1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.

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1 AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and the y are all bases to the eponent. When you take a power to another power, multiply the eponents. Answer: / / 4 /4 4 8 y. Also, ( ) 6 = 6 = = 8. a) You can only take the log of a positive number, not zero. Remember, in correct domain notation, brackets indicate inclusion. Answer: ( 6, ) b) Look at the denominator, factor it. ( 6)( + ), Answer: (, ) (, 6) ( 6, ) c) You can take to any power, once again, the denominator can t be zero.,0 0, Answer: ( ) ( ) d) Here s one where you have to look at both. The numerator can t be negative, the denominator can t be zero. The top s domain is 4.5 or greater and the denominator can t be Answer: [ 4.5, ) e) You must factor first, then make a sign chart. ( 7)( ) +. When is smaller than -, 5 4 is positive. When is in between - and 7, 5 4 is negative. When is larger than 7, 5 4, 7, is positive. Answer: ( ] [ ) f) The numerator of a fraction can be anything, so look to the denominator. The denominator can π π π π not be zero. Answer, from [ 0, π ) : 0,,, π. a) Answer: = ( ) = ( + 6)( 5) b) Look at this problem as two terms, factor out what is common from each term. ( ) ( ) ( ) ( ) + is common in both terms. If you take those out, you are left with + + which equals. Answer: ( ) ( + ) ( ) 0y + 50y = 0y + 5y = 5y 5y = 5y c) Answer: ( ) ( )

2 4. a) Factor, solutions are 4 and -4, make a sign chart. + ( 4 ) ( 4 ) + Also check to see if the inequality sign has a bar under it to determine if you use [ ] or ( ). Answer: (, 4) ( 4, ) b) Same procedure as 4a. Answer: (, 8) (, ) c) Subtract the 0 first. Then it is like 4a and 4b. This one is inclusive. Answer: [,5] d) Same as 4c. Answer:, 5. a) Down 4 b) Right 4 c) First reflect it in the -ais (upside down), then left (Remember, always stretch/reflect first) d) First vertically stretch by 5, then move up e) Horizontally shrink by e) All parts of the graph that are below the -ais need to be reflect up. No part of the graph should be below the -ais. = and = 0 7 b) Distribute AND subtract the, you must always have a 0 on one side of the equals sign = = + + = + + = 0, Answer: 6. a) ( 7 ) = 0, Answer: = and = c) You must use the Quadratic Formula. Answer: = 0.76 and = 5.6 d) This one doesn t factor and gives nonreal solutions. Answer: No solution = + 6 = + = 0, Answer: = and = e) f) Since fractions are involved, try to get common denominators, then use proportional rules. + + =, =, =, = 0, ( )( ) = Answer: = and = 8 = 0, Answer: = ± 8 and ± g) h) ( / )( / ) 9 = 0, Answer: = 8 and = = 6, = 6, 6 + = 0, + = 0, Answer: = and = i) ( )( ) 7. a) Vertical asymptotes are found at values that make ONLY the denominator zero. If a value of makes both the denominator and numerator zero, there is a hole. Horizontal asymptotes are also known as end behavior. Answer: VA: =, HA: y = b) For the horizontal asymptote, since the bigger eponent is in the denominator, the function will always approach zero. Answer: VA: = and =, HA: y = 0 c) Answer: VA: none, HA: y = 0 d) Factor before deciding either. Answer: VA: = 0 and = 6, HA: y = 0 ( ) ( ) + + y = = , at =, there is a hole. e) Since the largest eponent is in the numerator and denominator, the horizontal asymptote can be found by looking at the leading terms, =. Answer: VA: =, HA: y =

3 8. a) Since there is an in every term, you can cross one off. Answer: b) + = = + +, Answer: + (You can not simplify any further) c) y y y y ( + y)( y) y = =, Answer: + y y + y y ( + y) y y or y d) Answer: + or + e) You can t do anything to this one. Answer: a) g ( ) f ( ) =, = 9, Answer: 9 b) f ( ) g ( ) = 4, 4 = 7, Answer: 7 c) h( ) f ( ) = 0.5, 0.5 = 0.5, Answer: 0.5 d) h( ) f ( ) g ( ) 0.5 =, =, =, Answer: 0. a) Common denominators and proportion/ratio rules. 5 =, 4 5 =, =, = Answer: = 6 b) + = + = = + = + = ( )( ) = Answer: = and = , 5, 5, 6 5, 5 6 0, 0 c) With this, multiply all terms by 6, that would get rid of the fractions. + 6 =, ( + ) ( ) = 6, = 0, = 0, Answer: = d) This should be the easiest of the group, cross-multiply and solve. 5 5 = +, = 8 Answer: = 4

4 . a) KNOW HOW TO FACTOR. ( ) cos cos = 0, cos cos = 0, Answer: π π =,,0 b) Give eact answers when possible. cos =, Answer: 5π 7π = and 6 6 c) Don t forget the plus/minus sign. sin = ±, Answer: π 5π 7π π =,,, d) Always have a quadratic trinomial set equal to zero before you try to factor. sin + sin = 0, sin sin + = 0, sin = and sin =, π 5π π Answer: =,, 6 6 e) sin ( cos + ) = 0, Answer: π 4π = 0, π,, f) Factor first (difference of squares), ( )( ) sin cos sin + cos = 0. If we set each factor equal to zero, it s hard to solve when there are two trig. functions involved. But, if you use a little common sense, and your unit circle, it s not that hard. sin = cos and sin = cos. π π 5π 7π There are 4 places where sine and cosine are equal or opposites. Answer: =,,, Function notation, this information is the same as (, -5) and (-, ). Answer: you want to get rid of all fractions, and write it in standard form, 6 + 5y = 6 y =, or if 5 5. Point-slope needs to be used as much as possible. It is the easiest form b/c the only math involved is finding slope. Many people make mistakes when finding b, that doesn t happen in this form. Answer: y = ( 5) 4. Use the slope formula, Answer: ( ) ( ) = = Rate of change means slope. Answer: B, C, A y =, y + = y, y y =, y = y + 6. Switch all s and y s, then solve for y. ( ) = = Answer: y g ( ) 7. Two ways to do this, use your calculator or use algebra (substitution). Since y is solved for, = + 6, + = + 6, 4 5 = 0, substitute into the other equation for y. ( ) remember, always get zero on one side before factoring. ( )( ) Answer: (5, 4) and (-, -) 5 + = 0, = 5 and =,

5 8. KNOW YOUR UNIT CIRCLE Answers: a) b) or c) d) - e) Undefined f) 9. Piecewise functions are not as confusing as they look. They are just pieces of functions put together over a given domain, to make one function. Remember, all functions must pass the vertical line test, so keep that in mind. When is less than -, use the parabola 5, when = -, put a point at 0, when is greater than -, use the line 6 4. Also notice the open and closed circles. Answer: 0. You can plug numbers in, or look at the graph. Answer: Left f ( ), Right f ( ) 0. a) Parabola opening upward with verte (, ), Answer: Domain (, ), Range [, ) b) Shape of a V, with verte (4, -), Answer: Domain (, ), Range [, ) c) You can take the cube root of any number, Answer: Domain = Range = (, ) d) The 5 is the amplitude, Answer: Domain (, ), Range [ 5,5] e) A positive base can be taken to any power, but will never yield a negative answer., 0, Answer: Domain ( ), Range ( ). Area: 9π 8.74 Circumference: 6π (always round or truncate (cut off) to decimal places). Average of the bases times height, must solve for height first. + 5 Answer: cm = h = 6 = 7, 4. Use law of cosines to find the missing side, then use law of sines to find the missing angles. c = cos 0 = 6.59, make sure to store this answer for later use sin 0 = sin a = sin b, Answer: Missing side is 6.59, missing angles are and Subtract the two volumes, volume of a cylinder is π r h, Answer: 9.05 ft 6. Answer: ( + ) ln ln 5

6 7. The missing side is 40, find the area of the rectangle. A ( ) solutions. Answer: = ft or = ft 8. Answer: a) 6 elk b) years c) 6 elk = 40 = 550, this problem has 9. If you can t factor or see identities, change everything into sine and cosine. Answer: sin sin or sin csc sin or sin sin sin cos sin cos 0. Answer:. Put one side into y, and other into y, find the point(s) where they intersect. Answer: = Solve for y and look at their slopes, if they are different, they must intersect. Answer: Yes. For those that didn t take accelerated precalc, don t panic. We will spend the first month talking about limits (lim). All that is asking if for the behavior or y-value near the given -value. So, if lim f it is asking for the y-value just to the right of 4. you see ( ) 4 Answers a) 0 b) - c) and - d) e) 0 f) DNE 4. a) 0000 b).605 years 5. a) ( ) ln 9ln b) ln + ln e = ln + 6. ln + b) ln y 4 z 7. a) b) Can not do c) 0 d)