Name:______________________ Calculus AB: I. First Semester Exam Topics And Review Definitions and theorems you need to know: The Fundamental Theorem of Calculus Definition of Limit Definition of Derivative (f’(c) and f’(x) forms) Definition of Indefinite Integral Definition of Definite Integral Mean Value Theorem and Rolle’s Theorem 1. Given a function, f(x) = x2 + 1 i. Use a symmetric difference quotient with =.1 to estimate f   2  ii. Determine f   x  algebraically: iii. Determine f   2  iv. Write the equation of the line tangent to f  x  at (2,f (2)) v. Apply the Mean Value Theorem to f  x  on (1,4) vi. Integrate f(x) to find an equation for g(x). vii. Find a specific equation (no plus c) for g  x  if g  3  1 . viii. Find  f  x  dx using: 4 1 (f(x) = x2 + 1) a) Trapezoid rule by hand: Use 6 trapezoids b) Trapezoid rule on TI: Use 100 trapezoids c) Riemann Sums: Find U6, M6, L6 and also sketch a graph of U6 d) fnInt (use your TI device) e) Use the Fundamental Theorem of Calculus (integrate algebraically) II. Limits of functions. (know the props of limits) x 2  3x  4 x1 x3  x a. Evaluate: lim 4 x 2 1 b. Evaluate: lim , x x  2 x 2 4  x2 x 2 3  x2  5 c. Evaluate: lim d. Evaluate: (BE CLEVER!) 3 3  lim x  h  2 x  2 . h0 h e. Which of the following statements are true about the function f(x) whose graph is shown below? I. II. III. lim f ( x)  1 x 1 f(1) = lim f ( x) x 1 lim f ( x)  1 x 1 (a) I and II (d) I only (b) (e) I and III I, II and III (c) II and III 5  x 2  Let f(x) =  x  3   x2 f. x2 x2 Find the following values, if they exist. (a) f(0) (b) f(2) (c) (d) (f) (e) lim f ( x) x  2 lim f ( x) x  2 lim f ( x) x 5 lim f ( x) x 2 Is f(x) a continuous function? Explain why or why not. g. (a) 2 3 (b) Evaluate lim x 1 3 h. (a) 0 (b) 1 i. 2 2 x2  x  3 3x 2  x  2 (c) 1 Evaluate lim x 1 (d) 0 (e) None of these x 1 x 1 (c) 2 (d) Show numerically that lim h 0 1 2 sinh 1 h (e) None of these x 2  2x  15  x 3 x 2  x  20 k) 3x  5  x 3 m) j) lim l) lim x 3 n) lim x  x x lim  x 3 lim x  2 x  2 x o) lim x 8 3x  5  x 3 7 x2  4x 2  256 = x 8 2 p) Solve for the value of K which would make f(x) continuous @ x = 2. x 2 if x  2  f (x )    K  x if x  2 Derivatives i. ii. iii. iv. v. vi. vii. viii. Chain Rule Power rule Implicit differentiation Trig functions Product rule Quotient rule Inverse trig functions Use derivatives to make a piece-wise defined function both continuous and differentiable. For these problems, find dy and simplify. dx 1. y = 3 cos (3x + 1) 2. y = tan 1  x 3. y  x sin x2 4. y  1  cos2 x  cos2 x 5. y  sin x x2 7. y = x tan x 6. y = 3x + ln(sec x) 8. y = x Arcsec(x) 9. y  sin 4 x x 4 10. y = x 7 e 3x 11. y = x3 + 4x2 - 7x + 11 12. y = sec(3x2) 13. y= 15. cos (xy) = 2x2 - 3y 17. y = 1  x2 ( x  2)( x  4) x 1 14. 16. y = 3tanxsinx y = tan-1(x-4) 18. If y  A. E. III. 3 dy , then = 2 4 x dx 3 2x B. 3x 1  x  2 C. 2 6x 4  x  2 D. 2 6 x 4  x  2 2 3 4  x  2 2 Integrals: Antiderivatives, indefinite integrals, definite integrals i. Product rule ii. u du (composite functions) iii. trig functions 1. Which of the following properties of the definite integral is true? b b a a  k f ( x) dx  k  f ( x) dx, k is a constant I. b b a a  x f ( x) dx  x  f ( x) dx II. b III.  a A. I only c b a c f ( x) dx   f ( x) dx   f ( x) dx B. I and II only C. I and III only D. II and III only E. I, II, and III 2. The Mean Value Theorem guarantees the existence of a special point on the graph of y = x2 between (0, 0) and (2, 4). What are the coordinates of this point? (a) (2, 1) (b) (1, 1) (c) (¼, ¼) (d) ( 1 2 , 1 2 ) (e) None of these 3. Evaluate the integral:  3 2 (1  sin x)3 cos x dx 4. The velocity of a moving object is given by: v  t   8  3t 0.5 where v(t) is in meters per second and t is in seconds. a. Sketch a detailed graph of v(t). Label all key parts. b. Write an integral in proper form for D, the displacement of the object from time t = 1 to t=10, by doing an operation that adds up the dD values and takes the limit as dt approaches zero. c. Solve the integral in b. 5. Evaluate the integral: 7 a)  1 (3x c)  12e 3  4)4 x 2dx 3x dx 6. Use this function f(x): 7 b)  1 (3x d) 6 3  4)4 dx 2 x dx 2  if x  2 x , f  x   3  ax  b, if x  2 Find the values of a and b that make f (x) differentiable at x=2. 7 Use the curve y  x 2  x  6 . Find the AREA of the region bounded by the curve and the x-axis. (show integral and a sketch is nice) 8. For y  x 2  6 . Find the VOLUME of the solid when the section of the curve along the positive x-axis from y = -6 to y = 14 is rotated around the y – axis. (sketch helps) Integral:____________________________ Antiderv: __________________________ Answer:___________________________ 9. Find the VOLUME of the first quadrant region bounded by the y-axis, the line y = 4 and the graph y  x 3 is rotated around the x – axis. (sketch helps) Integral:____________________________ Antiderv: __________________________ Answer:___________________________ 10. RELATED RATES: a) The length of a rectangle is decreasing at 4 cm/sec. and the width is increasing at 3 cm/sec. When the length is 240cm. and the width is 100cm; is the diagonal increasing or decreasing? Explain your answer. b) Suppose a liquid is to be cleared of sediment by pouring it through a cone shaped filter. Assume the height of the cone is 16 centimeters and the radius at the base of the cone is 4 cm. If the liquid is flowing out of the cone at a rate of 2 cubic centimeters/min, how fast is the depth of the liquid decreasing when the level is 8 inches deep? c) A shuttle is rising straight up from Cape Canaveral. NASA is tracking the shuttle a range finder from a distance of 5,280 ft away from the lift-off point. The angle of elevation between range finder and the rocket is found to be  at a particular instant, and that angle is 4 increasing at 0.42 rad/sec. What is the rocket’s instantaneous rate of change at the moment?