Learning Outcomes
- Recognize the meaning of the tangent to a curve at a point.
- Calculate the slope of a tangent line using definition, and find the equation of tangent line.
- 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
- 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 $x=-2$.
- 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
- $\rhd$ 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.
- $\rhd$ Secant lines and average change of rate (5:16) Consider $y=x^2$ and find the average rate of change over $[1,3]$.
- * Practice: Derivative notation review. (one problem with a guiding text)
- $\rhd$ 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 $g$.
- * Practice: Derivative as slope of curve. (4 problems)
- $\rhd$ 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.
- * Practice: The derivative and tangent line equations. (4 problems)
