South County Secondary School AP Calculus BC Summer Assignment For students entering Calculus BC in the Fall of 8 This packet will be collected for a grade at your first class. The material covered in this assignment will be tested sometime during the first three weeks of school. Please do not wait until the last minute to begin the assignment or to receive clarification of the assignment. During the end of August, teachers will be busy preparing for the upcoming school year and may not be able to respond to your last minute questions. You are now an AP student. Lesson # - Do not procrastinate!
FUNCTIONS. Given f() = - + - Sketch a. Domain: b. Range: c. f() = d. f(+5) = e. f() = - then =. a. Graph the piece-wise function, g, 4, 4 b. g(-) = c. g() = d. g() = e. Is g() a continuous function?. Given the graph of g() on the right a. Estimate g(6) - g() = 6 - b. The ratio in part (a) is the slope of a line segment joining two points on the graph. Sketch this line segment.
4. The rate at which water is entering a tank ( t > ) is represented by the given graph. A negative rate means that water is leaving the tank. State the interval(s) on which each of the following holds true: a. The volume of water is constant. b. The volume of water is decreasing. c. The volume of water is increasing. d. The volume of water is increasing fastest. 5. Q() a. Where is this function discontinuous? b. State the equation of the vertical asymptote = c. State the equation of the horizontal asymptote y = d. Sketch the graph. e. Write the equation of the inverse of Q() (Switch the & y and then rewrite as y =) 6. Use these functions: f ( ) g( ) h( ) 5 7 a) h b) g(5) c) f ( ) 4 h( ) d) g( ) e) gh ( ()) f) g( f ( )) 7. Simplify (no negative eponents, a single fraction): a) 9 7 b) 8 8. Rationalize the denominator: 4 a) b) 5 c) 5 5 d) 9
9. Simplify f h f h, where a) f b) f f c). The graph of the function y = f() is given as follows: Determine the graphs of the functions: a) f b) f c) f d) f TRIGONOMETRY What you need to know: Trig functions and inverse trig functions for all special angles (unit circle) Fundamental trig identities (reciprocal, quotient, Pythagorean) Graphs of sin, cos, tan Domain and range of sin, cos, tan How to solve trig equations. Evaluate without use of a calculator. (a) tan( 6 ) (b) cos( ) (c) sin() (d) csc( ) (e) sin - (f) cos - (g) arctan(- ) (h) sec - (-) (i) arcsin(- ) (j) sin(cos -.6). Sketch the following graphs without the use of a calculator. Show at least two periods and mark your aes. a) sin b) sin c) sin d) e) y tan 4 sin 4 4
SIMPLIFY COMPLETELY: EXPONENTS 8 *4 cd 6 cd 4. y.. 7 5 *9 4. 5. 7 5 4 6. * * 7. 7 8. 9. LOGARITHMS & EXPONENTIALS Solve for without a calculator. (#8 & 9 are calc)... log 4. log8 = 4. log(-9) = 5. log6 = -4 6. log log 4 4log 5 7. = 8..4 =.84 9. ln(+5)=- ln(-)-ln(+) SOLVING EQUATIONS. Solve the following equations for the indicated variable. a) y z, solve for a a b c c) V ab bc ca, solve for a b) A r rh, solve for positive r 5
SOLVE FOR X.. 5 9. 6. 4 7. cos. 8. 5 e 4.. ln ln 5. 4t t 8t 4. sec 4 6. 4 4 4. sin cos 7. 5. 4 solve by completing the square 8. e 7e Determine the equation of the following lines (leave in point-slope form). 6. line through (-, ) and (, -4) 7. the line through (-, ) and perpendicular to the line y 5 8. The equation has a solution =. Find all other solutions. 6
LIMITS The limit of a function is the y-value that you are getting close to as gets close to some number in the domain. In the limit process you never get to the limit, ecept for the limit of a constant function. We write lim f ( ), which is read the limit of f() as approaches a domain value a of a. The limit must be the same as approaches a from both the left and the right. To find the limit, substitute in values very close to a on both left and right and see if the y-value is approaching a single value. The limit does eist at a hole in a graph, but does not eist at a vertical asymptote or a jump in the graph. Limit eist at No limit at The graphs of some functions are pictured below. Do you think that limit does eits, state its value. lim f( ) eists? If you think the No limit at. f. f 4 4. 4. f 4 4 7
State the value of each of the following: 5. lim 5 4 6. lim 7. 8 lim 8. lim 9 9. sin lim. lim DERIVATIVES Find the derivative using the definition of derivative. f ( ) f ( ) f '( ) lim. f() =. f() =. f() = / Find the derivatives using the power rule. 4. f() = 5. f() = 6. f() = 6-7. f() = 4/ 8. f() = -4 9. f 5 8
Word Problems ) Epress in terms of the other variable in the picture. ) a) Find the ratio of the area inside the square but outside the circle to the area of the square in the picture (a) below. c) Find a formula for the perimeter of a window of the shape in the picture (b) above. ) A water tank has the shape of a cone (like an ice cream cone without ice cream). The tank is m high and has a radius of m at the top. If the water is 5m deep (in the middle), what is the surface area of the top of the water? 4) Two cars start moving from the same point. One travels south at km/hour, the other west at 5 km/hour. How far apart are they two hours later? 5) A kite is m above the ground. If there are m of string out, what is the angle between the string and the horizontal? (Assume that the string is perfectly straight). 9