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(Exercises for Section 4.5: Graphing) E.4.5

SECTION 4.5: GRAPHING

1) For each part below, sketch the graph of y = f x( ).

Find the domain of f . State whether f is even, odd, or neither, and incorporate any corresponding

symmetry in your graph.

Find the y-intercept, if any. You do not have to find x-intercepts in a), but findany x-intercepts in the others.

Find and indicate on your graph any holes, vertical asymptotes (VAs),horizontal asymptotes (HAs), and slant asymptotes (SAs), andjustify them using limits.

Find the critical numbers (CNs), if any.

Find all points at critical numbers (if any). Indicate these points on your graph.

Use the First Derivative Test to classify each point at a critical number as alocal maximum point, a local minimum point, or neither.(The next instruction may help.)

Find the intervals on which f is increasing / decreasing, andhave your graph show that.

Find the Possible Inflection Numbers (PINs), if any.

Find the x-intervals on which the graph of y = f x( )is concave up / concave down, and have your graph show that.

For each PIN, state whether or not the corresponding point on the graph is aninflection point (IP). Find any IPs.

a) f x( )= x4 + 4 x3 + 48 x2 + 112 x 500 .Hint: 2 is one of the critical numbers.We will discuss the x-intercepts in Section 4.8.

b)

f x( )= x2

3 x + 4( )2

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(Exercises for Section 4.6: Optimization) E.4.7

SECTION 4.6: OPTIMIZATION

1) (Open box problem). If a rectangular cardboard box with a square base and anopen top is to have a volume of four cubic feet, find the dimensions of the box that

require the least amount of cardboard. (Disregard the thickness of the cardboard.)2)

(Closed box problem). Repeat Exercise 1, except the box must have a closed top.

3) (Trash can problem). A metal cylindrical trash can with an open top is to have acapacity of one cubic meter. Find the dimensions that require the least amount of metal.

4) (Six pigpen problem). A 2 3 array of six congruent rectangular pigpens (that alllook the same from above) will be in the overall shape of a rectangle R. We mayuse 1000 feet of fencing to form the boundaries of the pigpens. Find thedimensions for a single pigpen that will maximize the total area of all the pigpens.(The fencing separating the pigpens has constant height, so we may ignore heightin our calculations. Also, assume that we do not double-fence the boundariesbetween pigpens; in other words, assume that the thickness of the fencing betweenpigpens is the same as the thickness of the fencing along the outer boundary, R.)

5) (Two ship problem). At noon, Ship A is 30 miles due south of Ship B and is sailingnorth at a rate of 15 miles per hour (mph). Ship B is sailing west at a rate of 10

mph. Find the time at which the ships are closest.6)

(Cheap building problem). We need to build a building in the shape of arectangular box with a capacity of 900 cubic feet. We require the width of the floorto be three-fourths of the length. The floor will cost $4 per square foot, the sides of the building will cost $6 per square foot, and the roof will cost $3 per square foot.What are the dimensions of the cheapest building that we can build? Round off theoptimal dimensions to the nearest hundredth of a foot.

7) (Closest point problem). Find the point on the graph of

y = x

2+ 1 that is closest to

the point

3,1( ).

8) (Big squares problem). Prove that, among all rectangles with fixed perimeter p,where p > 0 , the largest in area is a square.

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