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Math 148.05A - Review for Final - Chapter 11: Differentiation
(1) 11.1 Idea of derivative, definition.
(a) Let f (x) = x2 + 2x + 3. Write a formula for the slope of the secant line from x = 2 to
x = 2 + h. (Your answer should depend on h.)
(b) Given your answer to the previous problem, write an expression (involving a limit)
for the slope of the tangent line to f (x) at x = 2. Evaluate your expression to get a
number.
(c) Suppose a function g(x) is differentiable on the interval (10, 100). Is g(x) continuous
on the interval (10, 100)?
(d) Suppose a function h(x) is continuous on the interval (−1, 1). Is h(x) differentiable on
the interval (−1, 1)?
(e) What does a function that is differentiable look like, in terms of its graph?
(f) What does a non-differentiable function look like, in terms of its graph? Give examples.
(2) 11.2 Derivative rules.
(a) Find the derivative of x3 using the definition of the derivative and using the power
rule. Check that they agree.
(b) If f 0 (2) = 4 and g 0 (2) = −1, what is (4f − 2g)0 (2)?
(c) Find the equation of the tangent line to y = x3 − x + 1 at the point (2, 7).
(d) Find the derivative of f (x) = (x2 − 2)(x − 1 − x1 .
(3) 11.3 Derivative as rate of change.
(a) If p(t) gives you the position of an object at time t, what does p0 (t) represent?
(b) If marginal revenue for q = 10 is −5, estimate how much revenue changes if quantity
changes by 0.75.
(c) Given a function f (x), describe the relationship between the average rate of change of
f over the interval (1, 1 + h) and the instantaneous rate of change of f at x = 1.
√
(d) A manufacturer models their costs using the function c(q) = 41 q 2 − 2q + q. What is
their average cost when q = 5? What is their marginal cost when q = 5?
(4) 11.4 Product rule and quotient rule.
(a) f (4) = 2, f 0 (4) = 5, g(4) = −1 and g 0 (4) = 21 . What are (f g)0 (4) and (f /g)0 (4)?
(b) Find the derivative of (x3 − 2x2 + 3x − 4)ex .
(c) If demand for your product is q, and the price of your product is p(q), then your
revenue is r(q) = qp(q). Find a formula for marginal revenue. (Your answer should
involve p.)
√
√
3x
(d) Find the derivative of xx+
2 +1 .
(5) 11.5 Chain rule.
(a) Write the chain rule in two different ways.
(b) If demand for your product at a time t is given by the function q(t), and the price for
your product is given by p(q), find a formula for revenue as a function of time r(t).
Find a formula for marginal revenue
as a function of time.
√
(c) Find the derivative of f (x) = x−1 + x ln |x−2 + 2|.
(d) Suppose selling q units brings in r(q) = q ln(q), and m employees can produce q(m) =
m + 10
m units. If you have 20 employees, estimate how much revenue changes if you
dr
hire one more employee. (Recall that dm
is called the marginal revenue product.)