S56 (5.3) Further Calculus.notebook March 24, 2016

Transcription

1 Daily Practice Today we will be learning how to differentiate using the Chain Rule. Homework Solutions Video online - please mark 2009 P2 Polynomials HW Online due We use the Chain Rule when we are differentiating expressions of the form (x + a) n and (ax + b) n Rather than having to expand out the brackets & differentiate, we instead 1. Multiply by the power 2. Keep everything in the brackets the same. 3. Reduce the power by 1 4. Multiply all by the derivative of what's in the brackets If y = (ax + b) n / = an(ax + b) n - 1 Differentiate the following 1. y = (3x + 5) 5 2. f(x) = (4x - 7) -6 Trickier Differentiate the following 4. y = 2(2x 4-1) 3 3. f(x) = 1 剹 (5x - 1) 2

2 Trickier Differentiate the following 2 5. y = (3t 3 + 4) 2 Trickier Differentiate the following 6. f(x) = 5 剹 (1-6x) Daily Practice Today we will be continuing to practise 2009 P2 differentiating using the Chain Rule. Homework Due Tuesday! Trickier Differentiate the following 7. f(x) =, find the value of f'(x) when x = 1 Trickier Differentiate the following 8. y = (2v 2 + 4v) 1 4 /

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4 Daily Practice Exemplar paper 2 Today we will be learning how to differentiate cos and sinx. Homework due Tuesday Mark your homework Differentiate the following 1. y = 3cosx

5 Differentiate the following 3. Find the equation of the tangent to 5cosx at x = Differentiate the following 2. Given f(x) = sinx - 2cosx, find f'( ) From HSN Differentiate the following 3 + 2x 5. 2 cosx y = x 2 Differentiate the following 4. y = 5x 2-2cosx Daily Practice P2

6 Using the Chain Rule Today we will be continuing to differentiate trig. functions 1. y = sin5x 2. y = cos(4x - 3) Homework Due Tomorrow. Topics: Relationships and Calculus Assessment 3. f(x) = sin 5 x 4. y = cosx polynomials and quadratics Solving trig. equations with double angle Differentiation with questions in context & differentiating trig. functions Integrating functions and trig. functions Daily Practice Examples using the Chain Rule: 1. y = sin5x 2. y = cos(4x - 3) 3. f(x) = sin 5 x 4. y = cosx

7 Leibniz Notation The Chain rule makes more sense when we write it in Leibniz Notation. Given y = (ax + b) n we can let u = ax + b therefore y = u n We can split the derivatives up Differentiate (i) y = sin5x 0 (ii) y = (3x 2 + 1) 3 Integrating Functions of the form (ax + b) n 1. Increase the power by 1 2. Divide by the new power Today we will be learning how to integrate functions of the form (ax + b) n Homework Due! 3. Also divide by the derivative of the expression inside the factor This rule only works when there are linear expressions inside the bracket. 4. Don't forget C! Integrating Functions of the form (ax + b) n Integrating Functions of the form (ax + b) n 2. Find 1. Find (5x + 4) 5

8 Integrating Functions of the form (ax + b) n 3. Find Ex. 14J Daily Practice Topic: Solving equations with wave function Given that 2sinx 0-6cosx 0 = 8cos(x - 150) 0 solve 2sinx 0-6cosx 0 = 2, for 0 x Today we will be learning how to integrate trigonometric functions. Evaluation Booklet available on Maths website schoolmathematics.weebly.com Further Calculus Homework Online due Once again, think about integrating as being the exact opposite process as what you did when differentiating. For example: y = sin4x 0 = 4cos4x 0

9 Daily Practice P2 Today we will be continuing to learn how to integrate trig. functions. Homework due Revise for assessment over Easter. 2. sin(2t - )dt 1. Find 3cos(6x + ) 1 3. Find ( cos(3x) - sin(⅓x+ 1) ) to 3 d.p A curve for which / = 3sin(2x) passes through the point ( Ă Ĉĉ π, 3). Find y in terms of x.

10 5. Find the value of (cos3x - x) π/2 π/6