Second Derivative 6A

Starter: use graph software to draw these graphs Second derivative KUS objectives BAT find and use the second derivative Starter: use graph software to draw these graphs 𝑦= 𝑥 3 + 3𝑥 2 +1 Can you see any connections between them? 𝑦= 3𝑥 2 +6x 𝑦=6𝑥+6

𝐈𝐧𝐯𝐞𝐬𝐭𝐢𝐠𝐚𝐭𝐞 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭 Notes Geogebra: second derivative 𝐈𝐧𝐯𝐞𝐬𝐭𝐢𝐠𝐚𝐭𝐞 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐠𝐫𝐚𝐝𝐢𝐞𝐧𝐭

Goes from negative to positve Notes Geogebra: second derivative - + - + - + Graph of f(x) Local minimum - + + Graph of f’(x) Goes from negative to positve + + + + + + Graph of f’’(x) Gradient of the gradient Is positive

To find the nature of a stationary point we can Notes To find the nature of a stationary point we can Find where the gradient (first derivative) is zero Find the second derivative If the 2nd derivative is < 0 → MAXIMUM If the 2nd derivative is > 0 → MINIMUM The bit you are likely to find difficult is solving equations to find coordinates Here are some tips: Write any negative powers as fractions Multiply through to get rid of fractions Simplify Factorise when possible

When x = 2, therefore (2, 3) is a local minimum WB20 Find the coordinates of the stationary point on the curve 𝑦=𝑥+ 4 𝑥 2 and determine its nature When x = 2, therefore (2, 3) is a local minimum

When x = ¼ , therefore (¼ , -¼) is a local minimum WB21 Find the coordinates of the stationary point on the curve 𝑦=𝑥− 𝑥 and determine its nature When x = ¼ , therefore (¼ , -¼) is a local minimum

Practice 1

Practice 1 solutions

Practice 2

Practice 2 solutions

Given that there is a stationary point where x=2, find the value of k WB 22 The curve 𝑦= 𝑥 3 −𝑘 𝑥 2 +2𝑥−5 has two stationary points Find 𝑑𝑦 𝑑𝑥 Given that there is a stationary point where x=2, find the value of k Determine whether this stationary point is a min or max Find the x-coordinate of the other stationary point

One thing to improve is – KUS objectives BAT find and use the second derivative self-assess One thing learned is – One thing to improve is –

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