SAT2数学考试题目：Making Your Calculator Work for You

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　　As we’ve already mentioned, the calculator is a very important part of the Math IC test. You need to have the right kind of calculator, be familiar with its operations, and, above all, know how to use it intelligently.

　　There are four types of questions on the test: those that are calculator-friendly, calculator-neutral, calculator-unfriendly, and calculator-useless. According to ETS, about 60 percent of the test falls under the calculator-neutral and -friendly categories. That is, calculators are useful or necessary on 30 of the 50 questions on SAT II Math IC. The other 20 questions are calculator-unfriendly and -useless. The trick is to be able to identify the different types of questions when presented with them on the test. Here’s a breakdown of each of the four types, with examples. If you’re not certain about the math discussed in the examples, don’t worry. We cover all these topics in this book.

　　Calculator-Friendly Questions

　　A calculator is extremely helpful and often necessary to solve calculator-friendly questions. Problems demanding exact values for exponents, logarithms, or trigonometric functions will most likely need a calculator. Computations that you can’t do easily in your head are prime candidates. Here’s an example:

　　If f(x) = , then what is f(3.4)?

　　(A) –18.73

　　(B) –16.55

　　(C) –16.28

　　(D) –13.32

　　(E) –8.42

　　This is a simple function question in which you are asked to evaluate f(x) at the value 3.4. As you will learn in the Functions chapter, all you have to do to solve this problem is plug in 3.4 for the variable x and carry out the operations in the function. But unless you know the square root and square of 3.4 off the top of your head (which most test-takers wouldn’t), this problem is extremely difficult to answer without a calculator.

　　But with a calculator, all you need to do is take the square root of 3.4, subtract twice the square of 3.4, and then add 5. You get answer choice C, –16.28.

　　Calculator-Neutral Questions

　　You have two choices when faced with a calculator-neutral question. A calculator is useful for these types of problems, but it’s probably just as quick and easy to work the problem out by hand.

　　If 8x = 43 23, what is the value of x?

　　(A) 2

　　(B) 3

　　(C) 5

　　(D) 7

　　(E) 8

　　When you see the variable x as a power, you should think of logarithms. A logarithm is the power to which you must raise a given number to equal another number, so in this case, we need to find the exponent x, such that 8x = 43 23. From the definition of logarithms, we know that if given an equation of the form ax = b, then loga b = x. So you could type in log8 (43 23) on your trusty calculator and find that x = 3.

　　Or, you could recognize that 2 and 4 are both factors of 8, and, thinking a step further, that 23 = 8 and 43 = 64 = 82. Put together, 43 23 = 82 8 = 83. We come to the same answer that x = 3 and that B is the right answer.

　　These two processes take about the same amount of time, so choosing one over the other is more a matter of personal preference than one of strategy. If you feel quite comfortable with your calculator, then you might not want to risk the possibility of making a mental math mistake and should choose the first method. But if you’re more prone to error when working with a calculator, then you should choose the second method.

　　Calculator-Unfriendly Questions

　　While it’s possible to answer calculator-unfriendly questions using a calculator, it isn’t a good idea. These types of problems often have built-in shortcuts—if you know and understand the principle being tested, you can bypass potentially tedious computation with a few simple calculations. Here’s a problem that you could solve much more quickly and effectively without the use of a calculator:

　　(A) .3261

　　(B) .5

　　(C) .6467

　　(D) .7598

　　(E) .9238

　　If you didn’t take a moment to think about this problem, you might just rush into it wielding your calculator, calculating the cosine and sine functions, squaring them each and then adding them together, etc. But take a closer look: cos2(3 63°) + sin2(3 63°) is a trigonometric identity. More specifically, it’s a Pythagorean identity: sin2q + cos2q = 1 for any angle q. So, the expression {cos2(3 63°) + sin2(3 63°)} 4/2 simplifies to 14 /2 = 1/2 = .5. B is correct.

　　Calculator-Useless Questions

　　Even if you wanted to, you wouldn’t be able to use your calculator on calculator-useless problems. For the most part, problems involving algebraic manipulation or problems lacking actual numerical values would fall under this category. You should be able to easily identify problems that can’t be solved with a calculator. Quite often, the answers for these questions will be variables rather than numbers. Take a look at the following example:

　　(x + y – 1)(x + y + 1) =

　　(A) (x + y)2

　　(B) (x + y)2 – 1

　　(C) x2 – y2

　　(D) x2 + x – y + y2 + 1

　　(E) x2 + y2 + 1

　　This question tests you on an algebraic topic—that is, it asks you how to find the product of two polynomials—and requires knowledge of algebraic principles rather than calculator acumen. You’re asked to manipulate variables, not produce a specific value. A calculator would be of no use here.

　　To solve this problem, you need to notice that the two polynomials are in the format of a Difference of Two Squares: (a + b)(a – b) = a2 – b2. In our case, a = x + y and b = 1. As a result, (x + y – 1)(x + y + 1) = (x + y)2 – 1. B is correct.

　　Don't Immediately Use Your Calculator

　　The fact that the test contains all four of these question types means that you shouldn't get trigger-happy with your calculator. Just because you've got an awesome shiny hammer doesn't mean you should try to use it to pound in thumbtacks. Using your calculator to try to answer every question on the test would be just as unhelpful.

　　Instead of reaching instinctively for your calculator, first take a brief look at each question and understand exactly what it's asking you to do. That short pause will save you a great deal of time later on. For example, what if you came upon the question:

　　If (3, y) is a point on the graph of f(x) = , then what is y?

　　(A) –3

　　(B) –1.45

　　(C) 0

　　(D) .182

　　(E) 4.87

　　A trigger-happy calculator user might immediately plug in 3 for x. But the student who takes a moment to think about the problem will probably see that the calculation would be much simpler if the function was simplified first. To start, factor 11 out of the denominator:

　　Then, factor the numerator to its simplest form:

　　The (x – 4) cancels out, and the function becomes f(x) = (x – 1) ? 11. At this point you could shift to the calculator and calculate f(x) = (3 – 1) ? 11 = 2/ 11 = .182, which is answer D. If you were very comfortable with math, however, you would see that you don't even have to work out this final calculation. 2?11 can't work out to any answer other than D, since you know that 2?11 isn't a negative number (like answers A and B), won’t be equal to zero (answer C), and also won't be greater than 1 (answer E).

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