Slope of the Tangent Line. Estimating with a Secant Line

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1 Slope of the Tangent Line Given a function f find the slope of the line tangent to the graph of f, that is, to the curve, at the point P(a, f (a)). The graph of a function f and the tangent line at a point looks like this. Thus, we only know one point on the tangent line, P(a, f (a)), but we need two points to compute a slope. Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 2/16 Estimating with a Secant Line We first try to estimate the slope by choosing a second point Q(x, f (x)) on the graph of f, different from P. The line joining points P and Q is a secant line and its slope is m PQ You will get the same expression for the slope of the secant line whether Q is to the right or to the left of P. Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 3/16

2 Estimating with a Secant Line Note that if we take Q to be the same as P, the slope of the secant line gives the indeterminate form 0. This should make you 0 think of using limits. P Q 1 Further, note that the graph of f is below the tangent line, and that the secant line for Q 1 has a smaller slope than the tangent line and the secant line for Q 2 has a larger slope. This is because the graph curves downward. It is said to be concave down. Q 2 Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 4/16 A Limit Giving the Slope of the Tangent Line To use a limit to compute the slope of the tangent line, we let the point Q move along the curve toward P. Like this. As Q approaches P, the x-coordinate of Q approaches a, or P Q So the slope of the tangent is defined as m T 1 a x This formula gives the definition of the slope of the tangent line to the graph of the function f at the point where x = a. Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 5/16

3 Example 22 Applying the Limit Definition Consider the function f (x) = x 3 (a) (b) (c) Find the slope of the tangent line to the graph of f at the point where x = 1. Find the equation of the tangent line, and sketch the graph of the f and the tangent line. On the sketch in part (b) draw two secant lines, one for the point Q with x = 0 and the other with x = 2. Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 6/16 Solution: Example 22(a) Applying the definition from Formula 1 with gives m T Direct substitution gives 0/0. So we must try to factor and cancel. Recall that x 3 a 3 This result follows from a more general formula. For any positive integer n x n a n Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 7/16

4 Solution: Example 22(a) Continued Thus, x 3 1 m T = lim x 1 x 3 1 x 1 Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 8/16 Solution: Example 22(b) & (c) Since is the y-coordinate of the point on the curve where x = 1, the equation of the tangent line is given by Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 9/16

5 Linearization Another way to think about the tangent line is to zoom in on the graph of a function f. Consider the function we looked at earlier and zoom in so that we look only at the part of the graph in the rectangle shown. The zoomed in portion of the graph looks like this. The graph is not as curved and is closer to the tangent line. Zooming in on a smaller rectangle gives this. Still less curved and even closer to the tangent line. P(a, f (a)) If we zoom in close enough, the graph will look completely straight and the tangent line will be indistinguishable from the graph of the function. So if we look at small enough segment of the graph it will look like a straight line, which is the tangent line. We say that the tangent line to the graph of f at the point (a, f (a)) is the linearization of the function f at a and that the slope of the tangent line is slope of the curve at (a, f (a)). Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 10/16 An Alternate Formula For the point Q(x, f (x)) in the computation of the slope of the secant line, let x = a + h, where h is a number (positive or negative) giving the horizontal distance between the point of tangency P(a, f (a)) and the point Q. See the diagram. Now, P Q so that the slope of the secant line joining P and Q is a x m PQ Further,, so that the slope of the tangent line is m T 2 Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 11/16

6 Example 23 Using the Alternate Formula Use Formula 2 to re-compute the slope of the tangent line to the graph of f (x) = x 3 at the point where x = 1. Solution: Here so that f (a + h) = 1 + 3h + 3h 2 + h 3 Thus f (1 + h) f (1) Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 12/16 Continuing Example 23 Then and finally f (1 + h) f (1) h lim h 0 f (1 + h) f (1) h Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 13/16

7 The Binomial Formula The expansion used in the last example is useful. For a positive integer n (a + b) n The coefficients of the terms in the binomial expansion above are given by Pascal s triangle as follows: n 0 1 Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 14/16 The Binomial Formula There is a formula for the binomial coefficients found using Pascal s triangle. The coefficient of the a n r b r term in the binomial expansion is ( ) n nc r = r where n! = factorial. Thus. This is called n (a + b) n Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 15/16

8 Example 24 A Binomial Expansion Expand (2x 3y) 4. Solution: Using the binomial expansion gives (2x 3y) 4 Clint Lee Math 112 Lecture 6: Tangent Lines and Rates of Change 16/16