Find the derivative of the following functions: (i) f (x

MIDTERM 1 FOR CALCULUS

Date: 1999, October 28, 4:10–5:00PM

Problem #1 is worth 12 point, #2 and #3 are worth 14 points each, the remainings are worth 10 points each.

1. Compute the following limits:

(i) limh→01−1/h² 1+1/h²

(ii) limx→01−cos 4x 9x²

2. Find the derivative of the following functions:

(i) f (x) = (x²− 1/x²)¹⁰ (ii) y = cos⁴√

x 3.

(i) Find f⁰⁰(0) where f (x) = _1−x¹ . (ii) Find _dx^d£

f¡_x−1

x+1

¢¤in terms of f⁰.

4. Use implicit differentiation to find dy/dx for sin(x + y) = xy.

5. Find the equations for the tangent and normal lines of the curve x = cos y at the point (¹₂,^π₃).

6. Let

f =

( A²x², if x ≤ 2;

(1 − A)x, if x > 2.

For which values of A is f continuous at 2?

7. An object that weights 150 pounds on the surface of the earth will weight 150(1 +₄₀₀₀^r )⁻² pounds when it is r miles above the surface. If the object’s altitude is increasing at the rate of 10 miles per second, how fast is its weight decreasing at the instant it is 400 miles above the surface?

8. Use intermediate value theorem to prove that the equation x³+ ax²+ bx + c = 0 has at least one real root where a, b, c are constant.

9. Let f be twice differentiable on the open interval (a, b) (i.e., f⁰⁰(x) exists for any x ∈ (a, b)). Prove that if f⁰⁰(x) 6= 0 for all x ∈ (a, b), then f has at most two zeros in (a, b).

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