Embedded Files
Derivatives
Chapter Overview
Chapter Overview
Connections
Connections
Shown here is an airplane lifting off after moving down the runway. The airplane speed at lift-off can be determined from the equation D = (10/9)t^2, where D is the distance traveled in meters down the runway before lift-off at time t in seconds after the plane’s brakes are released. If the airplane becomes airborne when its speed down the runway reaches 200 km/hr, how long will it take it to be airborne and what distance will it have traveled by that time? This problem can be solved with the information covered in Section 3.4.
Chapter 3 Overview
Chapter 3 Overview
In Chapter 2, we learned how to find the slope of a tangent to a curve as the limit of the slopes of secant lines. In Example 4 of Section 2.4, we derived a formula for the slope of the tangent at an arbitrary point (a, 1/a) on the graph of the function ƒ(x) = 1/x and showed that it was -1/a^2.
This seemingly unimportant result is more powerful than it might appear at first glance, as it gives us a simple way to calculate the instantaneous rate of change of ƒ at any point.
The study of rates of change of functions is called differential calculus, and the formula -1/a^2 was our first look at a derivative. The derivative was the 17th-century breakthrough that enabled mathematicians to unlock the secrets of planetary motion and gravitational attraction—of objects changing position over time. We will learn many uses for derivatives in Chapter 5, but first, in the next two chapters, we will focus on what derivatives are and how they work.
Section 3.1: Derivative of a Function
Section 3.1: Derivative of a Function
Lesson Objectives
Determine the derivative of a function using the definition of a derivative.
Calculate the derivative of a function at a point.
Sketch a graph of the derivative of a function when given its graph.
Determine if a function is differentiable on a closed interval.
Check out the song Differentiabul!
Lyrics:
f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
To otherwise state: instantaneous rate.
f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
To find the slope at one point.
Infinitesimals dy over dx,
Why he wrote it I can’t say,
Leibniz just liked it better that way!
So,
f of x plus h minus f of x
all over h as h drops to zero
is the formula to find the derivative.
With this I will have to learn to cope!
Leibniz found the limit of the slope!
Complete the following exercises, beginning on page 107:
1–11 (odd problems only), 13–16 (all problems), 22, 26
03 - Definition of the Derivative.pdfDefinition of Derivative
Section 3.2: Differentiability
Section 3.2: Differentiability
Lesson Objectives
Estimate derivatives using graphs and numerical approximation.
Identify different types of non-differentiable points, including discontinuities, vertical tangents, corners, and cusps.
Section 3.3: Rules for Differentiation
Section 3.3: Rules for Differentiation
Lesson Objectives
Use the power rule to find derivatives.
Use the product rule to find derivatives.
Use the quotient rule to find derivatives.
Calculate second derivatives and higher-order derivatives using rules of differentiation.
Calculate instantaneous rate of change using the derivative.
03 - Power, Constant, and Sum Rules.pdf03 - Product Rule.pdf03 - Quotient Rule.pdf03 - Higher Order Derivatives.pdf
Section 3.4: Velocity and Other Rates of Change
Section 3.4: Velocity and Other Rates of Change
Lesson Objectives
Use derivatives to solve problems involving motion in a straight line.
Solve real-world problems involving rates of change using derivatives.
Section 3.5: Derivatives of Trigonometric Functions
Section 3.5: Derivatives of Trigonometric Functions
Lesson Objectives
Determine derivatives of trigonometric functions.
Unit 3 Test
Unit 3 Test
Page updated
Report abuse