DIFFERENTIAL CALCULUS

1. limx(ex x) Ans.

2. limx0(1 sinx)1/x Ans. 1/e

3. limx3(3/(x 3)) Ans. Does not exist

4. limx(cos(2/x)x2) Ans. e-2

5. limx0(1 + ax)b/x Ans. eab

6. What is the derivative of 2cos(2+x3) with respect to x?

7. Find y in the curve y = (1-2x)3 at the point (1,-1).

8. Find the slope of r = tan when = /4.

9. What is the equation of the tangent line to the curve

9x2+25y2-225 = 0 at (0,3)?

10. Find the equation normal to the curve x2 = 16y at (4,1).

11. Find the derivative with respect to x of (x+1)3/x.

12. Find the slope of the curve x = 6y3-2y2 at point (6,3).

13. Find the curvature of the curve y2 = 16x at the point

(4,8).

14. Find the radius of curvature of the ellipse 3x2+y2 = 12

at the point (1,3).

15. Find the radius of curvature of r = tan when =

3/4.

Problem:

A particle moves along the path whose parametric

equation are x = t3 and y = 12t2.

16. What is the velocity when t = 2sec?

17. What is the acceleration when t = 2sec?

18. Car A moves due east at 30kph at the same instant car

B is moving S30E with a speed of 60kph. The distance

from A to B is 30km. Find how fast is the speed between

them are separating after 1hr.

19. Determine the diameter of a closed cylindrical tank

having a volume of 11.3cm3 to obtain a minimum surface

area.

20. The three sides of triangle ABC are AB = 7cm, BC =

5cm and CA = 9cm. Determine the width of the largest

rectangle that can be inscribed in it such that the longer

side of the rectangle is on the 9cm side of the triangle.

21. A ferriswheel has a radius of 10m. Its center is 12m

above the ground. When the passenger is 17m above the

ground, he is moving at the rate of 1.81m/s. What is the

speed of rotation of the wheel in rpm?

22. Find the minimum length of the line segment

intercepted between the positive coordinates axes is a

minimum that passes through (3,4).

23. A revolving searchlight in a lighthouse 2km offshore is

following a car traveling slowly along the shore. When the

car is 1km from the point on the shore that is closest to

the lighthouse, the searchlight is turning at the rate of

0.25rev/hr. How fast in kph is the car traveling at this

moment?

24. Water is flowing in a conical cistern at the rate of

8m3/min. If the height of the inverted cone is 12m and the

radius of its circular opening is 6m. How fast is the water

rising when the water is 4m in depth?

25. A balloon is leaving the ground 18m from the observer

rises 3m/s. How fast is the angle of elevation of the line of

sight increasing after 8 seconds?

26. What is the minimum vertical distance between

parabolas x = y2 and y = 1+ x2.

Problem:

The upper end of the ladder 5m long leans against a

vertical wall. Suppose the foot of the ladder slips away

from the wall at the rate of 0.1m/min.

27. How fast is the top of the ladder descending when its

foot is 3m from the wall?

28. When will the top and the bottom of the ladder move at

the same rate?

29. When is the top of the ladder descending at the rate of

0.15m/min?

30. How fast is the angle at the foot of the ladder

decreasing when the foot is 3m from the wall?