Important Terms and Concepts

  • Difference quotient
  • Definition of derivative at a point and as a function
  • Tangent line
  • Local linearity
  • Power function
  • Exponential function
  • Linear combination
  • Differentiation
  • Antiderivative
  • Sinusoid
  • Period
  • Phase displacement
  • Sinusoidal axis
  • Composite function
  • Chain rule
  • Squeeze theorem
  •  Natural exponential function
  • Logarithmic function
  • Symmetric difference quotient
  • Forward difference quotient
  • Backward difference quotient
  • Tolerance
  • Derivative function
  • Displacement
  • Velocity and Speed
  • Acceleration
  • Second derivative
  • Indefinite integral
  • Inside function
  • Outside function
  • Amplitude
  • Vertical displacement
  • Horizontal translation
  • Horizontal dilation
  • Vertical dilation
  • Reciprocal function
  • Implicit diferentiation

Learning Outcomes

In this chapter, students will...
  • Investigate and apply the formal definition of derivative ƒ'(x) and ƒ'(c).
  • Differentiate the power function and sine and cosine functions algebraically.
  • Learn and apply the power rule, and that it  cannot be applied to functions where the variable is in the exponent.
  • Learn and apply the chain rule.
  • Be introduced to the derivatives of natural exponential and logarithmic functions.

Chapter 3 Sections

3-1 Graphical Interpretation of Derivative
Objective: Given the equation of a function, calculate the derivative at a given point by taking the limit of the quantity (change in y)/(change in x)
Homework: Problems 1-8 all

3-2 Difference Quotients and One Definition of Derivative
Objective: Given the equation of a function, and a value of x, use the definition of derivative to calculate algebraically the value of the derivative at that point, and confirm your answer numerically and graphically.
Homework: Q Problems 1-10, 1, 2, 3-11 odd

3-3 Derivative Functions, Numerically and Graphically
Objective: Given the equation of a function, graph the function and its (numerical) derivative function on the same set of axes, and make conjectures about the relationship between the derivative function and the original function.
Aligned Instructional Resources: Interactive Flash "Derivative Grapher" Demonstration
Homework: Q Problems 1-10, 1-13 odd, 14-17 all

3-4 Derivative of the Power Function and Another Definition of Derivative
Objective: Given a power function, ƒ(x)=x^n, where n stands for a constant, or given a linear combination of power functions, find an equation expressing ƒ'(x) in terms of x.
Homework - Day 1: Q Problems 1-10, 1-18 all, 19-23 odd
Homework - Day 2: 25, 26, 29-32 all

3-5 Displacement, Velocity, and Acceleration
Objective: Given an equation for the displacement of a moving object, find an equation for its velocity and an equation for its acceleration, and use the equations to analyze the motion.
Homework - Day 1: Q Problems 1-10, 1, 2, 5, 6 and 7
Homework - Day 2: 9, 11, 12, 15-18 all, 21, 22

3-6 Introduction to Sine, Cosine, and Composite Functions
Objective: Work with your study group to form conjectures on how to differentiate a composite function.
Homework: Problems 1-7 in class

3-7 Derivatives if Composite Functions - The Chain Rule
Objective: Given the equation for a composite function, write the equation for its derivative function.
Homework - Day 1: Q Problems 1-10, 1-10 all 
Homework - Day 2: Problems 11-20 all
Homework - Day 3: Problems 21-30 all

3-8 Proof and Application of Sine and Cosine Derivatives
Objective: Be able to derive algebraically the formulas for the derivatives of sin x and cos x. Find rates of change of sinusoidal functions in real-world problems.
Homework - Day 1: Q Problems 1-10, 1, 3, 7
Homework - Day 2: Problems 8, 13, 14

3-9 Exponential and Logarithmic Functions
Objective: Given an exponential or logarithmic function, find its derivative function algebraically.
Homework - Day 1: Q Problems 1-10, 1, 3, 5-10 all 
Homework - Day 2: Problems 2, 4, 11-16 all
Homework - Day 3: Problems 17-30 all