(D) TRUE. The limit exists because the function approaches the same value from the left and right at 5.
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1 AP Practice Test # Part I: Multiple Choice (no calculator) 1) If y = x sin x, then dy/dx = A) sin x + cos x B) sin x + xcos x C) sin x xcos x D) x(sin x + cos x) E) x(sin x cos x) Answer B ) If f (x) = 7x 3 + ln x, then 1 A) 4 B) 5 C) 6 D) 7 E) Answer E 3) The graph of function f is shown. Which of the following statements is false? (E) The function f is continuous at x = 3. Comments: (A) TRUE. The limit exists because the function approaches the same value from the left and right. Note that the value of the function is different than the value of the limit. So, as an aside, we note that the function is not continuous at. (B) TRUE. The limit exists because the function approaches the same value from the left and right at 3. (C) FALSE. The limit does not exist because the function approaches different values from the left and right at 4. (D) TRUE. The limit exists because the function approaches the same value from the left and right at 5. (E) TRUE. The function is continuous because a) the function approaches the same value from the left and right at 3 and b) that limit is the value of the function at 3. Answer C Page 1 of 11
2 AP Practice Test # 4) If y = (x 3 cos x) 5, then y = Answer E for 5) for Let f be the function defined above. For what value of k is f continuous at x =? A) 0 B) 1 C) D) 3 E) 5 at all values of To get the value of at for which the function is continuous, calculate the limit as. lim lim 1 lim 1 1 Answer E Page of 11
3 AP Practice Test # 6) If 4 and 3, then the derivative of f (g (x)) at x = 3 is Use the chain rule: First, find : Then, find : 3 3 Finally, substitute 3 into the chain rule expression for: Answer A 7) Notice that the expression is the definition of a derivative: ln4ln 4 lim ln Answer B Page 3 of 11
4 AP Practice Test # 8) The function f is defined by. What points (x, y) on the graph of f have the property that the line tangent to f at (x, y) has slope of? Recall that the slope of the tangent line of a function at a point is equal to the derivative of the function at that point. Then, 1 1 The problem tells us that this is equal to a slope of. So, ,4 To get the points, substitute the values into to get the points:,,, Answer C 9) The line y = 5 is a horizontal asymptote to the graph of which of the following functions? Horizontal asymptotes occur at the limits as and as. Let s try limits as. (A) lim 0 (B) lim 5 (C) lim (D) lim 0 (E) lim lim 5 using L Hospital s Rule using L Hospital s Rule Answer E Page 4 of 11
5 AP Practice Test # 10) If f ( x) 16 x then f "(4) is equal to A (B) -3 (C) -4 (D) - (E) Answer A 11) If,what is the value of at the point 3, 0? Notice that, at the point (3, 0), 1 Now, substitute in,, Answer A Page 5 of 11
6 AP Practice Test # 1) d e sin x dx A (B) (C) cos (D) cos (E) sin sin cos Answer D 13) If x dy y, then 3x 1 dx = 9 (A) (3x 1) 7 (B) (3x 1) 6x 5 (C) (3x 1) 7 (D) (3x 1) 7 6x (E) (3x 1) Answer B Part II: Graphing Calculator is Allowed 14) Let f be a function that is continuous on the closed interval [, 4] with f () = 10 and f (4) = 0. Which of the following is guaranteed by the Intermediate Value Theorem? A) f (x) = 13 has at least one solution in the open interval (, 4). TRUE since 13 is between 10 and 0. One way to think about it is that to get from 10 to 0, the function must go through 13. Answer A B) f (3) = 15 FALSE. 3 can be anything, including values below 10 or above 0. C) f attains a maximum on the open interval (, 4). FALSE. may have a maximum at 4. For example, consider the line connecting (, 10) and (4, 0). It has a maximum at 4, not in the open interval (, 4). D) f (x) = 5 has at least one solution in the open interval (, 4). TRUE, but this is not required by the Intermediate Value Theorem, which deals with the values of the function, not the values of the derivative. E) f (x) > 0 for all x in the open interval (, 4). FALSE. The curve could slope downward in some sub interval between and 4. Intermediate Value Theorem: If a function,, is continuous on the interval,, and is a value between and, then there is a value in, such that. Page 6 of 11
7 AP Practice Test # 15) 15 ft Let: the distance of the person from the streetlight the length of the shadow 6 ft x y Using the above drawing, we are looking for when 4 ft/sec. Comparing the small rectangle to the large triangle, from Geometry: Note that: So, / Answer B 16) A particle moves along the x-axis so that its position at any time t (in seconds) is acceleration of the object at t = seconds is: st () t 3t 5. The (A) 19 units/s (B) 11 units/s (C) 16 units/s (D) 4 units/s (E) 0 units/s Acceleration is the second derivative of the position curve (the first derivative is velocity) Note that the acceleration is the same for all values of. So, / Answer B Page 7 of 11
8 AP Practice Test # Free Response: Part I: No Calculator x1, x 17) (1981 AB) Let f be defined by f( x) 1 x k, x. (A) For what value(s) of k will f be continuous at x =? Justify with the definition. (B) Find the average rate of change of the function on the interval [1, 4]. (C) Find the instantaneous rate of change of the function at x = 4. (A) Continuity: A function,, is continuous at iff: a. is defined, b. lim exists, and c. lim In the problem given, this boils down to:. When : 1 becomes: 5 (B) The average rate of change is the slope of the line connecting the endpoints of the interval: 1: : Slope (C) The instantaneous rate of change at 4 is the slope of the curve at 4. For : 1 3, Page 8 of 11
9 AP Practice Test # Part II: Calculators may be used. 3 18) Let f be the function defined by the equation f( x) x x 6x. (a) Find the equations of the lines tangent and normal to the graph at the point,. (b) Find the equation(s) of the line(s) tangent to the graph of f and parallel to the line 73. (c) At what value of x, if any, is the tangent line horizontal? (A) Find the equations of the lines tangent and normal to the graph at the point, Tangent Equation is: (using point slope form) Normal Equation is: slope) (normal slope is opposite reciprocal of tangent Page 9 of 11
10 AP Practice Test # (B) Find the equation(s) of the line(s) tangent to the graph of and parallel to the line 73. We want the points where the slope of the curve is (i.e., the slope of 73) Use the quadratic formula to determine that:.,. so the points are:.,... and.,. Note: to get the values of the points, substitute the values into the original equation: 6 Tangent Equation 1 (point slope form):.. Tangent Equation (point slope form): (C) At what value of x, if any, is the tangent line horizontal? The tangent line is horizontal when the derivative is zero Using the quadratic formula, ,. Page 10 of 11
11 AP Practice Test # 19) (a) (b) So, the tangent line has slope Tangent Equation is: and goes through the point 3, 4. (using point slope form) (c) Continuity: A function,, is continuous at iff: a. is defined, b. lim exists, and c. lim (from above) lim 5 4 lim 74 left and right limits Since the limits from the left and right are the same at and are equal to the value of, which exists, we conclude that is continuous at. Page 11 of 11
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