MATH153 22 Calculus Problems and Difference Quotient Analysis Questions

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ebelzngm1698

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Mathematics

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MATH153

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Complete all problems in the following document. All are calculus problems and some will have certain specifications for each problem please be neat and show all work.

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MATH 153 NAME___________________ _ __ _ __ Show all work for credit. Simplify/reduce. State exact forms of answers unless rounding is specified. 1) For each function, find and fully simplify the difference quotient a) 𝑓(𝑥) = 5 𝑥 b) 𝑓(𝑥) = √𝑥 −3 2) For the function 𝑓(𝑥) = 3𝑥2 − 2𝑥 a) Find/fully simplify the difference quotient 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ b) Calculate the value of the difference quotient for the following pairs of x and h-values: i) x = -1 and h = 3 ii) x = -1 and h = 2 iii) x = -1 and h = 1 iv) x = -1 and h = 0.5 v) x = -1 and h = 0.1 3) Sketch the graph of the function 𝑓(𝑥) = 3𝑥2 − 2𝑥 and the secant lines from parts i) and iii) above (only those two secant lines) 4) For the function 𝑓(𝑥) = 8𝑥2 + 4𝑥 − 1, find the average rate of change between 𝑥1 = −3 and 𝑥2 = −1 5) For the function 𝑓(𝑥) = 𝑥2 + 3𝑥 − 2: a) Find/fully simplify the difference quotient b) Find the limit: lim 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ→0 ℎ c) Find the average rate of change in 𝑓(𝑥) from 𝑥1 = 1 to 𝑥2 = 1.1 d) Find the instantaneous rate of change in 𝑓(𝑥) at 𝑥 = 1 6) For the function 𝑓(𝑥) = 1 3 𝑥 3 + 2, a) Find/fully simplify the difference quotient b) Find the limit: lim 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ→0 ℎ c) Find the slope of the secant line to the graph of 𝑓(𝑥) from 𝑥1 = 1 to 𝑥2 = 1.1 d) Find the slope of the tangent line to the graph of 𝑓(𝑥) at 𝑥 = 1 7) Find the limit by graphing the function and using TRACE or TABLE to examine the graph near the indicated x-value. Make a rough sketch of the graph to support your answer. 64 −8 lim 𝑥 𝑥→8 8 −𝑥 8) For the piecewise linear function f(x) graphed below, find each limit or function value. (If an answer does not exist, state “DNE.”) a) lim 𝑓(𝑥) = ____________ − _ b) lim 𝑓(𝑥) = ____________ + _ lim 𝑓(𝑥) = ____________ _ 𝑥→2 𝑥→2 c) 𝑥→2 d) 𝑓(2) = ____________ _ 9) For the piecewise linear function 𝑓 (𝑥) = { 8−𝑥 14 − 2𝑥 𝑖𝑓 𝑥 ≤ 2 𝑖𝑓 𝑥 > 2 a) Sketch the graph b) Find each of the following. (If an answer does not exist, state “DNE.”) lim 𝑓(𝑥) = ____________ _ lim 𝑓(𝑥) = ____________ _ 𝑥→2− 𝑥→2 lim 𝑓(𝑥) = ____________ _ 𝑓(2) = ____________ _ 𝑥→2+ 10) Complete the table and use it to find the given limits. Round calculations to four decimal places if necessary. x9 − 1 x−1 x 0.9 0.99 0.999 1.1 1.01 1.001 lim 𝑓(𝑥) = _________________ 𝑥→1− lim 𝑓(𝑥) = ____________ _ lim 𝑓(𝑥) = ____________ _ 𝑥→1+ 𝑥→1 11) Find the limit by constructing a table including at least three points on either side of the limit or by examining the graph (include a rough sketch). lim (1 + 5𝑥 )1/𝑥 𝑥→0+ 12) Find the limit algebraically (without a graph or table): a) lim 𝑥→−3 2𝑥3 −6𝑥2−36𝑥 𝑥2 +3𝑥 b) lim 𝑥+4 𝑥→16 √𝑥 c) lim 𝑥→5 2𝑥2 −50 𝑥−5 13) Determine each of the following by using a calculator to generate a graph, make a rough sketch of the graph to support each answer: a) lim 6𝑥+3 𝑥→∞ 2𝑥 b) lim −𝑥 𝑥→−∞ 2𝑥−8 c) lim 1 𝑥→3+ −𝑥−3 d) lim− 𝑥2 2 𝑥→1 𝑥 −1 14) Find the derivatives, using formulas (“shortcuts”): a) 𝑓(𝑥) = 𝑥280 b) 𝑓(𝑥) = 𝑥6 1 3 c) 𝑓 (𝑥 ) = 6 √𝑥 d) 𝑓(𝑥) = 1 𝑥2 e) 𝑓(𝑥) = 8𝑥2 − 6𝑥 + 2 15) For the function 𝑓(𝑥) = 𝑥2− 2𝑥 + 2 a) Find the derivative 𝑓′(𝑥) [use formula(s) (“shortcuts”)] b) Find the equation of the tangent line to the graph of 𝑓(𝑥) at the point where 𝑥 = 5 c) Sketch the graph of the function 𝑓(𝑥) and the tangent line from part b) 16) Businesses can buy multiple licenses for a data compression software at a total cost of approximately 𝐶 (𝑥) = 87𝑥2/3 dollars for x licenses. a) Find the marginal cost function 𝑀𝐶 (𝑥) b) Evaluate 𝑀𝐶 (8) (round to the nearest hundredth if necessary) c) Circle the correct interpretation of the answer to part b): i) Eight licenses cost this amount ii) Each license costs this amount iii) When eight licenses are purchased, the additional cost to purchase the ninth approximately this amount iv) When eight licenses are purchased, the average cost of each license is this amount 17) It has been estimated that the number of people who will see a newspaper advertisement that has run for x consecutive days is 𝑁(𝑥) = 600000 − 275000𝑥−1. a) Find the average rate of change in 𝑁(𝑥) from 𝑥1 = 2 to 𝑥2 = 6 (the average rate of change in the number of people who have seen the ad over the period of time from week 2 to week 6). b) Find 𝑁′(𝑥) [use derivative “shortcuts”] c) Find the instantaneous rate of change in 𝑁(𝑥) at 𝑥 = 4 (the instantaneous rate of change in the number of people who have seen the ad at week 4) 18) Sal’s Auto Sales, a used car dealership, finds that the number of cars that it sells on day 𝑥 is 1 given by 𝑠(𝑥) = − 𝑥2 + 10𝑥 . Find 𝑠′(𝑥) and find the instantaneous rate of change in the 3 number of people who have seen the ad on day 𝑥 = 4 For questions 19, 21, and 22, use the appropriate rule (product, quotient, or generalized power rule): 19) Find/simplify the derivative: 𝐹(𝑥) = (4𝑥3 − 2)(𝑥4 − 2𝑥 + 1) 20) Find/simplify the derivative: 3𝑥2 𝐹(𝑥) = 3−𝑥 21) Find/simplify the derivative: 𝐹(𝑥) = √5𝑥 − 1
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Explanation & Answer

Attached.

MATH 153

NAME___________________ _ __ _ __

Show all work for credit. Simplify/reduce. State exact forms of answers unless rounding is specified.
𝑓(𝑥+ℎ)−𝑓(𝑥)
1) For each function, find and fully simplify the difference quotient
ℎ
a) (𝑥) =

F(x +h) =

5
𝑥

5
𝑥

( +ℎ)−5/𝑥
ℎ

5𝑥−5𝑥−5ℎ

= 1/h( (𝑥+ℎ)∗(𝑥)

5

F(x)1 =- (𝑥+ℎ)(𝑥)

b) (𝑥) = √𝑥 −3
f(x+h) =

√x+h−3+√x−3
ℎ

= 1/h((

1

x+h−3)−(x−3)
√x+h−3+√x−3)

1

F(x)1 =- (√x−3+√x−3)) = (2√x−3))

2) For the function (𝑥) = 3𝑥2 − 2𝑥

a) Find/fully simplify the difference quotient
F(x)1=

3(𝑥+ℎ)2 −2(𝑥+ℎ)−3𝑥 2 −2𝑥

F(x)1= 6x +3h -2

ℎ

=

ℎ(6𝑥+3ℎ−2
ℎ

b) Calculate the value of the difference quotient for the following pairs of x and h-values:
i)

x = -1 and h = 3
F(x)1= 6x +3h -2
F(x)1= 6(-1) +3(3) -2 = -1

ii)

x = -1

and h = 2

F(x)1= 6x +3h -2
F(x)1= 6(-1) +3(2)-2 = -2

iii)
x = -1 and h = 1
1
F(x) = 6x +3h -2
F(x)1= 6(-1) +3(1)-2 = -5

iv)

x = -1

and h = 0.5

F(x)1= 6x +3h -2
F(x)1= 6(-1) +3(0.5)-2 = -6.5

v)

x = -1 and h = 0.1
F(x)1= 6x +3h -2
F(x)1= 6(-1) +3(0.1)-2 = -7.7

3) Sketch the graph of the function 𝑓(𝑥) = 3𝑥2 − 2𝑥 and the secant lines from parts i) and iii) above
(only those two secant lines)
(-1,-1) and (-1,-5)

4) For the function 𝑓(𝑥) = 8𝑥2 + 4𝑥 − 1, find the average rate of change between
𝑥1 = −3 and 𝑥2 = −1
X1 = -3
h=2
x +h = -1
𝑓(−1)−𝑓(−3) 3−57
f(x)1 =
= 2 = -27
2

5) For the function (𝑥) = 𝑥2 + 3𝑥 − 2:
𝑓(𝑥+ℎ)−𝑓(𝑥)
a) Find/fully simplify the difference quotient
ℎ

f(x)1 =

f(x)1 =

f(x)1 =

(𝑥+ℎ)2 +3(�...