AP Calculus AB / Mr. Hansen
12/16/2004

Name: _________________________

Test on Chapter 5
You must show work for credit on problems 6-8, even though it is possible to achieve the answer by calculator alone. Point values are shown in parentheses.

1.
(9 pts.)

The two central topics of our course, namely the _____________________ calculus and the _____________________ calculus, are linked by an extremely important result known as the __________________________________________________ (abbreviated _______ ).

 

 

2.
(6)

The definite integral (Riemann integral) is defined as the _____________ of all possible _____________ sums as the norm of the partition approaches _____________ .

 

 

3.
(19)

The simplest way of numerically estimating the definite integral is to divide the interval [a, b] into a large number of _____________ of equal width, using either the left endpoint or the _____________ endpoint of each one to provide the function value that tells the _____________ . These are called the left endpoint and right endpoint methods, respectively, and they can be averaged to provide a result that equals the number provided by the _____________ method. However, the TI-83 and TI-83 Plus calculators are programmed to use a more sophisticated method for estimating definite integrals, namely _____________ _____________ . The advantage of this method over the simpler methods is that for a given level of accuracy (e.g., 10–5), the TI-83 method is _____________ . Briefly explain why this is so:

_________________________________________________________________

_________________________________________________________________

_________________________________________________________________

 

 

4.
(16)

State both forms of FTC. Be sure to state any conditions on f that may be required.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.
(10)

Show that one form of FTC implies the other. (You do not need to prove both directions. Choose whichever direction you prefer. As you may recall, one of the directions is considerably easier than the other.)

 

6.
(12)

Use FTC to compute . Show all work and proper notation. You may check your work with your calculator.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.
(12)

Find G(7), correct to 3 decimal places, if G is an antiderivative of f (x) = 11ecos x – 7x + cos4(sin x) and G(–2.1) = –3.118. You must write an expression for G(x) before you start using your calculator.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.(a,b)
(16)

Use the Trapezoidal Rule, then Simpson’s Rule, to estimate the total distance traveled during the first 10 seconds by a rocket whose velocity at times t = 0 seconds through t = 10 seconds is given by 0, 10, 25, 40, and 60 m/sec. Measurements are made at 2.5-second intervals. Show formula, plug-ins, and answers. If you prefer, you may show your work by means of a table in which the columns are clearly labeled.