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The diagram shows the graph of the function $f(x) = 6x - x^2$ in the domain $0 < x < 6$, $x eq ext{R}$ - Leaving Cert Mathematics - Question Question 1 - 2013

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Question Question 1

The-diagram-shows-the-graph-of-the-function-$f(x)-=-6x---x^2$-in-the-domain-$0-<-x-<-6$,-$x--eq--ext{R}$-Leaving Cert Mathematics-Question Question 1-2013.png

The diagram shows the graph of the function $f(x) = 6x - x^2$ in the domain $0 < x < 6$, $x eq ext{R}$. (a) Find $f(0)$, $f(1)$, $f(2)$, $f(3)$, $f(4)$, and $f(... show full transcript

Worked Solution & Example Answer:The diagram shows the graph of the function $f(x) = 6x - x^2$ in the domain $0 < x < 6$, $x eq ext{R}$ - Leaving Cert Mathematics - Question Question 1 - 2013

Step 1

Find $f(0)$, $f(1)$, $f(2)$, $f(3)$, $f(4)$, and $f(6)$

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Answer

f(0)=6(0)−(0)2=0f(0) = 6(0) - (0)^2 = 0
f(1)=6(1)−(1)2=5f(1) = 6(1) - (1)^2 = 5
f(2)=6(2)−(2)2=8f(2) = 6(2) - (2)^2 = 8
f(3)=6(3)−(3)2=9f(3) = 6(3) - (3)^2 = 9
f(4)=6(4)−(4)2=8f(4) = 6(4) - (4)^2 = 8
f(5)=6(5)−(5)2=5f(5) = 6(5) - (5)^2 = 5
f(6)=6(6)−(6)2=0f(6) = 6(6) - (6)^2 = 0

Step 2

Use the trapezoidal rule to estimate the area of the region enclosed between the curve and the x-axis

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Answer

The trapezoidal rule formula for estimating the area is given by:
A=h2[y0+2y1+2y2+...+2yn−1+yn]A = \frac{h}{2} [y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n]
Where h=1h = 1 (the width of each subinterval) and:

  • y0=f(0)=0y_0 = f(0) = 0
  • y1=f(1)=5y_1 = f(1) = 5
  • y2=f(2)=8y_2 = f(2) = 8
  • y3=f(3)=9y_3 = f(3) = 9
  • y4=f(4)=8y_4 = f(4) = 8
  • y5=f(5)=5y_5 = f(5) = 5
  • y6=f(6)=0y_6 = f(6) = 0
    Estimating the area gives:
    A=12[0+2(5)+2(8)+2(9)+2(8)+2(5)+0]=12[0+10+16+18+16+10+0]=12[70]=35A = \frac{1}{2} [0 + 2(5) + 2(8) + 2(9) + 2(8) + 2(5) + 0] = \frac{1}{2} [0 + 10 + 16 + 18 + 16 + 10 + 0] = \frac{1}{2} [70] = 35
    Thus, the estimated area is 3535.

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