Honors AP Calculus / Mr. Hansen	Name: _______________________________
9/26/2012	READ INSTRUCTIONS IN EACH PART! ______

Test #1 (100 points): Textbook 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.

	Part I: Fill-Ins 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 formal system, i.e., a ___________-based system in which answers can be obtained by mere ___________ 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.	The rule of ___________ allows us to think of concepts, or attack problems, in a number of different ways: graphically, numerically (i.e., through tables and calculator button-pushing), analytically (i.e., using algebraic manipulation), and ___________ . For example, the derivative can be thought of graphically as the ________________________________________________ , numerically as the one number that the ___________ ___________ appears to be approaching when the step size to the left and right is reduced again and again, and analytically by its definition, namely the two-sided ___________________________________________________________ .
3.	Give an informal verbal description of each of these terms, in your own words, except that you must use the word “instantaneous” in the second one: (a) limit: ____________________________________________________________________         _______________________________________________________________________ (b) derivative: __________________________________________________________________
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, . (a) In symbols: __________________________________ (b) In words: __________________________________________________________________ (c) Why would we never write as “All students do not love the calculus”?     _________________________________________________________________________

5.	The real numbers, have a property called density. To say that the real numbers are dense means, formally, that for any real numbers a and b, if then there exists a real number c such that a < c < b. Note: The compound inequality a < c < b can and should be thought of as the expression .
	(a) Rewrite the density definition in symbols only (no English words).     _________________________________________________________________________ (b) Write the negation of the definition in symbols, with the ~ pushed as far as it can go, and with     no “~” symbol, no use of the word “not,” and no negative expression whatsoever in the final     line of your derivation. Implications, if any, must be converted to “or” statements before     being negated. Show intermediate steps. You may not need as many lines as are given here.     _________________________________________________________________________     _________________________________________________________________________     _________________________________________________________________________     _________________________________________________________________________     _________________________________________________________________________     _________________________________________________________________________
	(c) Write the negation of the definition in words. Use informal or formal wording (your choice).     _________________________________________________________________________     _________________________________________________________________________
	Part II: Calculator Skills Fill in the blanks with the requested answers. Show work only for #8.
6.	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 0.25 units of sin^2 = 0.75. Answer: = ____________
7.	seems to be ______________
8.	Use a trapezoid rule approximation with 4 steps of size 0.75 to estimate the area under the curve y = f (x) = x^2 from 0 to 3, i.e., Note: Those of you who had Ms. Dunn last year, which is most of you, already know that the exact answer is 9. THAT IS NOT WHAT IS BEING ASKED. Use a trapezoid rule approximation with 4 steps of size 0.75. Show work, and circle your answer. Final answer must conform to AP rules, and yes, you are already supposed to know what that means. Write “OVER” if you need to continue on the back side.

	Part III. Free Response
9.	State the formal, correct definition of You may use words or symbols (your choice).
10.	Use the definition of derivative (not the power rule, not your calculator, and not any other shortcuts you may have learned) to compute where f (x) is the function y = f (x) = x^3. The work is what is graded here. The answer, as you will see in #11 below, is already known.
11.	Write the exact calculator keystrokes (every single one of them!) needed to get your calculator to compute the answer to #10, which is 27. (And no, you can’t write “2, 7, ENTER.”)
12.	Many calculators actually display something like 27.000001 instead of 27 in #11. Why?