1. CHAPTER 2 Linear Approximations animation 2.4 Continuity The tangent line at (a, f(a)) is used as an approximation to the curve y = f(x) when x is near a. An equation of this tangent line is y = f(a) + f’(a)(x – a) and the approximation is f(x)  f(a) + f’(a)(x – a) is called the linear approximation or tangent line approximation of f at a.

2. Example Find the linear approximation for the function f(x) = ln x at a = 1.

3. Example Atmospheric pressure P decreases as altitude h increases. At a temperature of 15’C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h=1 km, and 74.9 kPa at h=2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km.

4. CHAPTER 2 2.4 Continuity What does f’ Say about f ? animation

5. CHAPTER 2 If f’(x) > 0 on an interval, then f is increasing on that interval. If f’(x) < 0 on an interval, then f is decreasing on that interval. 2.4 Continuity If f”(x) > 0 on an interval, then f is concave upward on that interval. If f”(x) < 0 on an interval, then f is concave downward on that interval.

6. CHAPTER 2 A point where a curve changes its direction of concavity is called an inflection point. 2.4 Continuity

7. Example Sketch a graph whose slope is always positive and increasing; sketch a graph whose slope is always positive and decreasing and give them equations.

8. Example Suppose f’(x) = x e-x3. a) On what interval is f increasing and on what interval is f decreasing? b) Does f have a max or a min value?

9. Quiz: Which is f(x), f’(x), and f’’(x)? animation animation

10. An antiderivative of f is a function F such that F’ = f. Example Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin.