Chapter 3 Derivatives

3.1 Derivative of a Function

Differentiate using the alternate definition.

Relationships between the Graphs of f and f’ Because we can think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f. We estimate the slope of the graph of f in y-units per x-unit at frequent intervals. We then plot the estimates in a coordinate plane with the horizontal axis in x-units and the vertical axis in slope units. Look at page 103

p.105 (1-19)odd, (26-28)

3.2 Differentiability How f’(a) Might Fail to Exist

A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability. What is linearization? Let’s discuss calculator derivatives.

Differentiability Implies Continuity The converse is not a true statement. Intermediate Value Theorem for Derivatives

p. 114 (1-37) odd

3.3 Rules for Differentiation

Find an equation for the line tangent to the curve at the point (1,2). Find the first four derivatives of

p.124 (1-47) odd

3.4 Velocity and Rates of Change

Motion Along a Line

Look at page 129 on how to read a velocity graph.

When a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x. The derivative f’(x) is a measure of this sensitivity.

p. 135 (1 -39) odd

3.5 Derivatives of Trigonometric Functions Let’s prove this using the definition of the derivative.

Find the derivative of

The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion.

Let’s derive the formula for tangent.

Find yn if

p. 146 (1-41) odd

3.6 Chain Rule

Power Chain Rule

Ex: (a) Find the slope of the line tangent to the curve at the point where (b) Show that the slope of every line tangent to the curve is positive.

p. 153 (1-39, 53-69)odd skip (57), also do (56, 58)

3.7 Implicit Differentiation

Implicit Differentiation Process

p.162 (1-57) odd and 54

3.8 Derivatives of Inverse Trigonometric Functions Derivatives of Inverse Functions Let’s go the exploration of page 166

How did we get this as a result?

Identity Functions Rules for differentiation

Ex: Ex: A particle moves along the x-axis so that its position at any time Is . What is the velocity of the particle when t = 16?

p. 170 (1-33) odd

3.9 Derivative of Exponential and Logarithmic Functions

Ex: At what point on the graph of the function y = 3t – 3 does the tangent line have a slope of 21? How can we show this?

Sometimes the properties of logarithms can be used to simplify the differentiation process, even if logarithms themselves must be introduced as a step in the process. The process of introducing logarithms before differentiating is called logarithmic differentiation.

p.178 (1-55)odd