INTERMEDIATE VALUE THEOREM

Transcription

1 THE BIG 7 S INTERMEDIATE VALUE If f is a continuous function on a closed interval [a, b], and if k is any number between f(a) and f(b), where f(a) f(b), then there exists a number c in (a, b) such that f(c) = k. 1. Illustrate this: 2. The intermediate value theorem is commonly used to show the existence of roots. How can you prove the existence of a root with this theorem? 3. Show that the function 3 2 f x x 3x x 4 has at least one real root on the interval [0, 4].

2 If a function f(x) is differentiable at x = a, then f(x) is continuous at x = a. 1. The contrapositive of this is also true. What is the contrapositive?: AB5/BC5: Let f be a function defined by f 2x 1, for x 2 x 1 2 x k, for x 2 2 a. For what value of k will f be continuous at x = 2? Justify your answer. b. Using the value of k found in part a, determine whether f is differentiable at x = 2. Use the definition of the derivative to justify your answer. c. Let k = 4. Determine whether f is differentiable at x = 2.

3 EXTREME VALUE Every continuous function on a closed interval [a, b] must have an absolute maximum and an absolute minimum somewhere on [a, b]. They may occur at the endpoints. 1. (a) What are some differences between a relative and absolute maximum or minimum. (b) Draw an example of a curve where a relative extrema is not the same as the absolute extrema (c) Draw an example of a curve where the relative extrema and absolute extrema are the same AB4/BC3: A function f is continuous on the closed interval 3, 3 and f 3 1. The functions f ' and f 3 4 such that f '' have the properties given in the table below. x 3 x 1 x 1 1 x 1 x 1 1x 3 f 'x Positive Fails to exist Negative 0 Negative f '' x Positive Fails to exist Positive 0 negative a. What are the x-coordinates of all absolute maximum and minimum points of f on the interval 3, 3. Justify your answer. b. What are the x-coordinates of all inflection points of f on the interval 3, 3? Justify your answer. c. On the axes provided, sketch a graph that satisfies the given properties of f.

4 ROLLE S If a function f(x) is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and if f(a) = f(b), then f (c) = 0 for at least one number c in (a, b) 1. A car on the interstate was traveling 70 mph at mile marker 10 and again was traveling 70 mph at mile marker 20. We know nothing about the car s velocity between those two points. What does Rolle s Theorem guarantee about the car s acceleration during this trip? AB3/BC3 t (hrs) Rt (gal/hr) The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table shows the rate as measured every 3 hours for a 24-hour period. 24 a. Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate Using correct units, explain the meaning of your answer in terms of water flow. R t dt. 0 b. Is there some time t, 0 t 24 such that R' t 0. Justify your answer. 1 2 c. The rate of water flow Rt can be approximated by Qt t t. Use Qt to 79 approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

5 THE MEAN VALUE FOR DERIVATIVES If a function f(x) is continuous on a closed interval [a, b] and is differentiable on the open interval (a, b), then there exists a number c in f b f a (a, b) such that f ' c. ba 1. Illustrate the Mean-Value Theorem on the graphs below (if possible): a. b AB3/BC1: Consider the curve y 4 x and chord AB joining points 4,0 B 0,2 on the curve. a. Find the x and y coordinates of the point on the curve where the tangent line is parallel to chord AB. A and b. Find the area of the region R enclosed by the curve and the chord AB. c. Find the volume of the solid generated when the region R, defined in part b, is revolved about the x-axis.

6 THE FUNDAMENTAL OF CALCULUS Suppose f is continuous on [a,b] h x I. If g x f tdt, then g ' x f hx h' x b a II. d dx f x dx F b F a, where F is any antiderivative if f, that a is, F = f. 1. (a) f x dx (b) f ' x dx d sin dt dx (d) 2x 2 x 3 (c) t 8 0 d dx 1x dx 99 AB5/BC5: The graph of the function f, consisting of three line segments is given. Let g x 1 x f t dt. a. Compute g 4 and 2 g. b. Find the instantaneous rate of change of g, with respect to x, at x = 1. c. Find the absolute minimum value of g on the closed interval 2,4. Justify your answer. d. The second derivative of g is not defined at x = 1 and x = 2. How many of these values are x- coordinates of points of inflection of the graph of g? Justify your answer.

7 THE AVERAGE VALUE (A.K.A. THE MEAN VALUE FOR DEFINITE INTEGRALS) If f is continuous on [a, b], then there must exists a number c in [a, b] 1 b such that f x dx ba f c, where a value. f c is the function s average 1. Estimate the value(s) of c which will satisfy the Mean-Value Theorem for definite integrals on the graph below. 98 AB5/BC5: The temperature outside a house during at 24-hour period is given by t F t 80 10cos, 0 t is measured in hours. a. Sketch the graph of F., where Ft is measured in degrees Fahrenheit and t b. Find the average temperature, to the nearest degree Fahrenheit, between t = 6 and t = 14. c. An air conditioner cooled the house whenever the outside temperature was at or above 78ºF. For what values of t was the air conditioner cooling the house? d. The cost of cooling the house accumulates at the rate of $0.05/hour for each degree the outside temperature exceeds 78ºF. What was the total cost, to the nearest cent, to cool the house for this 24 hour period?