Elementary Functions and Calculus I Math 131 (Sec 42), Autumn 2004 Practice Mid-term 2

1. Recall the definitions of the following.

(a) That a function f : R → R has a limit l at a point x.

(b) That a function f : R → R be continuous at a point x.

(c) That a function f : R → R be differentiable at a point x.

2. Recall the following results (without proving them). State clearly any assumtions that are required for each result to hold.

(a) The intermediate value theorem.

(b) The product rule.

(c) The quotient rule.

(d) The chain rule.

(e) The power rule.

3. Apply the intermediate value theorem to show that p(x) = x^{7}+10x^{3}+2 has at least one root.

4. Use any of the definitions or theorems you have quoted above to find the derivative f′ of f when defined by the following formulae. State clearly which results you are using, and prove any steps which are not direct applications of the theorems.

(a) f(x) = 10x^{25} + 50x^{2} − 4x + 20x^{3}

(b) f(x) = (x^{3} + 40x)^{308}

(c) f(x) = \frac{x^{9}+50}{x^{2}+4}

5. Given that y : R → R is differentiable and satisfies x^{2}y(x) = 1+y^{2}(x)x find y′(c) in terms of c and y(c).

6. Let g : R → R be defined by g(x) = x(x+1) if x ≤ 0 and g(x) = ax+b if x > 0 for some given a, b ∈ R. Find g′(y) for any y ≠ 0. For what values of a and b is g continuous at zero? For what values of a and b is g differentiable at zero? What must g′(0) be if it exists? Explain your answer.