US5244US5244Demonstrate Calculus Skills

Gradients of FunctionsGradients of FunctionsMany real life situations can be modelled by straight lines or curves (functions)

e.g. The cost of hiring a taxi can be modelled by a straight line where the slope (gradient) represents the cost per kilometre

e.g. For the distance travelled by a ball, the gradient represents the velocity of the ball

e.g. For a graph of a roller coaster’s profile, the gradient can represent its steepness at any particular point.

Distance (km)

Cost ($)

Time (s)

Height (m)

Height (m)

Time (s)

Gradients FunctionsGradients Functions

Below is the function y = x2

x-4 -2 2 4

y

2

4

6

8

10

To find the gradient at any particular point you need to calculate the gradient of the tangent to that point.

x Gradient

-3

-2

-1

0

1

2

3

The formula to find the gradient at any point is the gradient function.

The gradient function of y = x2

= 2x

-6

-4-2

0

2

4

6

Finding Gradients Functions (Differentiating)Finding Gradients Functions (Differentiating)

Through calculating gradients of other functions, the following results can also be found.

function gradient function

y = x3 dy/dx = 3x2

y = x4 dy/dx = 4x3

f(x) = x5 f’(x) = 5x4

f(x) = x6 f’(x) = 6x5

It is through these results that a pattern emerges:

If the function is written y = the gradient function is dy/dx = If the function is written f(x) = the gradient function is f’(x) =

If y = xn then dy/dx = nxn-1

If f(x) = xn then f’(x) = nxn-1

Two other important results can also be established

If f(x) = axn then f’(x) = n×axn-1

If f(x) = g(x) + h(x) then f’(x) = g’(x) + h’(x)

e.g. Find the gradient functions (differentiate) of the following y = x3 + 4x - 5 f(x) = 2x4 – 5x3 + 3x2 - 4

dy/dx = + 4 4×2x4-1

f’(x) = 8x3

3x2 f’(x) = – 3×5x3-1 + 2×3x2-1

– 15x2 + 6x

Sketching Gradients FunctionsSketching Gradients FunctionsThese sketches show how the gradient changes for a function1. Gradients of Straight Lines

x-4 -2 2 4

y

-4

-2

2

4

With a straight line, the gradient is always constant.

For the above example, the gradient is always 2 so we draw a horizontal line through 2.

x-4 -2 2 4

y

-4

-2

2

4

For the above example, the gradient is always -3 so we draw a horizontal line through -3.

2. Gradients of Quadratics (Parabolas)The gradient function of a quadratic is always a straight lineIf the coefficient of x2 is positive, the gradient function is positive.If the coefficient of x2 is negative, the gradient function is negative.

x-4 -2 2 4

y

-4

-2

2

4

- Look for when the gradient is 0 and mark the point on the x-axis

- The line goes above the x-axis where the quadratic has a positive slope, and below where it is negative

x-4 -2 2 4

y

-4

-2

2

4

- Mark the point on the x-axis where the gradient is 0

- The line goes above the x-axis where the quadratic has a positive slope, and below where it is negative

3. Gradients of CubicsThe gradient function of a cubic is always a quadratic (parabola)If the cubic goes from bottom to top, the gradient function is positiveIf the cubic goes from top to bottom, the gradient function is negative

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

- Look for when the gradient is 0 and mark the points on the x-axis

- Look for when the gradient is 0 and mark the points on the x-axis

- The parabola goes above the x-axis where the cubic has a positive slope, and below where it is negative

- The parabola goes above the x-axis where the cubic has a positive slope, and below where it is negative

Antidifferentiation or IntegrationAntidifferentiation or IntegrationThis is the reverse process to differentiation

e.g. 2x dx = x2

3x2 dx = x3

4x3 dx = x4

We know however, that when we differentiate, any number (constant) disappears, therefore when integrating we must always add in a constant (c)

In general: xn dx = xn + 1 + c n + 1

e.g. 7x6 dx =

(9x2 – 6x + 3) dx =

(2x3 + 3x2 - 8x - 5) dx =

7x7 7

= x7 + c

9x3

3 = 3x3 - 3x2 + 3x + c

2x4

4 = 1x4 + x3 - 4x2 + c

2

+ c

+ c

+ c

+ 3x- 6x2

2

+ c

+ c

+ 3x3

3

- 8x2 2

+ c