MATH 108: Mathematics for Business Calculus II Final Exam Spring 2023
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MATH 108: Mathematics for Business Calculus II
Final Exam
Spring 2023
1. (10 points) For this question, let u(x), v(x) be differentiable functions such that f (x) = Find the derivative of f (x) without using the Quotient Rule for derivatives.
2. (10 points) Recall that if b is any positive real number ( b > 0 ) then b = eln b . Use this fact to find the derivative for f (x) = bx .
3. (10 points) A computer parts manufacturer sells processors. The owner of the company ask her assistant, Adriana, to get the average revenue for the sale of 2, 000 units and leave it on her desk before the next meeting this afternoon. Adriana knows that the revenue function is R(x), so she computes R\ (2, 000), writes it on a piece of paper, and leaves the note for her boss. Is Adriana’s calculation correct? Explain why or why not. If Adriana’s calculation was incorrect, what should she have computed?
4. (10 points) An artisan guitar maker sells custom guitars. He knows his total cost function C(x), where x is the number of guitars. To find the marginal cost when making 60 guitars , he computes C(61) - C(60). Is the guitar maker correct? Explain why or why not. If the guitar maker was incorrect, what should have he computed instead?
5. (10 points) Using the formal definition of a derivative, compute the derivative for f (x) = (x - 2)3 .
6. (10 points) Find an equation of the tangent line to the curve y = x3 + 2x + 5 at the point (2, 17).
7. (10 points) Find the second derivative, h\\ (t), for h(t) = ln ╱ 、. [.i.e. First find h\ (t) and then take the derivative of h\ (t).]
8. (10 points) A company has a profit function (in dollars)
P (x) = 1200x + 45x2 - x3 - 28, 000
where x is the number of items sold. (Include the correct units with your answers.)
(a) (6 points) Using calculus, how many items should be sold to maximize profit? (You must show all your work for credit.)
(b) (4 points) What is the maximum profit?
9. The monthly marginal profit function for a certain product is given by
MP = 3x2 + 4x + 3.
10. (10 points) Evaluate the indefinite integral (15x4 - 8x + 27e) dx.It is also known that if no items are produced the company incurs a monthly loss of $10 , 000. (i.e. P (0) = -10, 000 ) Find the monthly profit function P (x).
x2 + 4
11. (10 points) Find the derivative of g(x) =
12. (10 points) Find the instantaneous rate of change of the function y = ^x3 + 5 at the point (2, 13).
13. (10 points) Using Calculus, find the intervals in which the function p(n) = (n + 5)(n + 2)(n - 3)(n - 7) is increasing. [Remember, you must show all your work to get credit.]
14. (10 points) Sketch the graph of a function that satisfies the given conditions
● f (0) = 0
● f\ (-5) = f\ (3) = f\ (7) = 0
● f\ (x) > 0 on (-,, -15), (-10, -5), (3, 7).
● f\ (x) < 0 on (-15, -10), (-5, 3), (7, ,).
● lim = 0
x→o
● lim = , x→ _15
● Label the axes. Indicate clearly where each condition is met. Your graph does not need to be to scale.
2023-05-24