What is the derivative of f(x) = x^3/4?

1.   ?    f '(x) = (¾)x^-1/4
2.   ?    f '(x) = (3/4)x^1/4
3.   ?    f '(x) = (4/3)x^1/4
4.   ?    f '(x) = (4/3)x^-1/4

What is the gradient of y = x⁵ at the point (1,1)?

1.   ?    1
2.   ?    0
3.   ?    -1
4.   ?    5

Differentiate y = 3x² + 6x – 2?.

1.   ?    dy/dx = 6x³ + 6x² -2x
2.   ?    dy/dx = 6x + 6
3.   ?    dy/dx = (3/2)x + 6
4.   ?    dy/dx = 6x² + 6

What is the gradient of the curve y = 4x³ + log_e2x at x = 1?

1.   ?    13
2.   ?    12
3.   ?    12.5
4.   ?    14

If revenue, R, and demand, Q, are related by R = 20Q – 0.002Q² what is the marginal revenue when Q = 1000?

1.   ?    18,000
2.   ?    18
3.   ?    16
4.   ?    0

Given that the demand function is P = 100 – 2√Q, calculate the price elasticity of demand when P = 20?

1.   ?    0.2
2.   ?    0.02
3.   ?    0.5
4.   ?    1.5

Which of the following statements is not necessarily true?

1.   ?    A local maximum is a turning point of a curve
2.   ?    At a local maximum, the gradient of the curve is zero
3.   ?    A local maximum is higher than all nearby points
4.   ?    A local maximum is the highest point of the curve

Find the values of x at which the curve y = 3x³ + 2x² has turning points.

1.   ?    0 and -4/9
2.   ?    0 and 3/5
3.   ?    -4 and 0
4.   ?    0 and 9

The curve y = 3x³ + 2x² has a turning point at x = 0. Is it a local maximum or minimum, and why?

1.   ?    local minimum since second derivative at that point is < 0
2.   ?    local minimum since second derivative at that point is > 0
3.   ?    local maximum since second derivative at that point is < 0
4.   ?    local maximum since second derivative at that point is > 0

Which of the following statements is not true?

1.   ?    A point of inflexion is a stationary point
2.   ?    At a point of inflexion, the first derivative is zero
3.   ?    At a point of inflexion, the second derivative is zero
4.   ?    A curve changes direction at a point of inflexion