# 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) = h

^{2}+ 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 = x ^{2} 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 = x ^{2} 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) = x^{5} + 4x^{3} + 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) x^{2}(x^{3} + 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/s^{2} or 32ft/s^{2}.

- The acceleration of an object falling due to gravity is 32ft^{2}/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) = x^{2}e^{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 =

x^{3} + ax^{2} + bx + 1.

- Let f(x) = 3x^{5} − 20x^{3}. (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).