AP Calculus Unit 1 Review Name __________________ Date __________ Block _____ s 1. Consider the circle x 2 + y 2 + 12 x − 4 y + 15 = 0. a. Find the center and the radius of the circle. Center: __________________ Radius: __________________ b. Find an equation of the line tangent to the circle at the point (−6, −3) Answer: _____________________ c. Find an equation of the line tangent to the circle at the point (−2,5) Answer: _____________________ d. Find the coordinates of the point where these two tangent lines intersect. Answer: _____________________ 2. Let point A = (-5, 9) and point B = (-1, -3). a. Find the length and the coordinates of the midpoint of segment AB . Simplify your answers. Length: __________________ Midpoint: __________________ b. Write an equation for the perpendicular bisector of segment AB . Express your answer in both pointslope and slope-intercept forms. Point-Slope Form: __________________ Slope-Intercept Form: __________________ 3. You are in a boat 2 miles from the nearest point on the coast. You are to go to a point Q located 3 miles down the coast and 1 mile inland. You can row at 2 miles per hour and walk at 4miles per hour. Write the total time T of the trip as a function of x Answer: _____________________ 4. Sketch the graph of each equation. Label the coordinates of all x and y intercepts and the equations of all asymptotes. Using proper notation, state the domain and range of each function. a. c. f ( x) = 6−2 x+2 b. f ( x) = − ln ( x + 2 ) Domain: _____________________ Domain: _____________________ Range: _____________________ Range: _____________________ f ( x) = 4 − x −1 d. f ( x) =−3 + 9 − x 2 Domain: _____________________ Domain: _____________________ Range: _____________________ Range: _____________________ 5. Using proper notation, state the domain and range of each relation. a. f ( x) =− 5 2 x3 + 5 x 2 − 14 x π b.= f ( x) 3csc x + 1 3 Domain: _____________________ Domain: _____________________ Range: _____________________ Range: _____________________ 6. Let f ( x= ) x 2 − 5 x and g ( x) =2 + x + 4 a. Find ( g f )(−4) b. Answer: _____________________ c. g Find ( x) and its domain. f Find ( f f )(1) Answer: _____________________ d. Find ( g f )( x) and its domain. Answer: _____________________ Answer: _____________________ Domain: _____________________ Domain: _____________________ 7. If= k ( x) sin 2 (2 x − 1) , find possible functions for f ( x) , g ( x) , and h( x) such that k ( x) = f ( g (h( x))) . Give two possible sets of functions. You may not use y = x for any of the functions. f ( x ) = _____________________ f ( x ) = _____________________ g ( x ) = _____________________ g ( x ) = _____________________ h( x ) = _____________________ h( x ) = _____________________ 8. Simplify each expression. 36 = _________ 3 x 2 = _________ 27 = _________ 9. Rewrite each function as a piecewise function without absolute value. a. f ( x) =4 + 2 x − 7 b. f ( x) = x + x + 5 10. Find the coordinates of all points of intersection of the graphs. = x 32 − y 2 y= x + 10 Answer(s): ____________________________________ 3 x3 = _________ x 11. Let f ( x) be the function whose graph is shown below. Sketch the graph of − f − 4 − 5 . Calculate the 2 transformed coordinates of points A, B, and C and label them on the graph. A = (-4, 1) C = (3, 0) B = (0, -3) 12. Use number line analysis to solve the polynomial inequality. Express your answer in interval notation. 3x3 − 7 x 2 ≥0 ( x + 2)3 Answer: ____________________________________ 13. Find the difference quotient and simply your answer. f ( x) =−4 x 2 + x − 5 Answer: ____________________________________ 14. The number of teeth of an average child is a linear function of the age of the child in months. According to published reports, an average child that is six months old has no teeth. An average child that is 30 months old has 20 teeth and has just completed growing his first set of teeth. a. Define the variables in this problem and write coordinate points for the data given. Find the linear equation (in slope-intercept form) that expresses the relationship between these variables. Answer: _____________________ b. Determine the units of the slope and explain what the slope means in the context of the problem. Units: _____________________ c. Use your equation to predict the average age of a child that has 5 teeth. Answer: _____________________ 15. Determine the limit based on the information given in the table below. x 1.75 1.9 1.99 1.999 2 2.001 2.01 2.1 2.25 f ( x) 44 290 29,900 2,999,000 ∅ 2,999,000 29,900 290 44 lim f ( x ) = __________ x→2 16. Determine the limit based on the information given in the table below. x 4 4.5 4.9 4.99 5 5.01 5.1 5.5 6 f ( x) 11 12 12.8 12.98 ∅ 13.02 13.2 14 15 lim f ( x ) = __________ x→5 17. Determine the symmetry of f ( x) = 2 x csc x . Use the algebraic tests to justify your answer. x5 cos( x) Answer: _____________________ 18. The product of two odd functions is _____________________. 19. The sum of two odd functions is _____________________. 20. Consider the function whose graph is shown below. a. State the values of x at which f(x) is discontinuous and name the type of discontinuity. b. Write the equations of all horizontal and vertical asymptotes of the graph. c. Find each limit or value based on the graph shown above. i. ii. iii. iv. lim f ( x) = __________ v. lim f ( x) = __________ ix. lim f ( x) = __________ __________ f (−3) = vi. f (0) = __________ x. f (2) = __________ lim f ( x) = __________ vii. lim f ( x) = __________ xi. __________ f (−2) = viii. f (1) = __________ xii. x→−3 x →−2 x →0 x →1 x→2 lim f ( x) = __________ x →−∞ lim f ( x) = __________ x →∞ − x 2 − 4 x 21. Consider the piecewise function = f ( x) 3 −2 x + 7 domain D= x≤0 0 < x ≤ 2 . Sketch the graph of the function over the 2<x≤5 { x : −5 ≤ x ≤ 5} . List all points of discontinuity and state the domain and the range. Discontinuities: ________________ Domain: ____________________ 22. Sketch the graph of one function with all of the following characteristics. lim f ( x) = 3 x → 2− lim f ( x) = 0 x → 2+ f (2) = 0 23. Sketch the graph of one function with all of the following characteristics. lim f ( x) = ∞ x → 0− lim f ( x) = −∞ x → 0+ f (0) is undefined 24. Sketch the graph of one function with all of the following characteristics. lim f ( x) = 0 x →∞ lim f ( x) = 0 x →−∞ lim f ( x) = −4 x →0 f (0) = 0 Range: _________________