Calculus I, Fall 2015

Midterm Exam 1

Oct 27, 2015, 10:10AM

1. (X pts) Applications of product rule, quotient rule and Chain Rule on prescribed functions (combination of elementary functions).

2. (X pts)

Similar problems and variant of Find lim_{y→0} \frac{sin y}{y}.

3. (X pts)

Give formal definition of lim_{x→···} f(x) = · · ·. Here ⋯ could be any real number, ±∞ or one-sided versions. Apply it to concrete examples of f and verity using the ε − δ argument.

4. (X pts) More advanced ε = δ arguments. such as the ε/2 arguments, and the proof of the continuity of composite functions

5. (X pts) Understand how to give definition of lim_{x→c} f(x) ≠ L and verify it with a prescribed c, f and L.

6. (X pts)

Find dy/dx where y = f(x)^{g(x)}, x > 0.

7. (X pts)

Find y′ and y′′, ⋯ at (x_{0}, y_{0}) where y(x) is implicitly given by F (x, y) = 0 near (x_{0}, y_{0}).

8. Statement and applications of Mean Value Theorem.

9. (X pts) Find \frac{d^{n}}{dx^{n}} (f(x)g(x))

10. (X pts) Find \frac{d}{dx}(f_{1}(x)f_{2}(x) ⋯ f_{n}(x)), \frac{d}{dx} determinant), etc

11. (X pts) Study the derivation of derivative of inverse functions. Take the inverse trigonometric functions as example. Start with proper choice of domain for the original trigonometric functions and proceed.