8. The diagram shows a sketch of the curve with equation, y=3^x + 1
The curve intersects the y axis at point A
a. write down the co-orinate of A
ok, i get this one, which would simply involve subbing in x=0 (0,2)
b i. Use the trapezium rule with 5 ordinates (four strips) to find an approximation for
∫[1 at top, 0 at bottom] (3^x + 1)dx, giving your answer to 3 significant figures.
Do i need to memorise the trapezium rule, and how do i use it?
b ii. By considering the graph of y=3^x + 1, explain with the aid of the diagram whether your approximation will be an over estimate or an underestimate of the true value of ∫[1 at top, 0 at bottom] (3^x + 1)dx
ok i dont get this either, i thought you subbed '1' into the integrated equation, then '0', then subtracted the first from the second
memorise the trapezium rule. i forget it now but its like you add up rectangles to get the area. or something
when you draw rectangles they will either overlap or 'underlap' the actual curve (since, you know, they cant curve since theyre rectangles). this is where the error in approximation comes in for b
At C2 level, you don't have the necessary tools to integrate this equation, so the method you use for estimating a result is to take several points and draw trapeziums within between them. Then add up the areas of trapeziums. Since you're integrating between 0 and 1, and they tell you to use 4 strips, find the y-values at x= 0,0.2,0.4,0.8,1. Then use the formula for area of a trapezium (a+b)/2 * h.
To make it quicker, h will be 0.2 for all of them (the width of the strip). a and b will be the y-values at specific points. The ends will only get counted once, but the middle y-values will be counted twice. So overall, the equation will be f(0)+f(1) + 2*(f(0.2)+f(0.4)+f(0.6)+f(0.8))/2 * 0.2
These questions are pointless and involve lots of numberwork
Then you have to work out if the trapeziums undercut or overcut the actual curve, and so whether the area you've worked out will be an overestimate or an underestimate.
ok thanks, i get the trapezium rule now, so pick x vlaues keeping the distance between them the same, then find the y values and sub in to the equation. Ok thanks.
At C2 level, you don't have the necessary tools to integrate this equation, so the method you use for estimating a result is to take several points and draw trapeziums within between them. Then add up the areas of trapeziums. Since you're integrating between 0 and 1, and they tell you to use 4 strips, find the y-values at x= 0,0.2,0.4,0.8,1. Then use the formula for area of a trapezium (a+b)/2 * h.
To make it quicker, h will be 0.2 for all of them (the width of the strip). a and b will be the y-values at specific points. The ends will only get counted once, but the middle y-values will be counted twice. So overall, the equation will be f(0)+f(1) + 2*(f(0.2)+f(0.4)+f(0.6)+f(0.8))/2 * 0.2
These questions are pointless and involve lots of numberwork
Then you have to work out if the trapeziums undercut or overcut the actual curve, and so whether the area you've worked out will be an overestimate or an underestimate.