3.4 - Calculus

Question 1

I know understand what “rate of change” is

Answer 1

The rate of change is how quickly one variable changes as you vary another.

For example, in graphs we look at the gradient of a line - this is how “fast” up or down a line moves as we go across the graph.

Question 2

I can differentiate x⁵

Answer 2

5x⁴

Question 3

I can differentiate 4x

Answer 3

4

Question 4

I can differentiate x² + 10

Answer 4

2x

Question 5

I can find the gradient of a line by differentiating

For example, what is the gradient of the line given by this equation:

y = 4x^2^ + 2

at the point when x = 3?

Answer 5

Differentiating gives the gradient of the line:

dy/dx = 8x
      = 8 × 3
      = 24

Question 6

What does it mean if the rate of change is 0?

Answer 6

The graph of the equation must be flat at this point - normally this is a turning point.

Question 7

I can find the turning points of a line by differentiating

For example, what are the turning points of this equation:

y = 2x^2^ - 8x + 3

Answer 7

Differentiate to find the gradient equation:

dy/dx = 4x - 8

Turning points are when dy/dx = 0, so

0 = 4x - 8
x = 2

Substitute back into original equation to get

y = 2x^2^ - 8x + 3
y = 2×2^2^ - 8×2 + 3
y = 2×4 - 16 + 3
y = 8 - 13
y = -5

Turning point is at (2, -5)

Question 8

What is the rate of change of distance with respect to time?

Answer 8

Velocity - as time passes, something with a high velocity will move further than something with a low velocity.

Question 9

What is the rate of change of velocity with respect to time?

Answer 9

Acceleration - As time passes, something with high acceleration will pick up more speed than something with low acceleration.

Velocity changes faster when acceleration is high