Introduction to Calculus 1: RATE OF CHANGE and FIRST PRINCIPLES Independent Practice: Oxford Text: Exercises p201, 7C; p202, 7D & p302, 7E Pearson Chapter 11 p368 - 379

Rate of Change (Slope) Average Rate of Change: Recall: Slope (gradient) of a line can be calculated by: Average Rate of Change: Slope of the line (secant) that connects two points on a curve Average rate of change from A to B = gradient from A to B Instantaneous Rate of Change: Slope of the line (tangent) that touches the curve at a point Instantaneous rate of change at x =1 = gradient of tangent at x =1 A tangent line is difficult to accurately draw. We will explore ways to accurately calculate the slope of a tangent

Finding the slope of a secant line The gradient of the secant line AB is written as Using the “formula” for the gradient of a secant line, write an expression for the gradient of a secant line for and simplify your answer.

Finding the slope of a tangent line We start by finding the secant slope between two points, one with an x-coordinate “x”, and the other, some distance “h” away, to give a second x-coordinate “x+h”. We substitute these x – values into the slope formula as follows: The function defined by is known as the derivative of f. The notation for the derivative is or so the derivate is defined by 𝑑𝑦 𝑑𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥 ℎ 𝑓′(𝑥)= lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥 ℎ

Finding the slope of a tangent using the derivative 1. Find the derivative of and hence find the gradient of the tangent at x = 3.

2. Find the derivatives of the following functions using first principles (in terms of x). (Substitute into the derivative formula & simplify, then find the limit as h ➝ 0)

3. Find the derivatives of the following functions using first principles (in terms of x). (Substitute into the derivative formula & simplify, then find the limit as h ➝ 0)

Looking for a general rule 𝑓 𝑥 = 1 𝑥 = 𝑥 −1 𝑓 𝑥 = 1 𝑥 2 = 𝑥 −2 𝑓 𝑥 = 𝑥 = 𝑥 1 2 From the derivatives above can you define a general rule for functions of this type? If then 𝑓” 𝑥 = 𝑓 𝑥 = 𝑥 𝑛 4. Using the general rule, find the derivatives of the following functions & express your answer with only positive exponents.