Spring 2018 Math 1316 Test 3 Review PDF

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Math 1316 Test 3 Review

The following formulas will be provided to you on a cover sheet.

FORMULAS Product Rule: if

f x   u x  vx  then f x   u x   v x   vx   u x 

Quotient Rule:

if f  x 

Chain (Power)Rule:

if

v x   u x   u x   v x  u x  then f  x   v x  vx 2

f x   u x 

Derivative of Natural Log:

n

f x   n u x 

n 1

then

if y = ln (x) then

y 

ln u ( x)

then

if y =

Derivative of Exponential Function: if if

1 x

y 

1  u ( x) = 𝑢′(𝑥) u(x) 𝑢(𝑥)

y  e x then y  e x ;

y  eu ( x )

then

y   e u( x)  u ( x)

 kdx  kx  C

x n  x dx  n  1  C if n ≠ -1

dx 1  x   x dx  ln x  C

 u( x)

1

n

∫

x x  e dx  e  C

1

u dx   u 1du  ln u  C  ln u( x)  C

𝑒 𝑘𝑥 𝑑𝑥

=

un 1 u (x )n 1  u( x) udx   u du  n  1  C  n  1  C if n  -1 n

e

u( x)

n

udx   e udu  e u  C  e u( x)  C

𝑞

Integration by Parts: ∫ u dv = uv - ∫ 𝑣 𝑑𝑢

1  n i 1

n

i  i 1

nn 1 2

𝑒 𝑘𝑥 𝑘

+C

where

du  udx

𝑞

PS = ∫0 0 [𝑝0 − 𝑆(𝑞)]𝑑𝑞

CS =∫0 0 [𝐷(𝑞 ) − 𝑝0 ]𝑑𝑞

n

 u x 

n

i 2  i 1

where

du  udx

n n  1 2n  1 6

and v = ∫ 𝑑𝑣

∑𝑛𝑖=1 𝑖 3

=

𝑛2 (𝑛+1)2 4

The test will consist of 20 - 25 multiple choice questions modeled after problems in your workbook and on your homework. You will need a scantron form 882-E and may use a 4 by 6 notecard as well as a calculator on the test. You will need to be able to do the following:

The Test Will Consist of the Following Types of Problems

 1 Optimal Level of Production problem  1 where given u(x) you need to be able to identify the correct value for du in a u-substitution integration problem using du =𝒖′𝒅𝒙  1 or 2 summation problems  10 – 15 Assorted indefinite integral problems including u-substitution problems, problems involving lns and exponentials, problems invoving radicals or negative exponents, and application problems where given MC and a cost point or MR you are asked to calculate C(x) and R(x)  5 – 8 Assorted definite Integral problems including application problems where you need to find the area between a positive curve and the x-axis between two given x values or where you need to calculate the area between two curves  1 problem where given the demand or supply curve and the equilibrium point you are asked to find the consumer or producer surplus  2 or 3 integration by parts problems where you need to be able to identify u, dv, v, and du as well as compute the resulting value of the integral.

1316 Test 3 Practice Test



6 1)  8x dx  _____________________________________________________________

ex 2)  dx  ________________________________________________________________ 9 

2 x 3)  8 e  5 x 



1 3

3 dx  ______________________________________________________ x

 

4)   e x  x 4 dx  __________________________________________________________



2 5) (3 x  5 x  2) dx  _____________________________________________________



6)  8dx  ______________________________________________________________

7)



10 dx = _______________________________________________________________ x

 

4 8)  x  5e 

9)

x

2 dx = _____________________________________________________ x

8

 6x  2 dx  ____________________________________________________________

 x2  dx = _______________________________________________________ 10]   2  5 x  20 x 

11)

 4 x  2 8dx = __________________________________________________________ 3

2x 12)  9 e dx  _____________________________________________________________

13) If R(x) is a revenue function then R(0) must equal what? _________________________

14) Suppose the marginal revenue is MR = 64x. What is the revenue generated at a production level of 10 units? 15) If the rate of change of cost for an item is MC = 4x-2 and the total cost of producing 6 items is $700, find the total cost function. 16) Find the optimal level of production given the marginal cost function for an item is: MC=5x + 50 and the marginal revenue function is: MR = 210 – 3x. 4

17)

 8  x dx  3

__________________________________________________________

1

10



18) 3dx =_________________________________________________________________ 2

2



19) 8xdx =_______________________________________________________________ 1

20) What’s the area under the curve y  8 3 x and above the x axis over the interval [1, 8] ? 21) What is the area under the curve y  6x and above the x axis over the interval [1, 3]? 2

22)∫(−3𝑒 2𝑥 )𝑑𝑥 =__________________________________________________________ 2

23) ∫(5𝑥𝑒 3𝑥 )𝑑𝑥 =__________________________________________________________ 24) Find R(2) if Marginal Revenue = 8x + 3.

25)



1 dx  __________________________________________________________ 3x  5

2 2 26)   e x  x 2 dx  ________________________________________________________ 9

9



27) Suppose the marginal revenue function for a product is MR=

40

𝑒 0.5𝑥

+ 10; find R(x).

28) What is the area found under the curve y  9 x and above the x axis over the interval [0, 1]? 29) Give that the fixed costs for producing an item are $270 and that MC = 18 x  4 ; find the total cost function. 30) The demand function for a product is p =D(x) = 100 – 4x. If the equilibrium point is (15,40) then what is the consumer surplus? 31) The supply function for a good is p  S (x )  4x  2x  2 . If the equilibrium point is (10,422) then what is the producer surplus? 32)Use integration by parts to evaluate the following integral: ∫(𝑥 ln(2𝑥))𝑑𝑥 . Be able to identify u, dv, v, and du as well as the solution to the integral. 33) Use integration by parts to evaluate the following integral:∫(10𝑥𝑒 2𝑥 )𝑑𝑥 . Be able to identify u, dv, v, and du as well as the solution to the integral. 2

34) ∫ (

−14 ) 𝑑𝑥 𝑥+2

= _____________________________________

2 35) ∫(4𝑥𝑒 −3𝑥 +7 )𝑑𝑥

3 2

36) ∫

5

7

( 𝑥 2 + 𝑥 4) 𝑑𝑥 = _______________________________________

6 8 ( ) 𝑑𝑥 1 3𝑥

37) ∫

3 0

38)∫

= _______________________________

= ___________________________________________

6𝑥

( 𝑥 2+1) 𝑑𝑥 = ______________________________________

39) Find the area between the x-axis and the function f(x) = ex over the interval [0, 2]. Round your answer off to 3 decimal places.

40) Find the total cost function C(x) given that the marginal cost function is:

𝑀𝐶 = 𝐶 ′ (𝑥) =

6

𝑥+1

and that the fixed costs for the product are $18. 2

2

41) Find the area between the 2 curves: y = x + 30 and y = x +40 from x = 1 to x = 5. 2

3

42) Find the area between the 2 curves: y = x and y = x from x = 0 to x = 1/2. 4

43) Use integration by parts to evaluate the definite integral: ∫1 (𝑥 ln(𝑥))𝑑𝑥 2 44) ∑40 𝑖=1 (3𝑖 − 7𝑖) =?

45.

3 2 ∑20 𝑖=1 (2𝑖 + 3𝑖 + 4𝑖 + 3 =?

46. In a u-substitution problem if u = 6x2 + 5x then du = _______________________?

Solutions 8 7 1]  x  C 7

1 x e C 9

2]

5 3

x 3 3] 8e  x  3 ln | x | C

𝑒 0.5𝑥

1

du = 𝑑𝑥

x5 1 4] e x   C 3 5

5] 𝑥 3 −

5𝑥2 2

𝑥2 2

x  5 ex  2 ln | x|  C 5

4 9] ln 6x  2  C 3 1 10] ln 5x 2  20x  C 10

1 4 11]  4 x  2  C 2 9 12] e 2x  C 2

13] 0 14] R( x)  32 x 2 ; R(10) = $3200 15] C( x)  2x 2  2x  640 16] x = 20 17] – 39.75 18] 24 19] 12 20] 90 square units 21] 52 square units 22] − 23]

3𝑒 2𝑥 2

2 5𝑒 3𝑥

6

+𝐶

+C

1

4

dv = 𝑒 2𝑥 dx

du = 10dx

v=

𝑒 2𝑥 2

5

5𝑥𝑒 2𝑥 − 𝑒 2𝑥 + 𝐶 2

34]−14 ln|𝑥 + 2| + 𝐶 35]

2 −2𝑒 −3𝑥 +7

3

+C =

−2

2 3𝑒 3𝑥 +7

+𝐶

36] 1.039 37] 4.77 38] 3 ln10 = 6.908 39] Area = 6..389 square units 40] C(x) = 6 ln|x + 1| + 18 41] Area = 40 square units 42] Area =.026 square units 43] 7.34 44] 60,680 45] 97,710

24] $22

2 3x  5  C 3 2 2 26] e x  x 3  C 9 27

𝑥2 2

ln 2𝑥 − 𝑥 2 + 𝐶

33] u =10 x

5

8]

𝑣=

𝑥

+ 2𝑥 + 𝐶

6] – 8x + C 7] 10 ln | x | C

−80

+ 10𝑥 + 80 28] 6 square units 29] C ( x)  12 ( x  4) 3 174 30] $450 31] $2766.67 32] u = ln 2𝑥 dv = x dx 27] R(x) =

25]

46] du = (12x + 5) dx

***************************************************************************** *****************************************************************************

Some Bonus Quiz Examples The following problems will not be included on test three itself but problems similar to these could show up on the bonus quiz and/or the final exam. For further help with the bonus quiz students should refer to their class notes, homework problems, and problems worked out in their workbook.

6x 4  6 dx  ___________________________________________________________ 1)  5 x 1  3  dx  ______________________________________________ 2)   3x  2 3x  6 x 2  e 3) Suppose that you are shooting paint balls into the air in such a way that the rate of change for 𝑑ℎ

the height of each ball t seconds after you shoot it is v = 𝑑𝑡 = 208 – 32t, in feet per second, and that 3 seconds after you shoot the ball it is 380 feet high in the air. Use this information to derive the function h(t) that describes the height of the ball at any time t. 4) Suppose the average rate of change for sales, in hundreds of dollars, x weeks after a company 𝑑𝑆

introduces a new product to the market is given by 𝑑𝑥 =

9𝑥2 +9

𝑥 3 +3𝑥+7

. If the company brought in $1100

in initial sales find the total sales 1 year after the product is introduced to the market. Round answer off to the nearest penny.

Solutions 6 5 6 x  xC 5 25 2] e  3x  1 ln 3 x 2  6 x  2  C 6

1]

3] h(t) = 208t – 16t2 – 100 4] $4072.70...