Math Revision 15.8 - Gradient of Curves

In these revision notes for Gradient of Curves, we cover the following key points:

• What is the gradient of a curve? How to calculate it?
• Does the gradient of a curve have the same value everywhere?
• How to find a general formula for calculating all gradients of the same type of functions?
• How to find the gradient of quadratic functions in two ways?
• The same for cubic functions.
• How to find the equation of the tangent line to a curve at a given point?

Gradient of Curves Revision Notes

The gradient is a very crucial concept in functions theory because it shows how fast the value of a function changes. We assume the x-coordinate of a function as always increasing and see what happens to the corresponding y-coordinate.

The tangency of a line with a curve occurs when the line touches the curve at a single point. The equation of tangent line with a curve is y = kx + t, where k is the gradient of the curve at the point of contact between the line and the curve.

Unlike in linear graphs where the tangent corresponds to the line itself and therefore it has the same equation as the line, the equations of various tangents in a curve are different, as all of them have different steepness from each other, and therefore, different gradients.

When dealing with graphs, we use the term gradient to describe the linear coefficient (inclination) of the tangent line with the graph at a given point. The procedure to calculate the gradient is always the same: after choosing two points from the graph, we divide the change in the y-coordinate by the change in the x-coordinate. However, since the curve changes continuously its angle to the horizontal axis, we must choose the two points as close as possible to each other, where the point of tangency lies between them.

In this way, we reach two important conclusions about the gradients of a curve:

1. Curves have different gradients at different points.
2. The closer to the point of tangency we choose the two points needed for calculating the gradient, the more accurate the result obtained is.

You can use the gradient formula

every time you need to find the gradient in a given point. However, this procedure is a bit annoying, as you have to calculate four times (two for the x-direction and two for the y-direction) the value of expressions that contain decimals raised to certain powers. This requires long and careful calculations. Therefore, it is better to use general formulas for each type of function that are applicable for all values of the independent variable in every specific function. Applying the general gradient's formula

we find that the gradient k of parent quadratic function f(x) = x² is k = 2x, and the gradient of the cubic parent function f(x) = x³ is k = 3x². Based on the above gradients and given that the gradient of a polynomial function is obtained by taking the sum of all individual gradients representing each term, we have:

1. The gradient of a second-degree polynomial function f(x) = ax² + bx + c is
   k = 2ax + b
2. The gradient of a third-degree polynomial function f(x) = ax³ + bx² + cx + d is
   k = 3ax² + 2bx + c

After calculating the gradient, you can find the equation of the tangent to any given point A(x_A, y_A)

where t is the line's constant. It is calculated by taking y = y_A and x = x_A in the above equation besides the gradient k found in the previous step.