Week

Term

2016

10

4

Theoretical Components STEP 1 Resources:

Maths Quest Year 11 Chapter 8

Knowledge Checklist:

what is a rate?

constant rates

variable rates

average rates of change

instantaneous rates of change

interpret graphs that illustrate rates of change

equations of tangents

what is a limit?

evaluating limits

what is a gradient function?

what is the x-intercept of a gradient function?

power rule

finding gradient functions by sketching

finding gradient functions by using the rule

finding gradient functions using your CAS

https://mathspace.co/learn/ac-methods-11/rates-of-change/the-derivative-as-a-limit-38714/the-limiting-chord-process-2518/

Practical Components

STEP 2 Complete the custom task on mathspace.co on Equations of Tangent Lines Relating the gradient function to the original function Exercise 8G: Q1 a, b, c, d; 2, 4 Complete the Time to Practice Questions The Limiting Chord Process https://mathspace.co/learn/ac-methods-11/rates-of-change/the-derivative-as-a-limit-38714/questions/ Rates of change of polynomials Exercise 8I: Q1a, b, c; 2a, b, c; 5: 9; SMM2 Q11

This week we are:

Finding equations of tangent lines

Relating the gradient function to the original function

Informally finding the rule for the gradient function

Goals

Learning Brief MM2/SMM2: Introduction to

Differential Calculus

STEP 3 Ensure you have completed Exercises 8A to 8E and the above practical before you attempt this investigation. See the following pages. Collect a hard copy from Jacqueline. This is a double investigation (ie for week 10 and 11) and will count as two “Is”, one for each week, when completed.

Investigation

Remember to scan in when you come to the Maths Area and when you leave.

QFO Quiz/Forum/Other

(Pick up paper copy from Maths Area) PART 1 Here is a graph of the cubic function 𝑦 = 𝑥3.

You are to draw (as best as possible by hand), tangents at the positions as indicated in this table, and calculate the gradient of those tangents (remember that gradient of a straight line is rise/run) and fill in the relevant row. X (coordinate) -2 -1 -0.5 0 0.5 1 2 Y (coordinate) -8 -1 -1/8 Gradient of the tangent (rise/run) 3dp accuracy

The next task is to draw a graph of x vs the gradient value. (On the x-axis use the same x-values, on the y - axis instead of drawing the graph 𝑦 = 𝑥3, you are going to draw a graph of the gradient). Plot the points (x, Gradient) - you have points. Write a comment here about what you have found, what does it look like?, What sort of function, can you find the equation of it?

Investigation

PART 2 Now consider a whole new function: 𝑦 = 𝑥3 + 3. What does the +3 do to this function? Will it change the gradient at the points evaluated above? Why/Why not? Write a statement about curves of the form 𝑦 = 𝑥3 + 𝑘, and what the gradient function would be like. What does the k do? Explain how you know this.

PART 3 Now consider a function of the form 𝑦 = 2𝑥3 What does the "2" in this function "do"? What do you think it will do to the gradient? (as you did before, find some points and then find the gradient of the tangents at those points)

X

Y

Gradient at x

What can you deduce about the "2" in relation to the value of the gradient?

Here are the graphs of some functions, and their matching "gradient functions". Follow the curves along simultaneously, can you see the correlation between the value of the gradient, whether it is positive or negative, and the original curve. What is special about the x-intercepts of the gradient function?

FUNCTION GRADIENT FUNCTION

𝑦 = 2𝑥

𝑦 = 2

𝑦 = 𝑥2 − 2

𝑦 = 2𝑥

𝑦 = −1

2𝑥2 + 4

𝑦 = −𝑥

𝑦 = 𝑥3 + 5

𝑦 = 3𝑥2

𝑦 = −𝑥2 − 2𝑥 − 3

𝑦 = −2𝑥 − 2

𝑦 = 𝑥4 + 4𝑥3 + 2𝑥2 − 4𝑥 − 3

𝑦 = 4𝑥3 + 12𝑥2 + 4𝑥 − 4 Can you see a pattern? Develop a rule for finding the gradient function of 𝑦 = 𝑎𝑥𝑛?