Differential Calculus
Final Exam
1. Find the derivative of ln cos 3x
a. 3tan 3x
b. -3tan 3x
c. 3cot 3x
d.-3cot 3x
2. Find the second derivative of the parametric equations: X=1-t2
a. – 1/ 4t3
b. – 1/ 4t2
c. 1/ 4t3
d. 1/ 4t
3. If the velocity after 1 second is 8 m/s upward, find the greatest distance above the starting point.
a. 17.8 m
b. 16.2 m
c. 12.5 m
d. 8.0 m
4. Differentiate partially with respect to X the function V=cot-1y/x
a. –x/x2+y2
b. x/x2+y2
c. y/x2+y2
d. –y/x2+y2
5. Find the derivative of y=x√b2-x2 + b2Arc sin x/b
a. 2 √b2-x2
b. √b2-x2
c. -2 √b2-x2
d. -4 √b2-x2
6. Find the radius of curvature of 3y=x3 at (1, 1/3 )
a. 1
b. 2
c. 2√2
d. √2
7. The motion of a particle is defined by the parametric equations x=t3 and y=2t2. Determine the velocity
when t=2 seconds
a. 14.42 m/s
b. 20.12 m/s
c. 22.42 m/s
d. 16 m/s
8. Find the height of the right circular cylinder of greatest volume that can be inscribed in a right circular
cone with a radius of 5cm and height of 12cm.
a. 3 cm
b. 4 cm
c. 6 cm
d. 8 cm
9. The rate of change of the area of a circle with respect to its radius when the diameter is 6 cm is:
a. 3∏
b. 9∏
c. 6∏
d. 36∏
c. 2.17182
d. 2.71828
10. Naperian logarithm has a base of:
a. 3.1416
b. 10
11. Simplify: (Sin4x – Cos4x) / (Sin2x – Cos2x)
a. Sin 2x
b.Cos2x
c. Sin x
d. 1
12. Evaluate the limit ( 1 + 2/x)x as x approaches infinity
a. 0
b. 1
c. e
d. e2
13. Find the total differential of xy+z2
a. xdy+ydx+2zdz
b. xdy+ydx+z2
c. xy+2zdz
d. xy+z2dz
14. If y=4 sin 2x, find dy/dx
a. 8 cos 2x
b. 4 cos 2x
c. 2 sin 2x
d. 4 sin x
15. It is a set of all admissible values of x in a function.
a. range
b. domain
c. function
d. limit
16. A man 6ft tall walked away from a lamp post 16 ft high at the rate of 5 mph. How fast does the end
of his shadow moved?
a. 5 mph
b. 8 mph
c. 6 mph
d. 30 mph
17. What is the limit of x3+27 divided by x+3 as x approaches -3.
a. 27
b. 18
c. 9
d. 0
18. Find the points of inflection of the curve y= 1/3 x3 – ½ x2 – 2x.
a. 3/2, -1/2
b. 1/2, 1/3
c. 1/2, -13/12
d. -13/12, 1/2
c. 2sec 2x
d. sec h2 2x
19. Find the derivative of y=tanh 2x.
a. 2sec h 2x
b. 2sec h2 2x
20. Find the volume of the largest box that can be made by cutting equal squares of the corners of a
piece of cardboard 15in. by 24in. and then tuning up the sides.
a. 246in3
b. 342in3
c. 486in3
d. 360in3
21. Determine the dimension of the largest rectangular field that can be enclosed with 240m of fence.
a. 60m by 60m
b. 80m by 30m
c. 40m by 60m
d. 60m by 80m
22. If 6000 people only will attend a basketball game at an admission price of P3.00, and if for every 5
centavos reduction in price 200 people will attend, what admission price per person will give the
maximum income?
a. P2.00
b. P2.25
c. P2.50
d. P2.75