Derivatives Worksheet 1 - Math 1A, section 103

0. If f(x) = 1, what is f'(2)?

1. Compute the derivatives of the following functions using the limit definition of derivative.

(a) f(x) = 2x + 3

(b) f(x) = x²

(c) f(x) = x³ - x + 1

(d) f(x) = √(1 - x)

(e) f(x) = x^3/2

(f) f(x) = 1/x²

(g) f(x) = x + 1/x

2. Describe two ways in which a derivative can fail to exist.

3. Where is the function f(x) = |x - 6| not differentiable?

4. In problem 1, you computed the derivative of f(x)= x². Check your answer by plotting the graph of f(x) and estimating the slope of the tangent line at x = 1.

5. Discrete derivatives: Suppose, instead of a function, we start with a sequence a₁, a₂, a₃, . . . of numbers. Then the discrete derivative of this sequence is defined to be the sequence

a₂ - a₁, a₃ - a₂, a₄ - a₃, . . .

consisting of the consecutive differences of entries. For example, the discrete derivative of 1, 2, 3, 4, 5, . . . is the sequence 1, 1, 1, 1, . . . .

(a) Find the discrete derivative of the sequence 3, 6, 9, 12, 15, . . . .

In general, what kinds of sequences have discrete derivatives which are constant sequences?

(b) Find the discrete derivative of the sequence 1, 4, 9, 16, 25, . . . consisting of the square numbers n². Can you find a formula for the nth term of this sequence? How does this relate to the derivative of f(x) = x²?

(c) Find the discrete derivative of the sequence whose nth term is n³-n+1. How many times do we have to take the derivative of such a sequence before getting a constant sequence?

(d) Find the discrete derivative of the sequence 1, 2, 4, 8, 16, . . . whose nth term is 2ⁿ. What about 3n?

6. Consider the function f defined by

Is f differentiable at 0? How about x · f(x)? How about x² · f(x)?