Math 251 Winter 2005 Midterm Examination - Prof. C. Phan, Exams of Calculus

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Math 251 Winter 2005 Chris Phan

First midterm examination

Name:

Instructions: Show all work. Be sure to state all theorems, laws, and properties used. Some limits and

derivatives may not exist; if this is the case, give a reason. GOOD LUCK!

1. The graph of a function f is given below.

(a) List all discontinuities of f , and classify them by jump discontinuities, removable discontinuities, and infinite discontinuities.

(b) List all points at which f is not differentiable. Explain why.

2. Let f (x) = x^2 − 2 x − 3.

(a) Find f ′(2), using the definition of the derivative (i.e, taking the limit of the difference quotient).

(b) What is the tangent line of f at x = 2?

3. A rock is dropped from the roof of a tall building. The height of the rock t seconds it is released is given in meters by the function g(t) = − 4. 9 t^2 + 50. What is the (instantaneous) velocity of the rock after 2 seconds? (If you take a derivative, do so by taking the limit of the difference quotient.)
4. Let p(x) = x^5 + 3x^4 + 2x^2 + x − 1. Prove that p has a real root.

5. Find

lim x→ 1 (x − 1)^2 cos

(

π x − 1

)

.

Be sure to prove your answer.

6. Let

Ω(t) =

{

t^4 if t < 1 t^6 if t > 1 , 0 if x = 1. (a) What is limt→ 1 Ω(t)? (Appealing to graphical evidence is insufficient. You must use the limit laws, although you may use direct substitution for continuous functions.)

(b) Is Ω differentiable at 1? Justify your answer.

END OF EXAM.