3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY

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1 MATH00 (Calculus).1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY Name Group No. KEYWORD: increasing, decreasing, constant, concave up, concave down, and inflection point Eample 1. Match the graph of in the left column with that of its derivative in the right column. Eample. The graphs of f, f and f are shown on the same set of coordinate aes. Which is which? Eplain your reasoning. Eample. Consider f( ) ( 1) Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the -coordinates of all inflection points.

2 Page Eample 4. Consider f () The derivative in graphing and applications Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the -coordinates of all inflection points. Eample 5. Suppose that water is flowing at a constant rate into the container shown. Make a rough sketch of the graph of the water level y versus the time t. Make sure that your sketch conveys where the graph is concave up and concave down, and label the y -coordinates of the inflection points.

3 MATH00 (Calculus). ANALYSIS OF FUNCTIONS II RELATIVE EXTREMA; GRAPHING POLYNOMIALS Name Group No. KEYWORD: critical point, stationary point, relative etremum (relative maimum and relative minimum), first derivative test, and second derivative test Eample 1. Use the graph of f to sketch a graph of f and the graph of f. Eample. Identify the real numbers 1,, and 4 in the figure such that each of the following is true. a. f( ) 0 b. f( ) 0 c. f () D.N.E. d. f has a relative maimum. e. f has a point of inflection.

4 Page 4 The derivative in graphing and applications Eample. Sketch the graph of a function f with all of the following properties: o domain of f is 0, 4 o f(0) 1, f(1) 0, f(), f(4) for 0 1 o f ( ) 0 for 1 o f ( ) 0 for 4 o f ( ) 0 o lim f( ) and 1 lim f( ) 1 Eample 4. Locate the critical points and identify which critical points are stationary points. 4 1) f ( ) ) f () 8 ) f( ) sin( )

5 MATH00 (Calculus). ANALYSIS OF FUNCTIONS III RATIONAL FUNCTIONS, CUSPS, AND VERTICAL TANGENTS Name Group No. KEYWORD: oblique or slant asymptote, curvilinear asymptote, and cusp Eample 1. Sketch the graph of the following and identify the locations of all relative etrema and inflection points. 1) f () 1 ) f () 4 ) f () 1 4) ( ) f () 5) f () 1 1 6) f () 5

6 Page 6 The derivative in graphing and applications Eample. Sketch the graph of the following. 1/ 1) f() 1/4 1/5 ) f() ) f() /5 4) f() 1/ 5) f() 6) f( ) 4 7) f( ) 8) f( ) tan( ) 9) f( ) tan( )

7 MATH00 (Calculus).4 ABSOLUTE MAXIMA AND MINIMA Name Group No. KEYWORD: absolute etremum (absolute maimum and absolute minimum), and Etreme-Value Theorem Eample 1. Consider the graph of y f(), shown below. For each of the following, compute the absolute maimum and absolute minimum values of f () on the given interval, if they eist. (Make reasonable assumptions about the behavior of the function outside of the shown interval.) a. (, ) b. 7,5 c. 7,0 d. ( 4,1) Eample. Sketch the graph of a continuous function, y f(), which has all of the following properties:. o domain of f is 1, 7 o f has an absolute maimum of 6 when and an absolute minumum of. 1 when 5 f for all in the domain of f, with the eception of where f () DNE. o ( ) 0

8 Page 8 The derivative in graphing and applications Eample. For each of the following, find the absolute maimum and minimum values of f () on the given interval. a. f () interval absolute maimum value absolute minimum value f( ) 4, b. f( ) ( 1) 1,4 c. f () ( 4) 4,1 d. f( ) cos( ) sin( ), e. f () 5 (, ) f. f() e (, )

9 MATH00 (Calculus).5 APPLIED MAXIMUM AND MIMIMUM PROBLEMS Name Group No. KEYWORD: Use the techniques from Chapter.4 to solve optimization problems, i.e. given a system of related quantities, find values of the quantities that optimize one of them (e.g. minimize a cost, maimize a volume, etc) PRACTICE PROBLEMS: EXERCISE SET.5 1. An open bo is to be made from a ft by 8 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maimum volume that the bo can have.. A closed rectangular container with a square base is to have a volume of 000 cm. It costs twice as much per square centimeter for the top and bottom as it does for the sides. Find the dimensions of the container of least cost.

10 Page 10 The derivative in graphing and applications 1. A cylindrical can, open at the top, is to hold 500 cm of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can. 51. Find the coordinates of the point P on the curve 1 y, 0 where the segment of the tangent line at P that is cut off by the coordinate aes has its shortest length.

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