US5244 Demonstrate Calculus Skills

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.

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 The formula to find the gradient at any point is the gradient function. The gradient function of y = x 2 = 2x

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 dy/dx =+ 44×2x 4-1 f’(x) = 8x 3 3x 2 f’(x) =– 3×5x ×3x 2-1 – 15x 2 + 6x

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.

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

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

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 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 c + 3x x c