Differential Calculus
1) The luminous intensity I candelas of a lamp at varying voltage V is given by: I = 4 × 10⁻⁴ V². Determine the voltage at which the light is increasing at a rate of 0.6 candelas per volt.
- 650
- 450
- 750
- 550
2) The length l meters of a certain metal rod at temperature θ°C is given by l = 1 + 0.00005θ + 0.0000004θ². Determine the rate of change of length in mm/°C when the temperature is 100°C.
- 0.33
- 0.23
- 0.13
- 0.43
3) The distance x meters described by a car in time t seconds is given by: x = 3t³ − 2t² + 4t − 1. Determine the acceleration when t = 0.
- -7
- 4
- 7
- -4
4) Supplies are dropped from a helicopter and distance fallen in time t seconds is given by x = 1/2gt² where g = 9.8 m/sec². Determine the velocity and acceleration of the supplies after it has fallen for 2 seconds.
- v = 18.6 m/sec, a = 8.8 m/sec²
- v = 19 m/sec, a = 10 m/sec²
- v = 9.8 m/sec, a = 19.6 m/sec²
- v = 19.6 m/sec, a = 9.8 m/sec²
5) A boy, who is standing on a pole of height 14.7m throws a stone vertically upwards. It moves in a vertical line slightly away from the pole and falls on the ground. Its equation of motion in meters and seconds is x = 9.8 t − 4.9t². Find the time taken for downward motions.
- 5
- 4
- 3
- 2
6) A ladder 10m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1m/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6m from the wall?
- 1/7
- 2/3
- 3/4
- 1/3
7) A car A is travelling from west at 50 km/hr. and car B is travelling towards north at 60 km/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 kilometers and car B is 0.4 kilometers from the intersection?
- 77
- 78
- 95
- 86
8) A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is being pumped into the tank at a rate of 2m³/min, find the rate at which the water level is rising when the water is 3 m deep.
- 8/9π
- 8/7π
- 6/5π
- 1/9π
9) Find the equations of the tangent to the curve y = x³ at the point (1,1)
- y = 3x + 1
- y = 3x + 2
- y = 3x – 2
- y = 3x – 1
10) Determine
- 10
- 11
- 17
- 15
11) Determine
- 3
- 6
- 5
- 4
12) Determine
- 20
- 35
- 45
- 25
13) Determine
- 0
- 9
- 3
- 1
14) Determine
- 5
- 3
- 4
- 9
15) Determine
- 4
- 2
- 1
- 3
16) Determine
- 6
- 5
- 7
- 4
17) Determine
- 32/6
- 52/6
- 57/6
- 37/6
18) Determine
- 0
- Not defined
- -1
- 1
19) Determine
- 0
- 6
- Does not exist
- 1
20) Determine
- 0
- Not defined
- -3
- 3
21) Determine
- 3
- -3
- Not defined
- 0
22) Determine
- √8/9
- √3/6
- √5/6
- √7/6
23) Given g(x) = 3x², determine the gradient of the curve at the point x = −1
- 6
- -6
- 4
- -8
24) Given the function f(x) = 2x² − 5x, determine the gradient of the tangent to the curve at the point x = 2
- 3
- 8
- -8
- 6
25) Determine the gradient of k(x) = −x³ + 2x + 1 at the point x = 1
- 3
- -1
- 5
- 6
26) Given: f (x) = −x² + 7. Find the average gradient of function f, between x = −1 and x = 3
- -2
- 7
- 5
- 6
27) Given: f (x) = −x² + 7, find the gradient of ‘f’ at the point x = 3
- -4x
- -7x
- -8x
- -2x
28) Determine the gradient of the tangent to g if g(x) = 3/x
- -3/a²
- -6/a²
- 6/a²
- 3/a²
29) Determine the equation of the tangent to H(x) = x² + 3x at x = −1
- y = - x +1
- y = -x – 1
- y = x – 1
- y = x +1
30) Use the rules of differentiation to find the derivative of y = 3x⁵
- 12x⁴
- 5x⁴
- 3x⁴
- 15x⁴