CHAPTER 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. Linear Approximations animation

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

Example Atmospheric pressure P decreases as altitude h increases. At a temperature of 15’C, the pressure is 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.

CHAPTER Continuity What does f’ Say about f ? animation

CHAPTER Continuity 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. 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.

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

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.

Example Suppose f’(x) = x e -x 3. a) On what interval is f increasing and on what interval is f decreasing? b) Does f have a max or a min value?

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

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.