AP Style Problem 1. The table below shows some points on a function f that is both continuous and differentiable on the closed interval [0, 20] 5 15 20 10 f(x) 10 22 40 22 10 A) f(x) is increasing in 0 0 C) f has at least one zero in the interval [0, 20] D) The maximum value of fon the interval [0, 20] is 40. E) For some value of x on the interval (0, 20), f'(x) = 0 2. Let f be a function that is differentiable on the open interval [1, 9]. If f( 1 ) = -3, f(4) = 3, and f(9) = -3, which of the following must be true? I. fhas at least two values of x such that f(x) = 0. II. The graph of f has at least one horizontal tangent. III. For some c, f(c) = 2 in the interval [4, 9]. A) I and II only B) II only C) III only D) I only E) I, II, and III3. Approximately how much less than 3 is 28) ? A ) I B ) 3 f(x ) = 3/ X ( 27 , 3 ) C) sip 3 3 X 27 D) 7 E) . y - 3 = 27 (X- 27) 28 4. If the radius of a circle increases from 5 to 5.2, what is the change in area using a linear approximation? A ) TO B) 27 C ) 0.2 D) 0. 1 5. Let f(x) be twice-differentiable function and f"(x) > 0 for all x. The graph of y = T(x) is the line tangent to the graph of fat x = 4. Which of the following is true? A) I ( 4. 2 ) = f(4.2) B ) I ( 4. 2 ) f ( 4 .2 ) D) none of theseEXAMPLE 11. Estimate the arctan (0.9) using a linear approximation. EXAMPLE 12. Estimate (4.2) using a linear approximation. Is the approximation overestimate or underestimate ? EXAMPLE 13. Use differentials to approximate v9.2 . Is the approximation overestimate or underestimate?AP Style Problem y = f' ( x ) - 3 - 2 1. The graph of f' is shown above. If we know that f(0) = 10, then the linearization for f(x) near x = 0 is A) x + 1 B) x - 1 C) x + 10 D) x - 10 E ) 1 2. Let f be a differentiable function such that g(2) = 3 and g'(2) = 6. If the tangent line to the graph of g at x = 2 is used to find an approximation to a zero of g, that approximation is A) 0 B) 0.5 C ) 1.5 D ) 7.5 E) 96. Suppose you have a function f(x) and all you know is that f(3) = 37 and the graph of its derivative f' is shown. yA 4 2 2 4 6 x Use linear approximation to estimate f(3.1). Is the approximation overestimate or underestimate? A) 36.8, underestimate B) 37.2, underestimate C) 37.1, underestimate D) 37.2, overestimate E) 36.8, overestimateEXAMPLE 7. The table below shows selected values of a differentiable function f. For [O, 10], 0 3 4 6 7 9 10 f(x) -2 3 - 2 0.5 1 1 what is the fewest possible number of zeros? IVT 3 2 what is the fewest possible number of x where f(.x) = 2? 2 Show that there exists at least one number c, 0 s x s 10, such that f (c)= 0 9 Answer: F is According to there must be at least one number c such that Show that there exists at least one number c, 0 0 E) none of these 4. Let f' be a continuous function, and the values of f' at selected values of x is given in the table below. Which of the following must be true for OS.*$10? 0 2 5 10 f (x) -10 -3 2 4 10 I. f ' is increasing in 0 S x - 10 II. There exists c, for 0 S c