Calculus for Business

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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