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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