Lesson 2-4: Rates of Change AP Calculus Mrs. Mongold
The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).
The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?
slope slope at The slope of the curve at the point is:
The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.
The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous example, the tangent line could be found using . If you want the normal line, use the negative reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)
If it says “Find the limit” on a test, you must show your work! Example 4: Let a Find the slope at . Note: If it says “Find the limit” on a test, you must show your work!
Example 4: Let b Where is the slope ?
Review: p velocity = slope These are often mixed up by Calculus students! average slope: slope at a point: average velocity: So are these! instantaneous velocity: If is the position function: velocity = slope p
Example Find an equation of the tangent line to the hyperbola at P(3, 1)
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Yes…. Complex fractions!! Example Yes…. Complex fractions!!
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Example Find slope of the curve at x=a and describe what happens to tangent at x=a as a increases. For y=x2 + 2.
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Example As a increases the slope will increase!
Your Turn Write an equation for the normal line to the curve f(x)=4-x2 and x=1.
Your Turn
Your Turn Write equation for the tangent line and normal line to at x=2
Your Turn Write equation for the tangent line and normal line to at x=2 Answer: Tangent y = -x + 3 Normal y = x - 1
Homework Pgs 87-89/ 2-32 Even