Math 5 Problem Set (Elec-Glu-Grav-Pho)Due: Sept 7, 2011, Wednesday, Class time. Write the complete solution to the following problems. Box your final answers, if applicable. Treat this problem set as a review for the upcoming exam. HPS 70pts

1. Evaluate the following limits (5pts each):

a . limx→ 0

sin34 x8 x3

b . limx→0

1−cos2πxx2

c . limx→−π

tan xπ+x

d . limx→0

x3 cos4

3√x3−x2

2. Discuss the discontinuities (on what point(s) are they continuous, and what is the type of discontinuity if they are not continuous at a point) of the following functions (5 pts each):

a . g ( x )={ 1 , x∈Z0 , otherwise b .h ( x ){ √ x+2, x←2

1|x|,−2≤ x≤2

−x2+ x+4 , x>2

3. On what union of intervals is the given function continuous? (10 pts)

f ( x )={ 1x+1

, x<0

⟦x ⟧ ,0≤ x≤3√ x2−16 , x>3

4. Find the equations of the tangent line and normal line to the function h ( x )=x3−x2+x−2 at x=1. (10 pts)

5. Use the definition of the derivative to get the derivative of the following functions (5 pts each)

a . f ( x )= x+1x−1

b . f ( x )=tan x

For the next item, choose one of the three (10pts) [Although it might be good to know the solution to all three questions]

6. What are the equations of the tangent lines to the function f ( x )=x2, which pass through the point (0 ,−2 )? (10 pts)

7. Suppose f ( x ) is a continuous function on [0,2], with f (0 )=f (2 ). Use IVT (or IZT) to show that for some c∈ (0,1 ), f ( c )=f (c+1 ). HINT: consider the function g ( x )=f ( x+1 )−f ( x ) (10 pts)

8. Suppose |f ( x )|≤M , ∀ x∈R, where M is a positive constant. Furthermore, suppose that limx→a

|g ( x )|=0. Use

the squeeze theorem to find the value of

limx→a

f ( x )g ( x )