Rates of Change and Tangent Lines Sec. 2.4 Rates of Change and Tangent Lines
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.)
limit ((1/(a + h) – 1/ a) / h, h, 0) Example 4: Let a Find the slope at . On the TI-89: limit ((1/(a + h) – 1/ a) / h, h, 0) Note: If it says “Find the limit” on a test, you must show your work! F3 Calc
On the Calculator: Let b Where is the slope ? Y= y = 1 / x WINDOW Example 4: On the Calculator: Let b Where is the slope ? Y= y = 1 / x WINDOW GRAPH
Example 4: Let b Where is the slope ? tangent equation
Given y=x2-4x… Find the slope of the tangent line at x=1.
Given y=x2-4x… Find the equation of the tangent line at x=1. Use m=-2 and the point (1,-3) to write a point-slope equation!
Given y=x2-4x… Find the equation of the normal line at x=1.
Review: These are often mixed up by Calculus students! average slope: slope at a point:
HOMEWORK P. 92-94 #1, 3, 7, 9, 11, 13-16, 19, 23, 25, 35-40