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.

To My Ma 20 Students: Some Notes on 2nd Long Exam

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Dear Ma 20 Students,

I arrived today (7/23/08) from my conference/meetings in France -- still having jetlag :(


For your 2nd Long Exam on Thursday, July 24, 2008, during our class period, you may focus your final review on the following:

1. Finding the derivative of a rational function using Chain Rule (15 pts.)

2. Problem involving marginal cost, marginal revenue, marginal profit and break even pts. (15 pts).

3. Problem involving continuity of a function (review theorems and properties) (15 pts.)

4. Graphing a rational function using the first and second derivative tests. Includes critical pts., y-intercept, x-intercept, horizontal and vertical asymptotes, concavity, and inflection points. (40 pts.)

5. Word problem involving maxima and minima (use of first and second derivative tests). (15 pts.)Pls. bring at least 8 sheets of clean bond paper and your scientific calculator (graphing calculators are not allowed).

I wish you all the best in your 2nd long exam on Thursday, July 24, 2008.

From,

Dr. Rafael Saldaña
in Quezon City
7/23/08

Ma 20 (Calculus): Some Review Questions

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For my Ma 20 (Calculus for Economists) class, here are some review questions for your forthcoming 1st Long Test scheduled on Thursday, 26 June 2008.

1. Using the graphical method, find the limit of the given function as x approaches zero. Show a complete solution. Given: f(x) = (absolute value of (3x - 2))/(x - 2).

2. Given: f(x) = 100x^5 - (square root of (x^2 + 2)) - pi. Find the limit of f(x) as x approaches 1. Cite the relevant limit theorems and properties.

3. Given: f(x) = [(x^2 + 7x +10)(x-1)/(2(x-2))]. Find the limit of f(x) as x approaches 2. Show a complete solution.

4. Given: f(x) = [(x^2 - 9)/(2(x + 3)]. Find the limit of f(x) as x approaches 1. Show a complete solution.

5. Given: f(x) = [5/(square root of (x^2 + 5))]. Find the derivative of f(x) using the limit definition of the derivative. Show a complete solution.

6. Given: f(x) = x^3 + 5x - 6, g(x) = 2(1 + x^2). Find the instantaneous rate of change of f(x) + g(x) at x = 1. Show a complete solution.

7. Given: f(x) = x^3 - 5, g(x) = x^2 + (2/x^2) - 100000, h(x) = f(x)*g(x). Find the derivative of h(x) using the product rule of differentiation. Show a complete solution. There is no need to simplify your final answer. [Note: the original f(x) = cube root of (x^3 - 5). I removed the cube root function to avoid the use of Chain Rule which has not yet been discussed in class... RPS, 6/23/08]

8. Refer to problem no. 7 for the value of f(x) and g(x). Let r(x) = f(x)/g(x). Find the derivative of h(x) using the quotient rule of differentiation. Show a complete solution. There is no need to simplify your final answer.

9. Find the equation of the tangent line to the curve y = 4x^3 - 2x +5, at x = 1. Show a complete solution.

10. (a) Give an example of a function with undefined limit. (b) Give an example of a function with a limit that is equal to negative infinity as x approaches zero.

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