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

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

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

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

#### 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)