2.4 Rates of Change and Tangent Lines Devils Tower, Wyoming Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington The slope of a line is given by: m y x y x The slope at (1,1) can be approximated by the slope of the secant through (4,16). y 16 1 15 5 x 4 1 3 We could get a better approximation if we move the point closer to (1,1). ie: (3,9) y 9 1 8
x 3 1 2 f x x 2 4 Even better would be the point (2,4). y 4 1 3 x 2 1 1 3 The slope of a line is given by: m y x y
x If we got really close to (1,1), say (1.1,1.21), the approximation would get better still y 1.21 1 .21 2.1 .1 x 1.1 1 How far can we go? f x x 2 y slope x f 1 h f 1 slope at 1,1 h 1 1 h f 1 h f 1
h 1 h lim h 0 2 1 h h 2 h 1 2h h 2 1 lim lim h 0 h 0 h h 2 The slope of the curve y f x at the point P a, f a is: m lim h 0
f a h f a h The slope of the curve y f x at the point P a, f a is: m lim h 0 f a h f a h f a h f a h 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 y y1 m x x1 . If you want the normal line, use the negative reciprocal of
the slope. (in this case, 1 ) 2 (The normal line is perpendicular.) Example 4: 1 Let f x x a Find the slope at x a . On the TI-89: limit ((1/(a + h) 1/ a) / h, h, 0) F3 Calc f a h f a m lim h 0 h 1 a a h 1 a a h a lim a h h 0 h a a h
1 a a h lim h 0 h a a h a a h lim 0 h 0 h a a h Note: If it says Find the limit on a test, you must show your work! 1 2 a Example 4: On the TI-89: 1 Let f x x b Where is the slope 1 ? 4 1
1 2 4 a a 2 4 a 2 Y= y=1/x WINDOW 6 x6 3 y 3 x scl 1 y scl 1 GRAPH Example 4: 1 Let f x x b Where is the slope 1 ? 4 We can let the calculator Ontangent: the TI-89:
plot the F5 Math Y= y=1/x A: Tangent ENTER WINDOW 2 ENTER 6 x6 3 y 3 x scl for 1 x = -2 Repeat y scl 1 tangent equation GRAPH Review: y average slope: m x f a h f a slope at a point: m lim h 0
h average velocity: Vave instantaneous velocity: velocity = slope total distance total time These are often mixed up by Calculus students! So are these! If f t is the position function: f t h f t V lim h 0 h