Multiple Choice Review Problems

Transcription

1 Multiple Choice Review Problems 1. (NC) Which graph best represents the position of a particle, st ( ), as a function of time, if the particle's velocity is negative and the particle's acceleration is positive? A) B) C) D) x 4 100x 39. (NC) If f ( t) dt =, then f ( t) dt is 5x ( A) ( B) ( C) ( D) 4 divergent

2 ( x) 3. (NC) Let f be a continuous function with the following values: f x 1 1 ( x) ( ) < > ( ) ( x) f ( x) ( ) = ( ) If f x 0 for all x 1 and f x is increasing for all x, then which of the following is possible? ( A) f is an odd function. ( B) is an odd function. ( C) f x 0 for all x 1, 1 ( ) ( D) f x dx = (NC) Let f be the function defined by f( x) =. x 4x+ What is the maximum value of f on the interval -1 x 3? ( A) ( B) ( C) ( D)

3 4 5. (NC) Let f be the function defined by f( x) =. x 3 x+ 7 What is the value of f ( x) dx or does the integral diverge? ( A) ( B) ( C) ( D) ln ln1 ln1 Diverges 6. (NC) At time t = 0, a piece of heated granite is placed in a container of water. o The internal temperature of the granite is 00 C at time t = 0. The internal temperature of the granite at time t minutes is modeled by the function dh 1 Ht ( ) that satisfies the differential equation = ( H 5), where Ht ( ) dt 5 is measured in degrees Celsius and H (0) = 00. What is the approximate temperature of the granite at time t line tangent to the graph at time t = 0? A) 5 B) 50 C) 150 D) 175 = 5 using the

4 7. (NC) At time t = 0, a piece of heated granite is placed in a container of water. o The internal temperature of the granite is 150 C at time t = 0. The internal temperature of the granite at time t minutes is modeled by the function dh 1 Ht ( ) that satisfies the differential equation = ( H 36), where Ht ( ) dt 6 is measured in degrees Celsius and H (0) = 00. What is the approximate temperature of the granite at time t line tangent to the graph at time t = 0? = 6 using the A) B) +36 C) D) 114 e e e = + During the time interval 0 t π, a particle moves along the polar curve r = f( θ) so that at time t seconds, θ = t. On what intervals of time t is the distance bewteen the particle and the origin increasing? A) 0 t 5 B) 0 t π C) 1 t 5 D) 0 t 1 and 5 t π 3. (NC) The function f( θ) θ 9θ 15 θ satisfies f( θ) 0 for θ 0.

5 9. (C) At time t =, a tank contains 30 gallons of water. For t 7, water πt flows into the tank at a rate of gt ( ) = tsin gallons per minute, and 6 3t water leaks out of the tank at a rate of Lt () = gallons per minute. How t + 4 many gallons of water are in the tank at time t = 7 minutes? ( A) 13.1 ( B) ( C) ( D) (NC) A parking lot is in the shape of a rectangle miles long and 5 miles wide. The density of cars at a distance of x miles from the mile side is given by β ( x) cars per square meter. Which expression gives the number of cars in the parking lot? ( A) ( B) ( C) ( D) β ( x) dx β ( x) dx 5 β ( x) dx β ( x) dx

6 1 1 tan (4 ) tan (4 ) 4 x a 11. (NC) If lim =, then a could equal x a x a 9 4 ( A) ( B) 1 ( C) ( D) 1. (NC) Which of the following are coordinates of the point where the slope kx 1 of the line tangent to the graph of f( x) = equals 1 when x= k? x + 4 (, 4) ( 4, ) (3, ) ( A) (1, 4) ( B) ( C) ( D)

7 13. (NC) Let f be a differentiable function such that f( x+ h) f( x) = 4xh+ 7h for all x, and f(3) = 6. What is the value of f()? ( A) ( B) ( C) 4 ( D) (NC) At which of the three points on the graph of y = f x below will f ( x) < f ( x)? ( A) A only ( B) C ( C) ( D) ( ) only A and B only A and C only in the figure

8 x t 15. (C) Let g be the function defined by g( x) = ( + 6 t + t )3 dt. Which of the following statements about g must be true? I. g is increasing on 1,3 ( ) ( ) II. g is increasing on 4, III. g() > 0 ( A) I only ( B) III only ( C) I and III only ( D) I, II, and III (NC) f(0) = 0 A function f has derivatives of all orders for 1 < x< 1. The f (0) = 4 derivatives of f satisfy the condition to the left. The Maclaurin n (0) = (0) for all 1 series for converges to ( ) for < 1. What are the first 4 non-zero terms of the series? ( n+ 1) ( n) f f n f f x x 4 3 x (A) 4x x + x 3 4 x x (B) 4x x x x (C) 4x+ x x x (D) 1 x + 3

9 n ( ) ( n+ ) 3 4 n x x x 1 x 17.(NC) h( x) = for n n Let P represent the nth-degree Taylor polynomial for h about x= 0 evaluated at 1 x=, where g is the function shown above. Use the alternating series erro r bound 1 to calculate the maximum error of P (a) (b) (c) (d)