Reference no: EM131082321
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) = x3/2
(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)?
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