Learning Outcomes

  1. Recognize the meaning of the tangent to a curve at a point.
  2. Calculate the slope of a tangent line using definition, and find the equation of tangent line.
  3. Calculate the derivative of a given function at a point using derivative definition.

Textbook

  • Chapter 3.1  Defining the Derivative  

Textbook Assignment

  • p. 228:   1, 3, 11-17 odd, 21-25 odd

WeBWorK Assginment

  • Derivatives-Limit Definition

Exit problems of the session

  1. Let f(x)=2-3x^2. Find the slope of the tangent line at x=-2 and find the equation of the tangent line at x=-2.
  2. Use the definition of derivative at a point to find f'(4) where f(x)=\sqrt{x}. 

 

 Key Concepts

  • The slope of the tangent line to a curve measures the instantaneous rate of change of a curve.
  • The equation of tangent line at the point (a, f(a))  with slope m is: y-y(a)=m(x-a). 
  • We can calculate the slope of the tangent line  by finding the limit of the difference quotient with increment h

\displaystyle m_{tan}=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}

  • The derivative of a function f (x) at a value a is found using the definition for the slope of the tangent line

\displaystyle f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}

 

Videos and Practice Problems of Selected Topics

  1. \rhd Derivative as a concept (7:16) Introduction to the idea of a derivative as instantaneous rate of change or the slope of the tangent line.
  2. \rhd Secant lines and average change of rate  (5:16) Consider y=x^2 and find the average rate of change over [1,3].
  3. * Practice: Derivative notation review. (one problem with a guiding text)
  4. \rhd Derivative as slope of a curve (6:09) Estimate the derivative f'(5) by analyzing the graph of f. Then compare the values g'(4) and g'(6) by looking at the graph of g.
  5. * Practice: Derivative as slope of curve. (4 problems)
  6. \rhd The derivative and tangent line equations (7:32) The derivative of a function gives us the slope of the line tangent to the function at any point on the graph. This can be used to find the equation of that tangent line.
  7. * Practice: The derivative and tangent line equations. (4 problems)

STEM Application

Limits and Continuity: Writing, Graphing And Analyzing Piecewise Functions From A Real Situation of yearly Income Tax