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Resources

Find derivatives of using multiple formulae

10-\mathrm{kg}

bucket is lifted from the ground to a height of

14

m at a constant speed with a rope that weighs

0.6 \mathrm{~kg} / \mathrm{m}

. Initially the bucket contains

42

kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the

14-\mathrm{m}

level. How much work is done? (Use

9.8 \mathrm{~m} / \mathrm{s}^{2}

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Show how to approximate the required work (in J) by a Riemann sum. (Let

x

be the height in meters above the ground. Enter

x_{i}^{* *}

x_{i}

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\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\boxed{98+35.28 x_{i}}\right) \Delta x

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Express the work (in J) as an integral in terms of

x

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\int_{0}^{14}(98+35.28 x

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Evaluate the integral (in J). (Round your answer to the nearest integer.)

One day, I went to my colony park and saw that many people were busy with their routines. I found that there were

150

people in total. I observed that

18

people were jogging,

24

were doing yoga,

36

were engaged in laughing therapy, and the rest of the people preferred to walk.

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(a) What percent of people were jogging?

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15\%

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12\%

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10\%

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20\%

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(b) What percent of people were doing yoga?

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12\%

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16\%

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18\%

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20\%

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(c) How many people preferred to walk?

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72

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62

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52

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82

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(d) What percent of people were doing laughing therapy?

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22\%

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24\%

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30\%

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26\%

8

\frac{4 x}{x^{2}-9}-\frac{x}{x^{2}-6 x+9}=-\frac{2}{x+3}

Which expression is equivalent to

(2g^{3}+4)^{2}

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49^{6}+16

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4g^{6}+18g^{3}+16

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49^{6}+8

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4g^{9}+16g^{3}+8

Which of the following is equivalent to the given expression?

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1

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\frac{a^{2}}{b^{2}}

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\frac{a^{2}}{c^{2}}

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\frac{a^{2}}{b^{2}c^{2}}

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\frac{a^{2}c^{2}}{b^{2}}

\frac{3n^{-6}m^{2}}{18m^{-3}n^{-5}}

f

be the function defined by

f(a)=\int_{0}^{1}(2z^{2}-15z^{2})\,dt

. On which of the following intervals is the graph of

y=f(x)

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(-\infty,0)

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(\rightarrow \infty,2)

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(0,\infty)

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(2,3)

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(3,\infty)

Which of these are equal to

1

? Select all that apply.

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Multi-select Choices:

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-\frac{7}{7}

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-\frac{-7}{7}

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\frac{7}{7}

Which of these are equal to

11

? Select all that apply.

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Multi-select Choices:

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\frac{44}{4}

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-\frac{44}{4}

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-\frac{44}{-4}

Which of these are equal to

2

? Select all that apply.

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Multi-select Choices:

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-\frac{4}{2}

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\frac{4}{2}

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-\frac{4}{-2}

Which of these are equal to

-11

? Select all that apply.

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Multi-select Choices:

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-\frac{22}{2}

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\frac{22}{2}

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-\frac{22}{-2}

Which of these are equal to

-9

? Select all that apply.

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Multi-select Choices:

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-\frac{45}{5}

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-\frac{45}{5}

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\frac{45}{-5}

Which of these are equal to

-4

? Select all that apply.

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Multi-select Choices:

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\frac{24}{-6}

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-\left(\frac{24}{6}\right)

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-\frac{24}{6}

Which of these are equal to

7

? Select all that apply.

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Multi-select Choices:

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\frac{35}{5}

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-\frac{35}{5}

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\frac{35}{-5}

Which of these are equal to

-8

? Select all that apply.

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Multi-select Choices:

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\frac{24}{3}

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-\frac{24}{3}

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-\frac{24}{-3}

2t^{3}v-10t^{2}v+12tv

Solve the equation.

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\frac{d y}{d x}=\frac{(x y)^{2}}{8}+y^{2}

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1

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y=-\frac{x^{2}+16}{16+C}

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y=-\frac{16}{x^{2}+16+C}

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y=-\frac{x^{3}+24 x}{24+C}

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y=-\frac{24}{x^{3}+24 x+C}

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y=\left(1+x^{2}\right) \operatorname{Arctan}{x}-x

Determine whether the statement is true or false.

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f(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}

g(x)=\frac{1}{1-x^{2}}=\sum_{n=0}^{\infty} x^{2 n}

f(x) g(x)=\sum_{n=0}^{\infty} x^{n} x^{2 n}=\sum_{n=0}^{\infty} x^{3 n}

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\left(\frac{x^{a}}{x^{b}}\right)^{b+c-a}\left(\frac{x^{c}}{x^{a}}\right)^{c+a-b}\left(\frac{x^{a}}{x^{b}}\right)^{a+b-c}

\left(\frac{n^{3}}{4 n}\right)^\frac{3}{4 }\

\frac{-9r^{2}s^{6}t^{4}}{54r^{5}s^{2}t^{8}}

x^{3}+3xy+2y^{3}=17

, then in terms of

x

y

(dy)/(dx) =

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((x^{2}+y)/(x+2y^{2}))

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((x^{2}+y)/(x+2y))

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((x^{2}+y)/(2y^{2}))

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((x^{2}+y)/(x+y^{2}))

1-x^{2}+\frac{x^{4}}{2!}-\frac{x^{6}}{3!}+\frac{x^{8}}{4!}+\cdots+(-1)^{n} \frac{x^{2 n}}{n!}+\cdots

converges to which of the following?

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\cos \left(x^{2}\right)+\sin \left(x^{2}\right)

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1+x \sin x

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\cos x

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e^{-x^{2}}

\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}

\frac{\sqrt{54 a^{3} b^{7}}}{\sqrt{6 b^{5}}}

Which equations define

y

as a nonlinear function of

x

? Select all that apply.

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Multi-select Choices:

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y = 4x^2 + 5x - 8

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y = -8 + 5x

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y = x^3 + 1

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y = x - 9^2

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y = \sqrt{x} - 3

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y = x^2

Simplify the expression:

\frac{x^{3}+3x^{2}y+3xy^{2}+y^{3}}{x^{3}+y^{3}}

Which of the following expressions are equal to

30,600

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\sum_{n=1}^{45} 30n-10

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\sum_{n=5}^{49} 20+30(n-5)

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\sum_{n=2}^{46} -10+30(n-2)

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D. I, II, and III

4 e^{\frac{7 \pi}{4} i} i+2 e^{\frac{3 \pi}{2} i}

\frac{-z x^{4} y^{0}}{-4 x^{4} y^{0} z^{2}}

\frac{-3 h k^{-6}}{-9 h^{-7} j^{6} k^{0}}

\left(x^{3} y^{5}\right)^{3} \cdot-x^{6} y^{5}

\frac{a^{5} b^{2}}{4 b} \cdot \frac{16 a^{9} b^{4}}{a^{3} b^{7}}

What could be the value of

x

in the following equation? Select all that apply.

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x^2 = 0.25

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Multi-select Choices:

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-\sqrt{0.25}

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-0.5

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\sqrt{0.25}

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0.5

Simplify the expression:

\frac{x^{2}+25}{x^{2}-10x+25}

The solution set for

2 x^{2}-7 x-4=0

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1

\{2,-1\}

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3

\{-2,1\}

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2

\left\{-\frac{1}{2}, 4\right\}

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4

\left\{\frac{1}{2},-4\right\}

-21a^{6}b

-3a^{2}b

, the quotient is

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1

7a^{4}

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2

-7a^{3}

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3

7a^{3}b

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4

7a^{4}b

-21 a^{6} b

b y-3 a^{2} b

, the quotient is

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1

7 a^{4}

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3

7 a^{3} b

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2

-7 a^{3}

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4

7 a^{4} b

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

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Multi-select Choices:

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y = 0

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y - 2x = 6

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2x - y = 6

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y = x^3

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y = x^2

Select all the expressions that are equivalent to

10^3 \times 6^3

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Multi-select Choices:

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60^3

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\frac{60^6}{60^3}

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60^9

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60^6

Select all the expressions that are equivalent to

7^3 \times 10^3

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Multi-select Choices:

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70^3

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\frac{1}{70^{-3}}

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70^9

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70^6

Select all the expressions that are equivalent to

10^2 \times 5^2

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Multi-select Choices:

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\frac{1}{50^{-2}}

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50^2

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\frac{1}{50^2}

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50^4

j^{3}+3j^{2}k+3jk^{2}+k^{3}

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Which of the following is equivalent to the given expression?

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1

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j^{3}+3(jk)^{2}+k^{3}

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j^{3}+6(jk)^{2}+k^{3}

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j^{3}+3jk(j+k)+k^{3}

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j^{3}+6jk(j+k)+k^{3}

Select all of the true statements.

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Multi-select Choices:

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6 \times 10^{-4}

300

times as much as

2 \times 10^{-6}

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7 \times 10^{4}

3.5

times as much as

2 \times 10^{4}

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2 \times 10^{-6}

0.01

times as much as

2 \times 10^{-8}

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9 \times 10^{4}

300

0

times as much as

300

1

The second derivative of a function

g

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g^{\prime \prime}(x)=x^{4}-x^{2}-x \text {. }

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On which interval is the graph of

g

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Use a graphing calculator.

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1

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(0,1.325)

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(-\infty, 0)

(1.325, \infty)

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(-\infty, 0.885)

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(0.885, \infty)

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(E) All real numbers

The radius of a cone is increasing at a rate of

3

centimeters per second and the height of the cone is decreasing at a rate of

4

centimeters per second.

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At a certain instant, the radius is

8

centimeters and the height is

10

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What is the rate of change of the volume of the cone at that instant (in cubic centimeters per second)?

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1

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\frac{224 \pi}{3}

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-\frac{224 \pi}{3}

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-\frac{736 \pi}{3}

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\frac{736 \pi}{3}

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a

cone with radius

r

h

\pi r^{2} \frac{h}{3}

\int \frac{x^{3}-3 x^{2}+5 x-3}{x-1} d x

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1

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\frac{x^{3}}{3}-x^{2}+3 x+C

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\frac{x^{3}}{3}-x^{2}+3 x+\ln |x-1|+C

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\frac{x^{3}}{3}-x^{2}+3 x-2 \ln |x-1|+C

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\frac{x^{3}}{3}-2 x^{2}+3 x+\ln |x-1|+C

\int \frac{2 x^{3}-3 x^{2}-3 x+2}{x-2} d x

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1

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\frac{2 x^{3}}{3}+\frac{x^{2}}{2}-x+C

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\frac{2 x^{3}}{3}-\frac{x^{2}}{2}+x+\ln |x-2|+C

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\frac{2 x^{3}}{3}+\frac{5 x^{2}}{2}-x+C

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\frac{2 x^{3}}{3}-\frac{x^{2}}{2}-x+\ln |x-2|+C

\int \frac{2 x^{3}+4 x^{2}-5}{x+3} d x

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1

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\frac{2 x^{3}}{3}-\frac{1}{2} x^{2}+6 x+13 \ln |x+3|+C

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\frac{2 x^{3}}{3}-x^{2}+6 x+13 \ln |x+3|+C

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\frac{2 x^{3}}{3}-x^{2}+6 x-23 \ln |x+3|+C

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\frac{2 x^{3}}{3}-x^{2}-6 x-23 \ln |x+3|+C

...