Honors AP Calculus / Mr. Hansen
12/14/2011 [problem numbering corrected]

Name: _____________________
Mr. Hansen’s use only: ________

Test (100 pts.): Calculator Permitted

 

Part I: Fill-Ins.

 

Write the word or phrase that best fits each blank. For names, a last name is sufficient.

 

 

1.

The term chaos may be mathematically defined as _______________________________

__________________________________________________________________________

2.

A mathematician who died in 2010 is remembered as the man most closely associated with the science of fractals. His name was ________________________________________ .

3.

IVT, EVT, and FTC1/2 all have as their hypothesis that we have a ________________ function on a(n) ________________ interval. MVT has the additional hypothesis that the function is _________________________ on a(n) ________________ interval.

 

 

4.

“A point of continuity at which  changes sign” is a suitable definition of
_______________________________________________________ .

 

 

 

Part II. Volumes.

 

5.

The green Play-Doh object is a ring having rectangular cross sections. You can think of it as “a hockey puck with a hole in it.” The rectangular cross sections are 2 cm wide by 1.5 cm high. The inner radius of the object is 1 cm, and the outer radius is 3 cm. Compute the volume by (a) plane slicing, (b) cylindrical shells, and (c) radial slicing. You may leave  in your answers, or you may give approximate answers to at least 3 decimal places. Most of the points are in the setups of the integrals, so be sure to show adequate work: unplugged formulas, plugged formulas, and the work (if any) leading to a final answer, which must be circled or boxed in each case. Sketches are recommended but not required.

 

 

(a)

Plane Slicing

 

(b)

Cylindrical Shells

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

Radial Slicing

 

6.

On Tuesday, Dec. 13, Mr. Hansen attended a Christmas musical party at which a blueberry coffee cake was served. Mr. Hansen was given exactly one quarter of the cake to take home (90 degrees of pure deliciousness). The cross sections are, let us say, ellipses having major axis of 3.1 inches and minor axis of 1.8 inches. The inner radius of the coffee cake, before it was cut, was 1.5 inches. Compute the volume of the one-quarter cake that is available for inspection. A sketch is recommended but not required.

 

7.

In #5, suppose that a factory is experimenting with various types of “hockey pucks with holes in them.” Let r denote the inner radius, R the outer radius, and h the height (thickness) of the puck. Prove by any calculus-based method you wish that the volume equals  cubic units. Note: This formula is easy to prove with freshman geometry, and you may certainly use that to cross-check if you wish, but you will not earn any points unless you use a calculus-based proof, wlog. A sketch is required this time.

 

8.

 

 

 

(a)

In #7, let us imagine that the factory insists that R, the outer radius, be fixed at 5 cm. However, the inner radius r and height h are allowed to vary in such a way that the volume is a fixed constant value of 70 cm3. In order to minimize the cost of the paint that must be applied to every exposed surface, the factory wishes to minimize the total surface area of the manufactured “hockey pucks with holes in them.” What do we call surface area in this situation? Answer: the __________________________ function.

 

 

(b)

Let f denote the surface area function. Write f as a function of r alone, and show the work leading to your answer. Simplification is not required.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

As you surely noticed, f (r) is a rather messy function. Without actually minimizing f, write a grammatically correct paragraph in which you describe how you would go about minimizing f. Standard abbreviations are permitted. Be sure to state the domain for r, and be sure to describe how the r value you expect to find actually gives a minimum value for f. The “hole in the hockey puck” is not allowed to vanish completely, nor can the hole extend all the way to the outer radius. Do not actually do the minimization. That would take too long.

 

 

 

9.

In geometry class, you learned that the volume of a right regular pyramid with square base equals  where s = side length of the base and h = altitude = perpendicular distance from apex to the plane containing the base. Prove, using the calculus, that the formula is equally valid for any “oblique square pyramid” of the type sketched on the board. Hint: The side length of the square cross sections tapers linearly from s at the base to 0 at the apex. A sketch is optional.