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Chapter 4. Applications of Derivatives. 4.1 Extreme Values of Functions. (0, 2). The Extreme Value Theorem . Local Extreme Values. Find the extreme values of . Find the extreme values of . p.193 (1-29, 35-41)odd. 4.2 Mean Value Theorem. Rolle's Theorem
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Chapter 4 Applications of Derivatives
4.2 Mean Value Theorem Rolle's Theorem Suppose that y = f(x) is continuous at every point of [a, b] and differentiable at every point of (a, b). If f(a) = f(b) = 0, then there is at least one number c in (a, b) at which f '(c) = 0. Proof: Being continuous, f assumes absolute max and min values on [a, b]. These can occur only: 1. at interior points where f ' is zero 2. at interior points where f ' does not exist 3. at the endpoints of the function's domain, in this case, a and b. By the hypothesis f has a derivative at every integer point of [a, b]. That rules option 2. If either the max or min occurs at a point c inside the interval, then f '(c) = 0 a previous theorem we have found a point for Rolle's Theorem. If both max and min are min are at a or b then the max and min values of f are both 0. Thus, f has the constant value 0 so f ' = 0, throughout (a, b) and c can be taken anywhere in the interval.
Proof: We picture the graph of f as a curve in the plane and draw a line through the points A(a, f(a)) and B(b, f(b)). The line is the graph of the function The vertical difference between the graphs of f and g at x is The function h satisfies the hypotheses of Rolle's Theorem on [a, b]. It is continuous on [a, b] and differentiable on (a, b) because f and g are. Also, h(a) = h(b) = 0 because the graphs of f and g both pass through A and B. Therefore h' = 0 at some point c in (a, b). This is the point we want. To verify we differentiate both sides of our previous equation with respect to x and then set x = c:
Ex: The function y = x2 is • Decreasing? • Increasing?
Example: A particle is moving along a horizontal line with position function t ≥0 Find the velocity and acceleration and describe the motion of the particle.
Procedure for Graphing y = f(x) by Hand. 1. Find y' and y". 2. Find the rise and fall of the graph. 3. Determine the concavity of the curve. 4. Make a summary and show the curve's general shape. 5. Plot specific points and sketch the curve. Note: Use zeros if you know them. Example: Sketch the graph of the function f(x) = x4 - 4x3 + 10 using the following steps. (a) Identify where the extrema of f occur. (b) Find the intervals on which f is increasing or decreasing. (c) Find where the graph of f is concave up or down. (d) Sketch a possible graph for f.
Let’s examine page 209 Example 4 to discuss how to graph a function based on its derivative. Let’s examine page 214 Example 9 to discuss how to graph a function based on its derivative.
Example: Find two numbers whose sum is 20 and whose product is as large as possible.
Example: A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the rectangle can have, and what dimensions give that area? Example: An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20 by 25 inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume?
Example: You have been asked to design a one liter oil can shaped like a right circular cylinder. What dimensions will use the least material? A drilling rig 12 mi offshore is to be connected by pipe to a refinery onshore, 20 mi straight down the coast from the rig. If underwater pipe costs $50,000 per mile and land-based pipe costs $30,000 per mile, what combination of the two will give the least expensive connection?
Ex: Suppose c(x) = x3 – 6x2 + 15x, where x represents thousands of units. Is there a production level that minimizes average cost? If so, what is it?
How rapidly will the fluid level inside a vertical cylindrical tank drop if we pump the fluid out at the rate of 3000 L/min? Example: A hot air balloon rising straight up from a level field is tracked by a range finder 500 ft from the liftoff point. At the moment the range finder's elevation angle is π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment?
Example: Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base of 5 ft. How fast is the water level rising when the water is 6 ft deep?
