TMTA Calculus and Advanced Topics Test 2010

Transcription

1 . Evaluate lim Does not eist TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {} {, -} 4. Which of the following is NOT equivalent to sin 60 sin 0 sin 0 cos 0 sin 45 cos 5 cos 45 sin 5 sin 90 cos0 cos 90 sin 0 cos0

2 5. Find an equation of the tangent line to the curve y 4 6 y 6 4sin at the point, y 6 y 4 6 y 6 y 6 6. Find the area of the region bounded by the given curves y and y The graph of y 4 4 is which type of conic section? parabola opening up parabola opening down circle ellipse hyperbola

3 8. Find the slope of the tangent line to the curve y + = at the point (-, ) Find f ( ) if f ( ) sin sin cos cos sin cos sin cos ( ) cos ( ) 0. Solve graphically in the interval, sin 0.5 e 0.5., , ,.. Round to the nearest hundredth.. In how many ways can 7 toys be divided between children if the youngest gets toys and each of the others get toys?,

4 . Integrate: 6 d c 4 c 4 c. Find the values of such that the vectors 8,,5 and,, = -, = =, = 8 = 4, = 6 = -, = 0 = -0, = - are orthogonal. 4. A ladder 5 feet long is leaning against the wall of a building. The base of the ladder is pulled away from the wall at a rate of 4 feet per second. At what rate is the angle between the ladder and the wall changing when the base of the ladder is 5 feet from the wall? 5 rad/s rad/s 6 rad/s rad/s rad/s 5. A motorboat leaves a dock and travels 70 m due west, then turns 4.0 to the south and travels another 50 m to a second dock. What is the displacement of the second dock from the first dock? 890 m, 4.0 S of W 756 m, 8.4 S of W 87 m,.6 S of W 779 m, 4. S of W 75 m,.7 S of W

5 6. The position function of a particle is given by r(t) = a minimum? t = 6 t = 0 t = t = 4 t = 4,,4 48 t t t t. When is the speed 7. Find the coordinates of the point on the curve which is perpendicular to the line (, 0), 9, 9 (-, ) (, ) y 4 y where there is a tangent line 8. The velocity function for a particle moving along a line is given by v( t) t. Find the distance traveled by the particle during the time interval 0t Find an equation of the tangent plane to the surface (,, -0). 6 9y + z = y z = y z = y + z = y z = 66 z y y 4 9 at the point

6 0. Find the maimum rate of change of 6 ln 6 ln f (, y) ln( y ) at the point (5, ).. If ( ) dt f t, find f ( ) t t t t t dt 0. A music class of 5 girls and 4 boys is having a recital. If each child is to perform once, how many ways can the program be arranged if the first two performers must be girls? 6,880 00,800,880 5,876 8,440. Evaluate the integral / + c sec + c ( ) + c tan sec tan + c d + c

7 4. Evaluate lim (csc ) Solve the differential equation y e e y e e 4 y e e y e e 5 5 y e e 5 5 dy d y e ; y(0) 6. Find the value of k if is a factor of 4 k k If f (), f () 6, f () 4, g(), g() , find f ( g ()) d d

8 8. Simplify 0 sin h sin h sin( h) sin cosh cos sin h sin( )cos( h) cos( )sin( h) 9. In a certain college, 5% of the men and % of the women are taller than 80 cm. Also, 60% of the students are women. If a student is selected at random and found to be taller than 80 cm, what is the probability that this student is a woman? 0 0. Find the area enclosed by the graphs y y 7 and.. Find the area of the region under the curve y for The area is infinite

9 . Find the volume of the solid formed by rotating the region bounded by y and y about the -ais Find the value of the real numbers a and b if 8 4y = 7 is a tangent to the curve a b y at =. a = -9 and b = 5 a = -9 and b = 8 a = 9 and b = 8 a = 9 and b = -8 a = -9 and b = The volume of a cube is increasing at a rate of 0 cm /min. How fast is the surface area increasing when the length of an edge is 0 cm? 4 cm /min 4 cm /min 5 cm /min 5 cm /min 4 5 cm /min

10 5. Find the Cartesian equation for the curve represented by the polar equation r r r 8 cos 6 sin 6 0 ( 4) ( y) 4 ( 4) ( y) 9 y 4 ( 4) ( y) 6 ( 4) ( y) An object is projected directly upward from the ground with an initial velocity of 96 feet per second. Its height, s, at the end of t seconds is given by velocity of the object when it hits the ground? 8 ft/s 80 ft/s - ft/s -96 ft/s -6 ft/s 7. Find the interval of convergence for the power series 4 ( ) ( ) ( ) s( t) 96t 6t. What is the 8. An airplane is climbing at a angle to the horizontal. How fast, to the nearest mile per hour, is it gaining altitude if its speed is 5 miles per hour? 87 mi/h 8 mi/h 68 mi/h 8 mi/h 9 mi/h

11 9. Determine the values of the number a for which the function f() has no critical numbers: f a a a ( ) ( 7 )cos ( ) a <.5.5 a 4.5 a 5 a > 5 a Find the sum of the following series e ( ) e e e e...!! 4! 5!

12 . b. e. e 4. a 5. b 6. b 7. e 8. d 9. a 0. d. e. c. a 4. a 5. c 6. e 7. e 8. e 9. c 0. a. e. b. d 4. e 5. c 6. b 7. b 8. d 9. d 0. a. d. c. d 4. b 5. b 6. d 7. e 8. c 9. b 40. a