Ma 19: Sample Long Exam 1 (2nd Semester, SY 2008-2009)

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SAMPLE LONG EXAM NO. 1. MA 19 (Applied Calculus for Busines)
(R. Saldaña)

Directions: Show complete solutions.

1. (15 pts.) Find the limit:
a.) lim f(x) if f(x) = | 3x - 2 |
x →-1⁺

b.) lim(((x²+7x+10)(x-1))/(x-2)),as x →2

c.) lim (((x+3)²+1)/(2(x-3)²-1)),as x →3

2. (15 pts.)

a. Find the derivative of f(x) using the limit definition of the derivative: f(x)= (5/(√(x²+5)))

b. Verify your answer in (2a) using a differentiation formula.

3. (15 pts.) Evaluate the derivative of f(x) at x = 1.

f(x) =((x³+5x-6)/((1+x³)²))

4. (15 pts.) Find the derivative of f(x).
f(x)=(([[3]√(x+3√x))]⁴)/(((2/(x³))+5)))

5. (15 pts.)Find the equation of the tangent line to the curve
y = 4x³ - 2x + 5, at x =1.

6. (25 pts.) The price-demand equation and the cost function for the production of a certain brand of computer are given, respectively, by:
x = 9,000 - 30p and C(x) = 150,000 + 30x

where x is the number of computers that can be sold at a price of p (in dollars) per computer, and C(x) is the total cost (in dollars) of producing x computers.

(a). Express the price p as a function of the demand x.

(b). Find the revenue function.

(c). Find the marginal revenue function.

(d). Find R'(3,000) and R'(6,000), and interpret the results.

(e). Find the marginal cost.

(f). Find the break even points.

(g). Graph the cost function and the revenue function on the same coordinate system for 0≼x≼9,000. Indicate the break-even points, and the regions of loss and profit.