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Applications of Differentiation

Secondary 4 Additional Math practice

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Applications of Differentiation practice questions

Secondary 4 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

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What this topic covers

Worked examples

Example 1

Find the yy-intercept of the tangent to y=−x3−3x+1y = -x^{3} - 3x + 1 at x=−2x = -2.
Show the worked solution
  • y=−x3−3x+1y = -x^{3} - 3x + 1
  • dydx=−3x2−3\frac{dy}{dx} = -3x^{2} - 3
  • At x=−2:m=−3(−2)2−3\text{At } x = -2: \quad m = -3(-2)^2 - 3
  • m=−15m = -15
  • y(−2)=−(−2)3−3(−2)+1y(-2) = -(-2)^3 - 3(-2) + 1
  • y(−2)=15y(-2) = 15
  • Tangent: y−15=(−15)(x+2)\text{Tangent: } y - 15 = (-15)(x + 2)
  • Set x=0:y=15−30\text{Set } x = 0: \quad y = 15 - 30

Answer: −15-15

Example 2

The side of a square increases at 5 cm/s. Find the rate of change of its area when the side is 4 cm.
Show the worked solution
  • A=s2A = s^2
  • dAds=2s\frac{dA}{ds} = 2s
  • At s=4:dAds=8\text{At } s = 4: \quad \frac{dA}{ds} = 8
  • dAdt=dAds×dsdt\frac{dA}{dt} = \frac{dA}{ds} \times \frac{ds}{dt}
  • =8×5= 8 \times 5
  • =40 cm2/s= 40 \text{ cm}^2\text{/s}

Answer: 4040

Example 3

Find the yy-coordinate of the maximum turning point of y=x3−27x+5y = x^{3} - 27x + 5.
Show the worked solution
  • y=x3−27x+5y = x^{3} - 27x + 5
  • dydx=3x2−27\frac{dy}{dx} = 3x^{2} - 27
  • Set dydx=0:\text{Set } \frac{dy}{dx} = 0:
  • 3(x+3)(x−3)=03(x + 3)(x - 3) = 0
  • x=−3 or x=3x = -3 \text{ or } x = 3
  • d2ydx2=6x\frac{d^2y}{dx^2} = 6x
  • At x=−3:d2ydx2=−18(<0⇒maximum)\text{At } x = -3: \quad \frac{d^2y}{dx^2} = -18 \quad (< 0 \Rightarrow \text{maximum})
  • y(−3)=(−3)3−27(−3)+5y(-3) = (-3)^3 - 27(-3) + 5

Answer: 5959

Questions parents ask

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Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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