Honors AP Calculus / Mr. Hansen	Name: _______________________________
9/21/2011	READ INSTRUCTIONS IN EACH PART! ______

Test #1 (100 points): Chapters 1-3 plus Class Discussions

General Instructions:

§ Calculator use is permitted throughout today’s test.

§ If you have spare batteries, raise your hand for a small bonus.

§ Questions marked with a star ( * ) can be dictated to Mr. Hansen or recorded on a digital recorder if you have a writing impediment.

	Part I: Fill-Ins (2 pts. per blank, 34 pts. in all). Write the name, word, or phrase that best fits. The blanks suggest the length of expected answers.
1.	The word “calculus” originally meant, in Latin, a physical object, namely a ___________ . Later the word came to refer to the types of rote computations that were facilitated by such objects. In the present age, we say “a calculus” to mean any ___________ system, i.e., a ___________-based system in which answers can be obtained by mere symbol manipulation, without regard to the underlying meaning. We say “the calculus” to mean our course, i.e., the _______________ calculus and the _______________ calculus invented about 350 years ago by Sir Isaac ___________ and Gottfried Wilhelm ___________ .
2.	In the early 1930s, a German ____________________________ [hint: the word means “someone who studies the mathematics of mathematics itself] named Kurt _________ stunned the world with his Incompleteness Theorem. He proved that all ___________ mathematical systems having at least the power of basic arithmetic are incomplete (i.e., must contain undecidable __________________ ), or equivalently, all mathematical systems of some power in which every possible proposition can be determined to be true or false are inconsistent.
3.	“Extreme sensitivity to initial conditions” is a working definition of the mathematical term _________ . The word definition refers to a ________________________ statement, i.e., an implication that runs in both directions, asserting that one side of the statement is both necessary and _______________ for the other.
4.	In our class, we learned the rudiments of the calculus of formal logic, an area of study that fits perfectly with computers because both areas involve formal, binary systems. We learned, for example, that the statement “All students love the calculus” is written in symbols as where x is a dummy variable for a student and L(x) is the statement that student x loves the calculus. The negation of is written in symbols as and in words as “It is not the case that for any student x, x loves the calculus.” Let us rewrite using the existential quantifier, instead of the universal quantifier, . In symbols: __________________________________ *In words: ____________________________________________________________________ *Why would we never write as “All students do not love the calculus”? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________
	Part II: Calculator Skills (6 pts. per blank, 24 pts. in all). Fill in the blanks with the requested answers. No work is expected to be shown in this section.
5.	If y = f (x) = sin^2 x, find the largest 3-decimal-place value of such that whenever x is in a punctured -neighborhood of , f (x) is within 0.01 units of sin^2 = 0.75. Answer: = ____________
6.	______________ (AP standard of at least 3 decimal places of accuracy)
7.	seems to be ______________
8.	Let h(x) denote the integrand in #6. Compute to AP standard accuracy: _____________
	Part III. Free Response (10 pts. each, except that #11 is double credit, 20 pts.)
9.	*Suppose that function g is a continuous function known to both you and me. You choose a small positive number and write it down on a piece of paper. I think for a while and then produce another small positive value such that whenever x is within my value units of 2.3, g(x) is within your value units of 7. We repeat this experiment 5 times, and each time, I succeed. Does this prove that Explain your answer carefully, using complete, grammatically correct sentences. Standard mathematical abbreviations are allowed.
10.	State the definition of the limit L of function f as x approaches z, where z denotes a finite real value. Important: For full credit, replace the absolute value notation used in class and in your book with interval notation for both the punctured -neighborhood of z and the -neighborhood of L. Also, be sure to include a double-headed arrow () or the word “iff” somewhere in your definition. (“Iff” means “if and only if.”)
11.	The acceleration function for a rocket fired straight upward is given by a(t) = 3.4 + 2(4 – t) for time t between 0 and 4 seconds, inclusive, and by a(t) = –9.8 for time t > 4 seconds. Units of the acceleration function are meters per second per second, denoted m/sec^2.
(a)	Compute for the domain Hint: The answer will be a family of functions, not a single function.
(b)	What are the units of your answer to part (a)? ________________ What English word starting with the letter v is the name of the antiderivative of the acceleration function? _________________
(c)	For the domain compute v(t) subject to the initial condition that v(0) = 1. Show work.
(d)	Use your calculator to compute . Give answer here, including units: ________________
(e)	*What does the answer to part (d) signify, in plain English? (Note that you can answer this question even if you could not get any of the other answers.) A sentence fragment is acceptable.
	Part IV. Parting Gift (2 pts.)
12.	Write the expression 494q + 949y – 35z: __________________________________