Question

1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove:

• f is differentiable at -2
• f is differentiable at 1

2)  Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c).

3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule.

4)  For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n <0 and x = 0. Hint: Use the product rule.

5)Suppose f : I→Ris bounded and g: I→Ris differentiable at c∈I and g(c) =g0(c) =0. Show that h(x) := f(x)g(x) is differentiable at c. Hint: You cannot apply the product rule.

6)Suppose f : I →R, g: I →R, and h: I →R, are functions. Suppose c∈I is such that f(c)=g(c)=h(c),gandharedifferentiableatc,andg0(c)=h0(c). Furthermoresupposeh(x)≤ f(x)≤g(x) for all x∈I. Prove f is differentiable at c and f0(c) = g0(c) = h0(c).

Answer 1

A. f:RR be detinad f)-2 Now i im K+ 2 lim X+21-0 t 2 fim -(R+2) (K+ 2) t -2,N+-2) and im font-fl-2) R+2 im X+21 (K+2) (n+2) (142) im f()-f(-) does net exib X42 Henu f(n) is net ditfenentiable at -2 im f(n)-fu) im 17+21-5 K1 tim (K+2)-3 C lim 157 n44 m fin-fu) enistand turedore fin) is diffeven tinble at -