1. US5244 Demonstrate Calculus Skills

2. Gradients of Functions Many 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.

3. Gradients Functions Below is the function y = x 2 To find the gradient at any particular point you need to calculate the gradient of the tangent to that point. xGradient -3 -2 0 1 2 3 The formula to find the gradient at any point is the gradient function. The gradient function of y = x 2 = 2x -6 -4 -2 0 2 4 6

4. Finding Gradients Functions (Differentiating) Through calculating gradients of other functions, the following results can also be found. functiongradient function y = x 3 dy/dx = 3x 2 y = x 4 dy/dx = 4x 3 f(x) = x 5 f’(x) = 5x 4 f(x) = x 6 f’(x) = 6x 5 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 = x n then dy/dx = nx n-1 If f(x) = x n then f’(x) = nx n-1 Two other important results can also be established If f(x) = ax n then f’(x) = n×ax n-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 = x 3 + 4x - 5f(x) = 2x 4 – 5x 3 + 3x 2 - 4 dy/dx =+ 44×2x 4-1 f’(x) = 8x 3 3x 2 f’(x) =– 3×5x 3-1 + 2×3x 2-1 – 15x 2 + 6x

5. Sketching Gradients Functions These sketches show how the gradient changes for a function 1. Gradients of Straight Lines 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. For the above example, the gradient is always -3 so we draw a horizontal line through -3.

6. 2. Gradients of Quadratics (Parabolas) The gradient function of a quadratic is always a straight line If the coefficient of x 2 is positive, the gradient function is positive. If the coefficient of x 2 is negative, the gradient function is negative. - 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 - 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

7. 3. Gradients of Cubics The gradient function of a cubic is always a quadratic (parabola) If the cubic goes from bottom to top, the gradient function is positive If the cubic goes from top to bottom, the gradient function is negative - 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

8. Antidifferentiation or Integration This is the reverse process to differentiation e.g.  2x dx = x 2  3x 2 dx = x 3  4x 3 dx = x 4 We know however, that when we differentiate, any number (constant) disappears, therefore when integrating we must always add in a constant (c) In general:  x n dx = x n + 1 + c n + 1 e.g.  7x 6 dx =  (9x 2 – 6x + 3) dx =  (2x 3 + 3x 2 - 8x - 5) dx = 7x 7 7 = x 7 + c 9x 3 3 = 3x 3 - 3x 2 + 3x + c 2x 4 4 = 1x 4 + x 3 - 4x 2 + c 2 + c + 3x- 6x 2 2 + c + 3x 3 3 - 8x 2 2 + c