph of $h$ have a relative maximum ? Choose 1 answer: (A) None (B) One (C) Two (D) Three

7. Consider the curve given by the equation $y^{3}-3 x y=2$ . a. Find $\frac{d y}{d x}$ .

Question 7
Let $f(x)=\sqrt{3 x+4}$ . Calculate the difference quotient: $\frac{f(x+h)-f(x)}{h}$

27-28 Explain why the Integral Test can't be used to determine whether the series is convergent.
27. $\sum_{n=1}^{\infty} \frac{\cos \pi n}{\sqrt{n}}$
28. $\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{1+n^{2}}$

Use the substitution $u=7 x^{2}+2 x$ to evaluate the following indefinite integral. $\int(14 x+2) \sqrt{7 x^{2}+2 x} d x$

Evaluate the given indefinite integrals. a) $\int \frac{6}{y^{5}} d y=\square+C$ , b) $\int\left(\frac{1}{u^{1 / 4}}+5 \sqrt{u}\right) d u=\square+C$ . c) $\int \frac{1}{6 x^{5}} d x=\square+C$ .

3. Use Green's Theorem to evaluate $\oint_{C} 2 x y d x+(x+y) d y$ . $C$ is the boundary of the region lying between the graphs of $y=0$ and $y=1-x^{2}$ , oriented counterclockwise.

Prove that $\sin ^{2} x+\cos ^{2} x=1$ using Calculus by: (1) Find the derivative of $f(x)=\sin ^{2} x+\cos ^{2} x$ (2) Use corollay to show that $f(x)=1$ on all of $[0,2 \pi]$ .

Let $f(x)=x^{2} \tan ^{-1}(8 x)$ $f^{\prime}(x)=$

Find the general antiderivative, $F(x)$ , of the function $f(x)=-\frac{2}{11} x^{5}+\frac{9}{7} x^{11}-\frac{6}{11} \frac{1}{x^{7}}$ $F(x)=$

Find the indefinite integral. (Use C for the constant of integration.) $\int(x-2)(x+4) d x$

$\lim _{x \rightarrow 4} \frac{x^{2}-16}{\sqrt{x}-2}$

9. Use an appropriate substitution to evaluate the definite integral $\int_{2}^{6} \frac{x}{\sqrt{2 x-3}} d x$ .

What is $\int_{0}^{1}(2 t+2) \, dx ?$

(1 point) Let $f(x)=(\ln (x))^{\operatorname{sed}(x)}$ . Find $f^{\prime}(x)$ . $f^{\prime}(x)=$

$\lim _{x \rightarrow \infty} \frac{\ln \left(x^{2}+1\right)}{\ln \left(3 x^{2}+7\right)}$

نعثر الدالة $f$ المعر فلة على