1. Tangent Line Approximations Section 3.9 Notes

2. Given a function, f (x), we can find its tangent at x = a. The equation of the tangent line, which we’ll call L(x) for this discussion, is,

3. Take a look at the following graph of a function and its tangent line. From this graph we can see that near x = a, the tangent line and the function have nearly the same graph. On occasion we will use the tangent line, L(x), as an approximation to the function, f (x), near x = a. In these cases we call the tangent line the linear approximation to the function at x = a.

4. Find the tangent line approximation of f (x) = 1 + sin x at the point (0, 1). Then use a table to compare the y-values of the linear function with those of f (x) on an open interval containing x = 0. Example 1

5. tangent line approximation

6. x-0.5-0.1-0.0100.010.10.5 f(x) = 1 + sin x y = x + 1.52057.5.99017.9.99000.99 1 1 1.0099 1.01 1.0998 1.1 1.4794 1.5 Because the tangent line values to the left of x = 0 are smaller than the actual values, the graph of f (x) is concave up. Because the tangent line values to the right of x = 0 are larger than the actual values, the graph of f (x) is concave down.

8. A.For, find the equation of the linear function that best fits f (x) at x = 8. Example 2 (8, 12) y – 12 = x – 8 y = x +4

9. B.Use the tangent line equation to approximate f (8.2). C.Find f (8.2) by using the function f (x). D.What is the error in your linear approximation? Since the approximation is larger than the actual function value, the function must be concave down near x = 8 and therefore f ′′(8.2) is negative. f (8.2) ≈ 8.2 +4 = 12.2

10. Let f be a differentiable function such that f (3) = 2 and f ′(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is A.0.4 B.0.5 C.2.6 D.3.4 C Equation of the tangent line at x = 3:

11. The function f is twice differentiable with f (2) = 1, f (2) = 4, and f (2) = 3. What is the value of the approximation of f (1.9) using the line tangent to the graph of f at x = 2? A.0.4 B.0.6 C.0.7 D.1.3 E.1.4 B Equation of the tangent line at x = 2: