AP CALCULUS BRAIN CAMP

1 AP CALCULUS BRAIN CAMP Part A: Basic graph shapes Review the following terminology. Some might be new to you. a) Is it a function? Is the inverse a ...
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1 AP CALCULUS BRAIN CAMP Part A: Basic graph shapes Review the following terminology. Some might be new to you. a) Is it a function? Is the inverse a function? Is it 1:1? b) domain, range, interval notation c) x intercepts, y intercepts d) symmetry around x axis, y axis, origin e) odd, even, or neither f) continuous, not continuous g) increasing, decreasing, neither h) concave up, concave down, neither Part A Problems. For the following, sketch the graph without using a graphing calculator, then describe it using the above terminology. Always use radians when doing calculus. Check the results using your calculator. 1) y = x

2) y = x2

3) y = x3

10) y = log(x)

11) y = ln(x)

12) y = ƒx„

13) y = sin(x)

14) y = tan(x)

15) y = csc(x)

16) y = arcsin(x)

2 Part B: Transformations, stretches, shrinks, reflections Review how changes in the equation causes changes to the graph. What effect is there on the graph of y = f(x) if you make these changes to the equation? a) y = f(x) + k b) y = f(x + k) c) y = cf(x) d) y = f(cx) e) y = -f(x) f) y = f(-x) g) y + k = f(x) h) cy = f(x) i) -y = f(x) Part B Problems. For the following, sketch the graph using what you know about translations, stretches, shrinks, and reflections. Don’t use a graphing calculator. Then give the new domain and range. Check the results using your calculator. 1) y = -x

2) y = x - 3

3) y = 2x

4) y = 2x - 3

5) y = (x + 4)2

6) y = x2 - 4

7) y = -x2

8) y =(-x)2

9) y = x3 + 2

10) y = (x + 5)3

21) y = -log(x) 22) y = log(-x)

23) y = 6sin(x)

24) y = -sin(x)

Notice in problem 26) that the changes are applied in the opposite order you do things when evaluating.

3 Part C: Further graph analysis Review what a vertical asymptote is and what a hole is, how to locate these from the equation, how to tell what the graph is doing on both sides of a vertical asymptote, and how to predict “end behavior.” Part C Problems. These problems are harder to do by transformations, stretches, shrinks, or reflections of basic types. Instead, do the following: a) Calculate the intercepts (x and y). b) Predict any symmetry the graph must have. c) Find any vertical asymptotes and holes. d) Predict the behavior of the graph on both sides of any vertical asymptotes. e) Predict the end behavior on both sides. f) Use all this to sketch the graph, then check the results using your calculator.

4 Part D: Finding equations of graphs Review slope, equations of vertical and horizontal lines, SI formula, PS formula, parallel and perpendicular lines, finding the equation of the inverse, and regression. Part D Problems. Do these problems from the book. 9:5-36,40a,41-53,56,57. On 5-8 do part b only. 19:1-4,13-20,31-34 using a calculator, 37-48,54 If time permits, also do the following problems from the book. 19:5-12,21-30,49,50

5 Part E: Composition of functions Review f(x) notation, composition of functions. Part E Problems. For 1) - 30) use the following formulas:

1) - 10) Find the value. 11) - 20) Simplify as much as possible. 21) - 30) Write a composition of the functions that would make the given result. 1) k[f(-1)]

2) f[k(-1)]

3) p[r(7)]

4) g[f(6)]

5) r[k(-8)]

6) k[r(-8)]

7) m[p(27)]

8) p(g(m(343)))

9) k(f(g(-6)))

10) f(g(k(r(-1))))

11) g[k(x)]

12) f[k(x)]

13) k[f(x)]

14) f[p(x)]

15) p[p(x)]

16) f[g(x)]

17) g[f(x)]

18) m[r(x)]

19) r[m(x)]

20) g(f(m(x)))

31) - 40) For each problem find a group of functions and a composition of them in the right order which would make that result.

6 Part F: Rates of change Differential calculus measures how rapidly something is changing. This is all based on the slope formula. If 2 points have coordinates (x, f(x)), (x + h, f(x + h)), then the slope of the line joining them is given by the formula

Part F Problems. For each of these, simplify the expression possible.

as much as

7 Part G: Exponents Review the properties of exponents, exponential growth and decay, exponential regression, and e as a limit of

.

Part G Problems. Do these problems from the book. 26:1-32,38-40

Part H: Logarithms Review the properties of logarithms, how to solve exponential equations, and how to solve logarithmic equations. Part H Problems. Do these problems from the book. 43:1-24,33-50,54-57

Part I: Trigonometry Review the arc length formula and the x, y, r definitions. Note: if you use degrees to work out a problem, convert any angles to radians before doing calculus with them. The calculus formulas involving trigonometry are all based on the argument being in radians, not degrees. Part I Problems. Do these problems from the book. QR 51:1-6 51:1-22,24-43,49-54

Part J: The method of exhaustion Integral calculus is based on the idea of adding up a number of things, especially an infinite number of things. In particular, areas of irregular shapes can be approximated by totaling a set of simple geometric shapes that together must be less than the area you are looking for, then totaling up another set of geometric shapes that must be more than the are you are looking for. The exact answer must be somewhere between the two totals. If you use more and more shapes in each total, the two separate answers usually get closer and closer to each other, so you get a better and better approximation for what the exact answer might be. Another example is the calculation of lengths of curved lines. They can be approximated by totaling the lengths of a group of line segments which must be less than the length you are looking for, then totaling the lengths of another group of line segments which must be less than the length you are looking for. The exact answer must be somewhere between the two totals. If you use more and more line segments in each total, the two separate answers usually get closer and closer to each other, so you get a better and better approximation for what the exact answer might be.

8 Calculus allows you to deduce what the exact answer would be when the number of line segments becomes infinite, but this method has been used for thousands of years to get very good approximate answers for large number of segments. It is how ancient mathematicians were able to figure out very accurate values for ð. In the problems below you will get some idea of how this was done. According to An Introduction to the History of Mathematics, Third Edition by Howard Eves (Holt, Rinehart, and Winston, copyright 1969), pages 89-90, around 240 B.C. the Greek mathematician Archimedes calculated that ð was between 223/71 and 22/7 by calculating the perimeters of regular polygons inscribed in a circle and circumscribed around the same circle. Eves does not spell out all the steps in the process Archimedes used, but he states that using regular polygons of 6, 12, 24, 48, and 96 sides would give you an idea of what Archimedes did, but the answers you will get are not exactly the same as his. Eves further states (on pages 154-155) that the first known tables similar to trig tables were created around 140 B.C. by the Greek mathematician Hipparchus. So Archimedes would not have had access to trig tables. However, he would have known geometry facts about ratios of sides in special right triangles, how to calculate approximate values of square roots, and geometry facts equivalent to the trig identities you have studied. Part J Problems. You may use your calculator instead of doing the arithmetic by hand the way the ancient Greeks had to. However, you may not use the trig keys on it. You may use the trig identities you have studied (instead of the other math formulas the Greeks would have used instead). Use n to stand for the number of sides of the regular polygon. Use è to stand for the central angle intercepting one side of the polygon. Use r for the radius of the circle. 1) Show that the perimeter of the inscribed polygon is given by the formula p = 2nr sin(è/2) or p = 2nr sin(180E/n). 2) Show that the perimeter of the circumscribed polygon is given by the formula p = 2nr tan(è/2) or p = 2nr tan(180E/n). 3) Use geometry to find the exact answers (in decimal form) for the sin, cos, and tan of 30E. Then use trig identities to find exact answers for the sin, cos, and tan of 15E. Then use trig identities to find exact answers for the sin, cos, and tan of 7.5E. Then use trig identities to find exact answers for the sin, cos, and tan of 3.75E. Then use trig identities to find exact answers for the sin, cos, and tan of 1.875E. 4) Calculate the perimeters of the regular polygons with 6 sides, 12 sides, 24 sides, 48 sides, and 96 sides, both inscribed and circumscribed, in a circle with diameter of 1. What do you notice about the answers you are getting?