math lab

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MATH 1325 NAME__________________________ CALCULUS FOR BUSINESS AND SOCIAL SCIENCES Lab 2 – Derivatives, Marginal Analysis, and Curve-Sketching Instructions: Please show all work, and write your solutions to the problems neatly on this handout. Draw box around your final answer. If you need help, please feel free to consult with me during office hours or go to the Math Lab and ask for assistance. 1. Find the derivative of each function. Be sure to simplify your answer. a) b) c) d) e) 𝑦 = ln (2𝑥) − 6𝑥) f)

3 6(7 2 )y x x= +

24 8 7y x= +

50.6 xy e=

2 32 5xy += ×

ln(9 3)y x= +

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2. Find the derivative of each function. Be sure to simplify your answer.

a)

b) c)

d)

e)

f)

2

3 4

6 (4 1) x xy x +

= +

3 2xy x e-=

(2 1) ln( 1)y x x= - +

3 8 ln xy x -

=

5 1

xey x

= +

2 3 4log ( 2 )y x x= +

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3. Suppose the demand function for x units of a certain product is , , where p is in

dollars. Suppose further that the cost function in dollars for q units of this product is .

a) Find the marginal revenue function. (Recall that revenue equals quantity times price.) b) Use calculus to approximate the revenue from the sale of 9th unit. c) Find the marginal profit function. d) Use calculus to approximate the profit from one more unit when 8 units are sold. 4. Assume that the total revenue (in dollars) from the sale of x television sets is given by .

a) Find the marginal revenue function, .

b) Find and interpret .

50100 ln

p q

= + 1q >

( ) 100 100C q q= +

( )2/32( ) 24R x x x= +

'( )R x

'(100), '(200), '(300)R R and R

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5. For the function , find

a) the critical numbers;

b) the open intervals where the function is increasing;

c) the open intervals where the function is decreasing. 6. The total profit (in thousands of dollars) from the sale of x units of a certain prescription drug is given by

for x in .

a) Find the total number of units that should be sold in order to maximize the total profit. b) What is the maximum profit?

4 3 2( ) 3 8 18 5f x x x x= + - +

( )P x

3 2( ) ln( 3 72 1)P x x x x= - + + + [0,10]

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7. Given , find the open intervals where the function is concave upward or concave downward. Find any points of inflection.

8. Let . Find the critical number(s) and determine whether they are local

extremum/extrema

9. Let 𝑓(𝑥) = 𝑥) + /0

1 𝑥1 − 18𝑥 − 1

a) Find f '(x) and f '' (x) . b) Construct a first derivative sign chart to determine the intervals where f is increasing or decreasing and find the coordinates of any relative maxima and minima.

3 2( ) 2 9 108 10f x x x x= - - + -

1( ) 9f x x x

= +

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c) Construct a second derivative sign chart to find the intervals where f is concave up or concave down and find the coordinates of any inflection points.

d) Fill out the chart below:

Intervals Sign of f ‘ Sign of f ‘’ f Increasing or Decreasing Concavity of f Shape of Graph on each interval

e) Graph the function using the graph below. Label all important points

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-8 -6 -4 -2 2 4 6 8 10

-5

-4

-3

-2

-1

1

2

3

4

10. Sketch a graph of a single function that has all of the following properties.

Ø Continuous and differentiable everywhere except at x = –3, where it has a vertical asymptote. Ø A horizontal asymptote at y = 1 Ø An x-intercept at x = –2 Ø A y-intercept at y = 4 Ø An inflection point at (4, 3)

Ø Ø Ø f ‘ (2) = 0

Ø Ø

Hint: A graph is allowed to cross over a horizontal asymptote in the middle of the graph, but not at the ends.

( ) ( ) ( )' 0 on , 3 and 3,2> -¥ - -f x ( ) ( )' 0 on 2,< ¥f x

( ) ( ) ( )'' 0 on , 3 and 4,> -¥ - ¥f x ( ) ( )'' 0 on 3,4< -f x