Query 7.2

I've been asked how to differentiate $y=3\times2^x+1$ by someone in 7.2. Remember you can only differentiate exponential functions of the form $e^{f(x)}$. \[ y=3 \times e^{(\ln 2 \times x)}+1 \\ \frac{dy}{dx}=3\ln 2 \times e^{(\ln 2\times x)}+1 \\= 3 \ln 2 \times2^x+1 \] So when $x=0$, $\dfrac{dy}{dx}=3\ln 2 \times 2^0 +1= 3\ln 2+1=\ln 8+1=\ln {8e}$

Joke from Rach in 2.4

What do you get if you divide the circumference of a pumpkin by it's diameter?

...Pumpkin Pi

4th Extension answers

Answers to 4th year extension

1. \[ \\13000 \times 0.85^? = 1000 \\0.85^? = \frac{1}{13} \\?=log_{0.85}{\frac{1}{13}}= 15.8 \]

   i.e. 16 years is needed (or you can do it by trial and error on the calculator) YOU DO NOT NEED TO KNOW THE LOG METHOD, trial and error using a calculator is sufficient at this stage

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2. a. \begin{align*} \\y^2 &= 5-3x^2 \\x^2 &= 5-y^2 \\x &=\sqrt{\frac{5-y^2}{3}} \end{align*}

   b. \begin{align*} \\ y(2x^2-1) &= 2x^2+3 && \textit{multiply by denominator} \\ 2x^2y-y&=2x^2+3 && \textit{expand the brackets} \\ 2x^2y-2x^2&=y+3 && \textit{get all the }x^2 \textit{ stuff on the left} \\ x^2(2y-2)&=y+3 && \textit{take out common factor of }x^2 \\ x^2&=\frac{y+3}{2y-2} \\ x&=\sqrt{\frac{y+3}{2y-2}} \end{align*}

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3. \[ \\3x -2(30-x)=55 \\3x -60+2x=55 \\5x -60=55 \\x =23 \]

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4. Rearranging gives \[y=\frac{3}{2}x + \frac{5}{2} \] Gradient is $\dfrac{3}{2}$, so for the new line: \[ y=\frac{3}{2}x+c \] Now use (2,-4) in here to get c ie \begin{align*} -4 = \frac{3}{2}\times 2 + c \\ \therefore c = -7 \\ \therefore y=\frac{3}{2}x-7 && \times \textit{both sides by 2 and subtract 3}x \\2y -3x = -14 && \textit{to get the required format} \\3x-2y = 14 && \textit{or this, either is fine} \end{align*}

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5. 
   

4th Year Extension Test Practice

Revision - 21 marks

1. A car depreciates (decreases) by 15% of its value every year. After how many years does a car bought for £13000 depreicate to less then £1000?

[3 marks]

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2. Make $x$ the subject:

1. $y=\sqrt{5-3x^2}$
2. $y=\dfrac{2x^2+3}{2x^2-1}$

[6 marks]

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3. A slug is climbing a wall. On an "upper" day he climbs upward 3m. On a "dozy" day he slides 2m downward. If after 30 days the slug is 55m up the wall, how many "upper" days did he have?

[4 marks]

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4. Find the equation of a line which is parallel to the line $2y-3x=5$ but goes through the point (2,-4). Give your answer in the form $ay+bx=c$

[4 marks]

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5. Shade regions

[4 marks]

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Additional topics include: Sets, ratio, percentages

Link to the answers

Promoting higher level thinking

John Mason and friends: ‘Questions and Prompts for Mathematical Thinking’ (ATM publication):http://www.atm.org.uk/Shop/Questions-and-Prompts-for-Mathematical-Thinking/dis002  

The ATM book ‘Thinkers’ is very good too.

Roger Nelsen’s ‘Proofs Without Words’ books: http://www.amazon.co.uk/Proofs-without-Words-Exercises-Classroom/dp/0883857006  

and http://www.amazon.co.uk/Proofs-without-Words-Classroom-Materials/dp/0883857219/ref=pd_bxgy_b_img_y