Higher Unit 3 www mathsrevision com Higher Outcome

Higher Unit 3 www. mathsrevision. com Higher Outcome 2 Differentiation The Chain Rule Further Differentiation Trig Functions Further Integration Integrating Trig Functions www. mathsrevision. com

The Chain Rule for Differentiating Outcome 2 www. mathsrevision. com Higher To differentiate composite functions (such as functions with brackets in them) we can use: Example

The Chain Rule for You have 1 minute 1. Differentiate outside the bracket. Differentiating to come up with www. mathsrevision. com Higher Outcome 2 same. 2. Keep the rule. the bracket Good News ! 3. Differentiate inside the bracket. There is an easier way.

1. Differentiate outside the bracket. The Chain Rule for Differentiating www. mathsrevision. com Higher 2. Keep the bracket the same. Outcome 2 Example 3. Differentiate inside the bracket. You are expected to do the chain rule all at once

1. Differentiate outside the bracket. The Chain Rule for Differentiating 2. Keep the bracket the same. www. mathsrevision. com Higher Outcome 2 Example 3. Differentiate inside the bracket. Drill

The Chain Rule for Differentiating www. mathsrevision. com Higher Example Outcome 2

The Chain Rule for Differentiating Functions www. mathsrevision. com Higher Outcome 2 Example The slope of the tangent is given by the derivative of the equation. Re-arrange: Use the chain rule: Where x = 3:

The Chain Rule for Differentiating Functions www. mathsrevision. com Higher Outcome 2 Remember y - b = m(x – a) Is the required equation

The Chain Rule for Differentiating Functions www. mathsrevision. com Higher Outcome 2 Example In a small factory the cost, C, in pounds of assembling x components in a month is given by: Calculate the minimum cost of production in any month, and the corresponding number of components that are required to be assembled. Re-arrange

The Chain Rule for Differentiating Functions Outcome 2 www. mathsrevision. com Higher Using chain rule

The Chain Rule for Differentiating Functions Outcome 2 Higher www. mathsrevision. com Is x = 5 a minimum in the (complicated) graph? Is this a minimum? For x < 5 we have (+ve)(-ve) = (-ve) For x = 5 we have (+ve)(0) = 0 For x > 5 we have (+ve)(+ve) = (+ve) Therefore x = 5 is a minimum x=5

The Chain Rule for Differentiating Functions Higher Outcome 2 www. mathsrevision. com The cost of production: Expensive components? Aeroplane parts maybe ?

Calculus Revision Differentiate Drill Chain rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Drill Differentiate Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Straight line form Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Straight line form Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Straight line form Chain Rule Simplify Bac k Quit Next

Trig Function Differentiation www. mathsrevision. com Higher Outcome 2 The Derivatives of sin x & cos x

Trig Function Differentiation www. mathsrevision. com Higher Example Outcome 2

Trig Function Differentiation www. mathsrevision. com Higher Example Outcome 2 Simplify expression where possible Restore the original form of expression

The Chain Rule for Differentiating 1. Differentiate outside the bracket. Trig Functions Higher Outcome 2 same. 2. Keep the bracket the www. mathsrevision. com 3. Differentiate inside the bracket. Worked Example:

The Chain Rule for Differentiating Trig Functions www. mathsrevision. com Higher Example Outcome 2

The Chain Rule for Differentiating Trig Functions www. mathsrevision. com Higher Example Outcome 2

Calculus Revision Differentiate Bac k Quit Drill Next

Calculus Revision Differentiate Bac k Quit Drill Next

Calculus Revision Differentiate Bac k Quit Drill Next

Calculus Revision Differentiate Bac k Quit Drill Next

Calculus Revision Differentiate Drill Straight line form Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Straight line form Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Bac k Quit Drill Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

Calculus Revision Differentiate Drill Chain Rule Simplify Bac k Quit Next

www. mathsrevision. com Higher You Integrating have 1 minute Composite Functions to come up with Outcome 2 the rule. Harder integration we get

1. Add one to the power. Integrating Composite Functions 2. Divide by new power. www. mathsrevision. com Higher Outcome 2 3. Compensate for bracket. Example : Drill

1. Add one to the power. Integrating Composite Functions 2. Divide by new power. www. mathsrevision. com Higher Outcome 2 Example 3. Compensate for bracket. Drill You are expected to do the integration rule all at once

Integrating Composite Functions www. mathsrevision. com Higher Example Outcome 2 Drill

Integrating Composite Functions www. mathsrevision. com Higher Example Outcome 2 Drill

1. Add one to the power. Integrating Functions 2. Divide by new power. www. mathsrevision. com Higher Outcome 2 Example 3. Compensate for bracket. Integrating Drill So we have: Giving:

Calculus Revision Integrate Drill Standard Integral (from Chain Rule) Bac k Quit Next

Calculus Revision Integrate Straight line form Drill Bac k Quit Next

Calculus Revision Use standard Integral Find (from chain rule) Drill Bac k Quit Next

Calculus Revision Integrate Drill Straight line form Bac k Quit Next

Calculus Revision Use standard Integral Find (from chain rule) Drill Bac k Quit Next

Evaluate Calculus Revision Use standard Integral Drill (from chain rule) Bac k Quit Next

Calculus Revision Evaluate Drill Bac k Quit Next

Calculus Revision Find p, given Bac k Quit Drill Next

Calculus Revision A curve for which passes through the point (– 1, 2). Express y in terms of x. Use the point Bac k Quit Next

Given the acceleration a is: Calculus Revision Drill If it starts at rest, find an expression for the velocity v where Starts at rest, so v = 0, when t = 0 Bac k Quit Next

Integrating Trig Functions Outcome 2 www. mathsrevision. com Higher Integration is opposite of differentiation Worked Example

1. Integrate outside the bracket Trig 2. Integrating Keep the bracket the. Functions same www. mathsrevision. com Higher Outcome 2 3. Compensate for inside the bracket. Special Trigonometry Integrals are Worked Example

1. Integrate outside the bracket Integrating Trig Functions 2. Keep the bracket the same www. mathsrevision. com Higher Outcome 2 Example 3. Compensate for inside the bracket. Break up into two easier integrals Integrate Drill

1. Integrate outside the bracket Integrating Trig Functions 2. Keep the bracket the same www. mathsrevision. com Higher Outcome 2 Example 3. Compensate for inside the bracket. Drill Re-arrange Integrate

Integrating Functions www. mathsrevision. com Higher Example Trig (Area) Outcome 2 The diagram shows the graphs of y = -sin x and y = cos x A a) Find the coordinates of A b) Hence find the shaded area 90 o 180 o S A T C 270 o 0 o

Integrating Trig Functions www. mathsrevision. com Higher (Area) Outcome 2 Drill

Integrating Trig Functions www. mathsrevision. com Higher Example Outcome 2 Remember cos(x + y) = Drill

Integrating Trig Functions www. mathsrevision. com Higher Outcome 2 Drill

Calculus Revision Drill Find Bac k Quit Next

Calculus Revision Drill Find Bac k Quit Next

Calculus Revision Drill Find Bac k Quit Next

Calculus Revision Integrate Drill Integrate term by term Bac k Quit Next

Calculus Revision Drill Find Integrate term by term Bac k Quit Next

Calculus Revision Drill Find Bac k Quit Next

The curve Calculus Revision passes through the point Find f(x) use the given point Bac k Quit Next

Calculus Revision passes through the point express y in terms of x. If Use the point Bac k Quit Next

A curve for which Find y in terms of x. Calculus Revision passes through the point Use the point Bac k Quit Next

Are you on Target ! Outcome 2 www. mathsrevision. com Higher • Update you log book • Make sure you complete and correct ALL of the Calculus questions in the past paper booklet. Differentiation Integration Drill