Math 20A Final Review Outline

Chapter 3: Differentiation

Section 3.1: Definition of the Derivative

• Know what is meant by the difference quotient , it is the the slope of the
secant line through the points (a, f(a)) and (a + h, f(a + h))

• Know the definition of the derivative at a point x = a and how it relates to the difference
quotient. The formula is

- If f(x) = (x + 1)² − 2, find f'(0) using the definition of the derivative.
- Let . Find f'(3) using only the definition of the derivative.

- Suppose f(x) is a function such that f(3) = 1 and f(3 + h) − f(3) = h² + he^h.

(i) Find f'(3) (ii) Find an equation for the tangent line to the graph of y = f(x) at x = 3.

- The line y = 3x + 5 is tangent to the graph of f at the point (1,8). (i) Find f(1) and (ii)
Find f'(1).

• Know that the derivative at a point is the slope of the tangent line to the curve at that point. - The slope, m, of the tangent line of y = x2 at x = 1 is 2. Visually, we have the picture to the right. • Know how to find the tangent line to a graph at a point. - Find the equation of the tangent line of y = x2 at (1,1). - If f is differentiable on (−∞,∞), f(0) = 5, and f'(x) ≥ 3, how small can f(4) possibly be?

• Know how to use the difference quotient to estimate the derivative at a point (use small values
of h)

• Know how to order various slopes - In the figure to the right, list the following quantities in increas- ing order: .

Section 3.2: The Derivative as a Function
• Know the definition of the derivative (as a function). That is,

• Know how to recognize some limits are just being derivatives in disguise
- Notice that is just the derivative of 2^x at x = 0.

• Know the different types of notation used for the derivative of y = f(x): (i) y', (ii) y'(x), (iii)


• Know the different types of notation used for the derivative of y = f(x) at a point x = a: (i)


• Know what the derivative tells us graphically:

If f'(x) > 0 on an interval, then f(x) is increasing over that interval.
  If f'(x) < 0 on an interval, then f(x) is decreasing over that interval.

• Note: Just because f(x) is increasing does not mean that f'(x) is increasing. It only means
that f'(x) > 0. It is easy to confuse the two.
• Know the power rule for taking derivatives: , where a is a constant and n
is any number.

- Find the derivative of f(x) = x⁵ + 4x³ + 27.

• As a special case of the above, if f(x) = c (a constant), then f'(x) = 0 for all x.

• Another special case of the power rule is .

• Note: The power rule only applies to the case where the variable (usually x) is raised to a
power. It is not true that the derivative of y = 3^x is y'(x) = x3^x-1!

• Know that f(x) may have a local max/min wherever f'(x) = 0.

• If you have a constant times a function that you are taking the derivative of, you can factor
out the constant.

• If you are taking the derivative of the sum/difference of two functions, you can take the
derivative of each piece

• Know that the derivative of y = b^x is proportional to b^x. (In fact, we see in Section 3.10 that


• Know that the derivative of y = e^x is e^x.

• Know the relationship between derivatives and continuity.

• If a function is differentiable, then it is continuous. The converse is not necessarily true!

- The function y = lxl is continuous, but not differentiable at x = 0. (Using the definition
of the derivative, we see that f'(0) does not exist since the limit from the left does not
equal the limit from the right.)

• So, that means that if f(x) is not continuous at x = a, then f(x) is not differentiable at x = a.

• f(x) is not differentiable at any sharp point, i.e. where we can t more than one tangent line

• f(x) is not differentiable wherever the function has a vertical slope

Section 3.3: The Product and Quotient Rules

• Know the product and quotient rules and how to apply them to solve problems.

Product Rule: .
Quotient Rule: .

• To remember the Quotient Rule, use the song "low-d-high minus high-d-low over low squared
and away we go"

• Note: The derivative of fg is not equal to f'g' and the derivative of f/g is not equal to
f'/g'
- Find the derivative of (i) x²(x³ + 1) using the product rule (ii) and (iii) .

Section 3.4: Rates of Change

• Know how that the derivative is the instanteous rate of change (often just mentioned as
ROC).

• Know that the rate dy/dx is measured in "units of y per unit of x".

• For motion, velocity v(t) is the ROC of position s(t) with respect to time. That is, s'(t) = v(t).
Acceleration a(t) is the ROC of velocity v(t) with respect to time. That is, v'(t) = a(t).

• , so can be a good estimate of the change in f due to a
one-unit change in x, when .

- Marginal cost is the cost of producing one additional unit. If C(x) is the cost of producing
x units, then the marginal cost at production level is . The derivative
C'() is then a good estimate for the marginal cost.

• Know Galileo's formulas for the height and velocity at time t of an object rising and valling
under the influence of gravity near the earth's surface

- , where = initial position, = initial velocity, and
g is the constant 9.8m/s² or 32ft/s².

- The acceleration of an object falling due to gravity is 32ft²/sec on the surface of the
Earth. An object is thrown upward from the surface of the Earth at an initial speed of
64 ft/sec. How high does the object go, and how long does it take to reach that height?

Section 3.5: Higher Derivatives

• Know what the second derivative tells us about f(x) and f'(x).

If f''(x) > 0, then f'(x) is increasing and f(x) is concave up.

If f''(x) < 0, then f'(x) is decreasing and f(x) is concave down.

• The following table should help to keep the relationships straight

• Know that f(x) may change concavity wherever f''(x) = 0.

• Note: f''(x) is the derivative of f'(x). So, the same properties that we discussed for the rst
derivative apply to these two functions.

• Given the graph of f'(x), know where f(x) is increasing/decreasing, concave up/down
- Let . Compute f''(x). On what interval(s) is f(x) concave down?

- Let f(x) = x²e^x. Where is f(x) increasing? Where is f(x) concave down?

- Find the values of a and b such that (1, 6) is a point of inflection for the curve y =
x³ + ax² + bx + 1.

- Let f(x) = 3x⁵ − 20x³. (i) Find all x-intercepts and y-intercepts of the graph of f. (ii)
Determine the interval(s) where f is increasing and the interval(s) where f is decreasing.
(iii) Find all local extrema of f, and determine which are local maxima and which are
local minima. (iv) Determine the interval(s) where the graph of f is concave up and the
interval(s) where the graph of f is concave down. (v) Find the point(s) of inflection of
the graph of f.

• Know that higher derivatives are defined by successive differentiation. That is, the third
derivative of f(x) is found by taking the derivative of f''(x).

• Returning to the equations of motion from the previous section, we have that s'(t) = v(t) and
since v'(t) = a(t), we have that s''(t) = a(t).