Mth Test 3 Review Stewart 8e Chapter 4. For Test #3 study these problems, the examples in your notes, and the homework.

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1 For Test #3 study these problems, the eamples in your notes, and the homework. I. Absolute Etrema A function, continuous on a closed interval, always has an absolute maimum and absolute minimum. They occur at Critical Points (where f (c) = 0 or f (c) dne) or at endpoints. A. Find the absolute maimum and absolute minimum values of f() on the given interval: f() = , [ 4, 2] B. Find the absolute maimum and absolute minimum values of f() on the given interval: f() = 2 + 2, [ 2, 2 ] II. The first derivative test for increasing/decreasing. Suppose that f() is continuous on [a, b] and differentiable on the open interval (a, b). If f () > 0 for all in (a, b) then f() increases on [a, b]. If f () < 0 for all in (a, b) then f() decreases on [a, b]. If f () = 0 for all in (a, b) then f() is constant on [a, b]. III. The First Derivative Test for Local Etrema. Let f() be a continuous function on [a, b] and c be a critical number in [a, b]. If f () 0 on (a, c) and f () 0 on (c, b), then f() has a local maimum of y = f(c) at = c. If f () 0 on (a, c) and f () 0 on (c, b), then f() has a local minimum of y = f(c) at = c. If f () does not change signs at = c, then f() has no local etrema at = c. IV. The Second Derivative Test for Concavity Let f() be a twice differentiable function on an interval I. If f () > 0 on I, the graph of f() over I is concave up. If f () < 0 on I, the graph of f() over I is concave down. V. Discontinuities A. The line = a is a vertical asyptote of f if either a a f() = ± OR f() = ± + You find a vertical asymptote of a function f() = N() by finding a value = a such that D() denominator equals zero D(a) = 0 AND the numerator is not zero N(a) 0. B. A hole (removable discontinuity) eists at = a if f(a) = 0 0.

2 VI. A Horizontal Asymptote describes the behavior of a function as gets very large. Limits at Infinity/Horizontal Asymptotes Let f() be the rational function given by f() = N() D() = a n n + a n n + + a + a 0 b m m + b m m + + b + b 0 where N() and D() have no common factors. The comparing the degrees of N() and D(). If n < m, then as a horizontal asymptote. f() can be determined by f() = 0 and the graph of f() has the line y = 0 (the -ais) If n = m then f() = a n and the graph of f() has the line y = a n b m b m as a horizontal asymptote. If n > m then f() = ± and the graph of f() has no horizontal asymptote. VII. Sophisticated graphing : Graphing using y and y : Steps: (a) Determine the points of discontinuity. (b) Determine the asymptotes (vertical, horizontal) (c) Determine the - and y- intercepts. (d) Determine the critical point(s). (Set f () = 0 and undefined). (e) Determine the intervals where the function f is increasing/decreasing. (f) Determine the local etrema. (g) Determine the possible point(s) of inflection. Set f () = 0 and undefined). (h) Determine the intervals where the function f is concave up/down. (i) Determine the inflection point(s). (j) Determine etra point(s) if necessary. (k) Sketch the graph using the information obtained above. Use f() to find intercepts, asymptotes, y-values Use f () to find critical pts, increasing and decreasing intervals, local ma and min Use f () to find possible pts of inflection, intervals of concavity, pts of inflection VIII. Problems: A. pg 298 # 8 B. Sketch the graph of a function that satisfies all of the given conditions. f (2) = 0, f(2) =, f(0) = 0, f () < 0 if 0 < < 2, f () > 0 if > 2, f () < 0 if 0 < < or if > 4, f () > 0 if < < 4, f() =, f( ) = f() for all 2

3 C. For f() = + 2, find the following: 2 (a) the vertical and horizontal asymptotes (b) the intervals of increase or decrease (c) the local maimum and minimum values (d) the intervals of concavity and the points of inflection. (e) Sketch the graph of the function. D. Use the sophisticated graphing guidelines to graph y = + 3 2/3 E. Use the sophisticated graphing guidelines to graph y = ln IX. Mean Value Theorem: Given f() continuous on [a, b] and differentiable on (a, b), then for at least one = c in (a, b) f(b) f(a) b a = f (c) Problems: See homework p. 286 #0, #24 X. Optimization Word Problems: Steps: (a) Read. (b) Draw a picture when applicable. (c) Determine if you are maimizing or minimizing the problem. (d) Summarize the information in the problem statement. (e) Determine the formula/function that applies. (f) Write the function in terms of one variable. (g) Determine the domain of the function. (h) Determine the critical point(s) (i) Test to determine the etrema. (j) Did you answer the question asked? XI. L Hopital s Rule Suppose that f() and g() are differentiable on an open interval I containing = a. If we have an indeterminate form of the type 0 0 or then f() a g() = f () a g () This rule can also work for 0,, 0 0, 0, and if it can be changed into 0 0 or application of sufficient algebra. with the 3

4 More Problems In problems 5-8, find the absolute maimum and minimum values of each function on the given interval. 5. g() = 4 2, f(θ) = sin θ, π 2 θ 5π 6 7. g() = sec, π 3 π 6 8. h() = 3 2/3, Give an eample of a function define on [0, ] that has neither a local maimum or a local minimum value at For what values of a, m, and b does the function 3, = 0 f() = a, 0 < < m + b, 2 satisfy the hypotheses of the Mean Value Theorem on the interval [0, 2]?. It took 4 seconds for a thermometer to rise from 9 C to 00 C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at eactly 8.5 C/sec. 2. Show that sin b sin a b a for any numbers a and b. 3. Let f() = p 2 + q + r be a quadratic function defined on a closed interval [a, b]. Show that there is eactly one point c in (a, b) at which f satisfies the conclusion of the Mean Value Theorem. For problems 4 and 5, (a) find the critical points of f. (b) On what intervals is f increasing or decreasing? (c) At what points, if any, does f assume local maimum and minimum values? 4. f () = ( 7)( + )( + 5) 5. f () = ( ) 2 ( + 2) 2 For problems 6, 7, and 8: (a) Find the intervals on which the function is increasing and decreasing. (b) Then identify the function s local etreme values, if any, saying where they are taken on. (c) Which, if any, of the etreme values are absolute? 6. f() = f() = f() = /3 ( + 8) 9. Sketch the graph of four differentiable functions y = f() through the point (, ) if f () = 0 and (a) f () > 0 for < and f () < 0 for > ; (b) f () < 0 for < and f () > 0 for > ; (c) f () > 0 for ; (d) f () < 0 for : 4

5 20. Sketch the graph of a continuous function y = g() such that: g(2) = 2, g < 0, for < 2 g () as 2, g > 0 for > 2, and g () as 2 + Use the steps learned in class (ie. first and second derivative tests, and y intercepts, asmymptotes ect...) to graph equations 9 through 24. Include the coordinates of any local etreme points and inflection points. 2. y = y = 2 3 2/3 23. y = y = If b, c, and d are constants, for what value of b will the curve y = 3 + b 2 + c + d have a point of inflection at =? Give reasons for your answer. 26. The figure shown below shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. B P (,?) A 0 (a) Epress the y-coordinate of P in terms of (You might start by writing an equation for the line AB.) (b) Epress the area of the rectangle in terms of. (c) What is the largest area the rectangle can have? 5

6 27. A rectangle has its base on the -ais and its upper two vertices on the parabola y = 2 2. What is the largest area the rectangle can have? 28. Find the volume of the largest right circular cone that can be inscribed in sphere of radius A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 meters of wire at your disposal, what is the largest area you can enclose? 30. An 25-ft 3 open-top rectangular tank with a square base feet on a side and y feet deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an ecavation charge proportional to the product y. If the cost is c = 5( 2 + 4y) + 0y what values of and y will minimize it? 3. Evaluate the its (a) (b) (2 + 0 e ) 4/ + ( π 2 π 2 ) tan (c) (3e / sin 2) 0 e (d) 0 e 2 2 (e) (f) ln( + 2e ) ( ) 2 6

7 Test 3 Review Problems Answers VIII. Sophisticated graphing : Graphing using y and y : A. a) (2, 4) (6, ) b) ma = 4, min = 2, 6 c) Concave up (, 3),(5, 7),(8, ) d) =, 3, 5, 7, 8 B. Sketch the graph of a function that satisfies all of the given conditions. y C. For f() = + 2, find the following: 2 (a) =, =, y = (b) increasing (0, ),(, ) decreasing (, 0),(, ) (c) the local minimum (0, ) (d) concave up (, ). concave down (, ), (, ) no inflection points y (e) D. E. More Problems 5. y = 0, y = 2 7

8 6. y =, y = 7. y = 0, y = 2 8. y = 0, y = f() = ( /2)( ) 0. a = 3, m =, and b = 4. Mean Value Theorem slope =9/4 = Show that sin b sin a b a 3. Mean Value Theorem. use MVT. For problems 4 and 5, (a) find the critical points of f. (b) On what intervals is f increasing or decreasing? (c) At what points, if any, does f assume local maimum and minimum values? 4. (a) = 5,, 7, (b) Inc ( 5, ) (7, ), dec (, 5), (c) (, 7), ma = min = 5, 7 5. =, 2, inc (, ), none For problems 6, 7, and 8: (a) Find the intervals on which the function is increasing and decreasing. (b) Then identify the function s local etreme values, if any, saying where they are taken on. (c) Which, if any, of the etreme values are absolute? 6. (a) Inc ( 2, 0), (2, ) Dec (, 2), (0, 2) (b) Ma (0, 6) min (±2, 0) (c) none 7. (a) dec (, 0) inc (0, ) (b) none (c) none 8. (a) dec (, 2) inc ( 2, ) (b) ( 2, 6(2 /3 )) (c) ( 2, 6(2 /3 )) 9. Sketch the graph of a differentiable function y = f() 20. Sketch the graph of a continuous function y = g() such that: b = (a) y = (b) A = (c) A = See quiz 8

9 28. Find the volume of the largest right circular cone that can be inscribed in sphere of radius 3. V = 32π A = 30. c = 5( 2 + 4y) + 0y 3. Evaluate the its (a) e 2 (b) (c) e 3 (d) 4 (e) (f) e 2 9

2.1. Definition: If a < b, then f(a) < f(b) for every a and b in that interval. If a < b, then f(a) > f(b) for every a and b in that interval.

2.1. Definition: If a < b, then f(a) < f(b) for every a and b in that interval. If a < b, then f(a) > f(b) for every a and b in that interval. 1.1 Concepts: 1. f() is INCREASING on an interval: Definition: If a < b, then f(a) < f(b) for every a and b in that interval. A positive slope for the secant line. A positive slope for the tangent line.