Posted: December 3rd, 2015

Calculus for Business

1. Find the following limits: a) 4

6 15 limx 4

x

o x

b)

2

4

2 8 limx 4

x x

o x

2. Determine which statement best describes the continuity of the function:

2 3 2

() 2 5 2 1

7 1

x if x

f x x if x

x if x

° d ®

° ! ¯

a) f(x) is Continuous for All Real values of x

b) f(x) is Continuous for All Reals except x = -2

c) f(x) is Continuous for All Reals except x = 1

d) f(x) is Continuous for All Reals except x = -2 and x = 1

3. Find the derivative of: 2 4 fx x x x () 7 4 5

4. Find the derivative of: 3

2 4

y

x x

5. Find the equation of the line tangent to 3 2 1 2 2

2 y x x at x

6. Use Marginal analysis to predict the estimated profits from the 221st unit sold

if Cx x ( ) 1200 5 and

2 Rx x x ( ) .04 32 200 .

7. Use Implicit Differentiation to find dy/dx for: 3 24 y x xy 5 63

8. Find derivatives of the following:

a) 2

3 ( ) 2

x h x

x

b)

6 3 y x 2 5 c) 4 Fx x x ( ) ln 8 2 d) 4 ( ) x hx e

9. Find the derivative of: 2 3 ( ) x hx xe

10. Find the Absolute Maximum and Absolute Minimum of 3 2 f x x x on ( ) 4 3 [ 1,5]

11. Find the derivative of: 2 3 hx x x ( ) ln 4 2

12. How long will it take an investment of $45000 to become $100000 if it earns

4.65% compounded

continuously?

13. How much does it take invested today to grow to $30000 at 4.8% compounded

continuously in 8 years?

14. Find the Elasticity function for the Demand function, Dx x ( ) 800 4 , and

then determine the elasticity

at x = 75. ( ) ( ) ( )

xDx E x

D x

c

15. Take first and second derivatives of 2 4 fx x x ( ) 54 , and then build

a sign chart to determine Intervals

of Concavity

16. Find the specific function f(x) satisfying the following conditions: 2 f x x x

and f c( ) 6 2 (4) 5

17. Evaluate the Integral: 3 3

1

4 43 x x dx ³

18. Find the value of the each Definite Integral:

a)

3 1 2 3

2

x x dx 3 ³ b)

1 3 2 1

2

x x e dx

³ c)

10

2 2

2

4

x dx

x ³

19. Use Integration by Parts: 4 ( 2) x x e dx ³

20. Find the Area bounded by 2 y x and y x x 5 6 11

21. Tim invests money in a 403b account according to the function .02 ( ) 15000 t

ft e . If the account earns

continuous gains at 5.5%, and the contributions are made for 25 years, what will

be the value of the

account? Use:

0 ( )

T kT kt FV e R t e dt ³ and round to the nearest $1.

22. An investor is looking at a business opportunity, which will pay him $100000

each of the next 8 years.

Assuming continuous growth at 6%, what is the Present Value of this income stream?

Use:

0 ( )

T kt PV R t e dt ³

23. Find

2

2

4 ( 3,2) ( , )

2

xy f if f x y

x y

24. Find Critical Points and then whether they are a Relative Max, Relative Min, or

Saddle Point.

22 3 f x y x y xy x (, )

25. The function 4 2 f x y x y xy ( , ) 2 12 has derivatives: 3 8 12 , 2 12 x y

f x yf y x . Find the

critical points and determine whether each is a Relative Max, Relative Min, or

Saddle Point.

26. Find the Partial Derivative xy f for 1 f xy x ( , ) ln

y

27. Find ALL 5 first and second order Partial Derivatives of:

a) 45 6 P x y x y xy (, ) 2 4 b) 2 (, ) 6

x y f xy

y x

28. Use the Method of Lagrange Multipliers to find the Maximum and Minimum of f x y

xy (, ) 2

subject to 2 2

x y 18

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