2. (10 marks The circle below represents a record which is rotating clockwise at 100/3 revolutions per minute. A bug is walking away from the centre of the record directly toward point P on the rim of the record at 1 cm/s. When the bug is at position R, 10 cm from O, the angle 8 is 7/4 radians. Find the rate at which the distance from the bug to the fixed point Q is changing when the bug is at R. Simplify as much as possible but your final expression may contain square roots or well-known constants (such as it or e).

(b) Find t') if f(x) = In ( sec x + tan x).

4. (6 marks] A bacteria culture initially contains 200 cells. After 1 hour, the bacteria population has grown to 400 cells. Assuming exponential growth, what will be the population after 4 hours? Simplify your answer as much as possible.

1. Find lim 2+0 sin 3.0 22 3.3 sin 4.r (0) - (d) 1 (e) The limit does not exist.

(c) 3 marks Find the critical points of f and the intervals of increase and decrease. Find any local maximum and/or minimum points.

11. Let In(2+1)-In 2 s() = { f(x)= a when x &gt; 0 when r = 0 when : &lt; 0 (eto If f is continuous at x = 0, then (a) a = In 2 and b=0 (b) a= 1 and b= 1 (c) a = 1 and 6 = 2 (d) a= In 2 and b== (e) a = 5 and 6= In 2

5. (a) (3 marks State the Mean Value Theorem.

6. Let f(x) = Vx+1 on (0,3). Find the number c that satisfies the conclusion of the Mean Value Theorem for f on (0,3). بن احمر بس احب ان اش بر | حبر خيانت

3. (5 marks] Use logarithmic differentiation to find in terms of r given y = (3x+5)*

2. For this question, simplify your final answers as much as possible. (a) Find f'(0) if f(1) = 3+ + 3-.

12. The graph of y= x(x - 347 has a point of inflection at &lt;=

3. Find lim 1-sin (It) 21 1 - cos(1 – t) (a) 0 (c) (d) (e) The limit does not exist.

(c) 14 marks] Suppose that f(0) = -3 and f'(I) &lt; 5 for all values of r. What is the largest possible value for f(2)?

4. Each side of a cube is increasing at a rate of 5 cm/s. At what rate is the volume of the cube increasing when the volume of the cube is 8 cm?

2z5 + 5r3 +1 1. Find the value of lim rn z+z2 - 6z5 ° ܝܕ|ܚ ܝ |ܘ ܗ 8

15. 1 - cos x - x sinx lim 0+0+ sina x + 2 cos x - 2 @ -o NIN

6. Consider the function f(x) = (3-1) and "C) 2(x2 - 6) (a) [3 marks) Show that f'() = ? and f" (30) =

8. The product of two positive numbers is 144. What is the minimum value of their sum? (a) 26 (b) 18 (c) 24 (d) 20 (e) 22

4. A point P on the curve y = x2 + 2x - 4 is such that the tangent line at P is parallel to the line y = 4x + 11. Find the y-coordinate of the point P. ® 3

5. The equation of the tangent to the curve 4y = r2 at the point on the curve where x = -2 is (a) x+y - 3 = 0 (b) x +y-1=0 (c) y-1 = 2x(x + 2) (d) 1 +y+1=0 (e)

(b) [3 marks) A car is traveling along a straight line. At 2:00 pm the car's speedometer reads 30 km/h. At 2:10 pm it reads 50 km/h. Show that at some time between 2:00 and 2:10 the acceleration was exactly 120 km/h?.

( 9 marks Sketch the graph of J. labeling Intercepts, asymptotes, max/min and inflection points

14. Two circles each with radius 1 are tangent to one another. One line passes through the centre of the first circle and is tangent to the second circle at the point P. A second line passes through the centre of the first circle and is tangent to the second circle at the point Q. Find the distance between P and Q. 10 న చ సల |

16. Let f(x) = { ?V= if 0 &lt;&lt; 2. Find the absolute minimum value of f(x) on (0,2). @e-1/e ® e-2/e © © e-1/2 e-2ve ® e-ve

5. Let f(2)= arctan(2x). Then f'(1) = War I CINCIN

3. Find the tangent line to the curve y+tan(ev - x) = r* - 1 at the point (1,0).

5. A bacteria culture initially contains 500 cells. After 1 hour, the bacteria population has grown to 1000. Assuming exponential growth, what will be the population after 3 hours?

3. Let f(x) = 2003 - 1 2.r3+1 Then the inverse function f-(x) = * +3 3+ 2r 1 + 1 r+2 3 - 5x

9. Let f(x) = ln x) cost. Then f'(.2) = (a) (cos x)(In 2)cos 2-1 (b) (In r)cos I In(In r) (c) – (In zowe (In(In ) sin x + cos2) (d) (In 2)cos (- sin &lt; In(In x) + Cuma) (e) The derivative is not defined. COST

10. If y = (c) satisfies y = cy, then when (a, y) = (2e, Ve), the derivative - - | వ

6. Let f(x) = x3 – 3x2 + 1. Let M be the absolute maximum of f(t) on the interval (-1,3] and let m be the absolute minimum of f(2) on the interval -1,3). Then M + m = (a) 20 (b) 14 (c) 17 (d) 18 (e) -2

6. Find the dimensions of a rectangle with perimeter 100 cm whose area is as large as possible.

(b) 13 marks Find the domain of f. Find the x and y intercepts and horizontal and vertical asymptotes, if any.

7. A spherical balloon is being filled with helium at a rate of 100 cm/s. At what rate is the radius of the balloon increasing when the radius is 10 cm? The volume of a sphere is nr.

1. (10 marks/ The top and bottom margins of a poster are each 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 50 cm”, find the dimensions of the poster with the smallest area. Check that the value obtained is indeed a minimum.

13. Consider the following 3 curves: I. y=sin x + cos2x – sin 3x + 7.0 - 5. 11. y = 325 - 33 + 8x – 5. III y = ex48 + 3x25 + 35x + 15. Which of the 3 curves has (have) at least one horizontal tangent line? @ I and III only. ® II and III only. © II only. III only © I and II only.

OC2735559

Emory University