Download
1 / 13
130 likes | 280 Views
The Derivative. Objectives: Discuss slope and tangent lines. Be able to define a derivative. Be able to find the derivative of various functions. Critical Vocabulary: Slope, Tangent Line, Derivative. I. Slopes of Graphs. What are the slopes of the following linear functions?.
E N D
The Derivative • Objectives: • Discuss slope and tangent lines. • Be able to define a derivative. • Be able to find the derivative of various functions. Critical Vocabulary: Slope, Tangent Line, Derivative
I. Slopes of Graphs What are the slopes of the following linear functions? How have you defined slope in the past?
I. Slopes of Graphs What about other functions? ________________________________________________________________________________
I. Slopes of Graphs To find the rate of change (slope) at a single point on the function, we can find a _______________ at that point Recall Circles: A line was _______ to a circle if it intersected the circle ______.
I. Slopes of Graphs To find the rate of change (slope) at a single point on the function, we can find a tangent line at that point Even though the tangent line is ________ the graph someplace else, we are only describing the slope at the _____________. Is the slope the same at each of these arbitrary points? With Curves (functions), it is a little different. We can be concerned with the line of tangency at a specific point, even if the line would __________ the function someplace else.
II. Defining a Derivative We will be using the idea of limits to help us define a tangent line. Important things to know: 1. __________________________________ 2. __________________________________ Let’s look at the slope formula: Define our points: A: ______________ B: ______________ Find the slope between points A and B This is the _______________
II. Defining a Derivative The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by: III. Finding the Derivative Example 1: Find the derivative of _______________ using the definition of the derivative. This is the general rule to find the slope at any given point on the graph.
III. Finding the Derivative The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by: Example 2: Find the derivative of _______________ using the definition of the derivative. What is the slope of the tangent line at the point (-1, 2)? (2, 5)? This is the general rule to find the slope at any given point on the graph.
III. Finding the Derivative Big Ideas • 2x was the _________ of f(x) = x2 + 1. This means it is the __________ for finding the slope of the tangent line to any point (x, f(x)) on the graph of f. • We write this by saying f’(x) = 2x. • We say this _______________________ 3. The process of finding derivatives is called ________________ Notations:
III. Finding the Derivative The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by: Example 3: Find the derivative of _______________ using the definition of the derivative. This is the general rule to find the slope at any given point on the graph.
III. Finding the Derivative The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by: Example 4: Find the slope of _________________ at (2, 1) This is the general rule to find the slope at any given point on the graph.
III. Finding the Derivative The slope of the tangent line at any given point on a function is called a derivative of the function and is defined by: Example 5: Find the equation of the tangent line to the graph of _____________ at the point (-3, 4).
