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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 + heh.

(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 2x 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 = 3x is y'(x) = x3x-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 = bx is proportional to bx. (In fact, we see in Section 3.10 that

• Know that the derivative of y = ex is ex.

• 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
- 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

• 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

The graph of f(x) Sign of f'(x) Sign of f''(x)
Increasing, concave up Positive Positive
Increasing, concave down Positive Negative
Decreasing, concave up Negative Positive
Decreasing, concave down Negative Negative

• 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) = x2ex. 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).