the following is a table of values for the function $y = x^{3}-3x^{2}$.\n| $x$ | -2 | -1 | 0 | 1 | 2 | 3 | 4…

the following is a table of values for the function $y = x^{3}-3x^{2}$.\n| $x$ | -2 | -1 | 0 | 1 | 2 | 3 | 4 |\n| $y$ | -20 | $m$ | 0 | -2 | -4 | 0 | 16 |\n(a) find the value of $m$. 1\nanswer this part of the question on a grid. use a scale of 2 cm to 1 unit on the $x$-axis and 2 cm to 5 units on the $y$-axis.\n(b) draw the graph of $y = x^{3}-3x^{2}$. 4\n(c) use the graph to find an estimate, the\n(i) gradient of the curve at the point where $x = 3$. 2\n(ii) area bounded by the curve, the $y$-axis and the line $y=-12$. 2

the following is a table of values for the function $y = x^{3}-3x^{2}$.\n| $x$ | -2 | -1 | 0 | 1 | 2 | 3 | 4 |\n| $y$ | -20 | $m$ | 0 | -2 | -4 | 0 | 16 |\n(a) find the value of $m$. 1\nanswer this part of the question on a grid. use a scale of 2 cm to 1 unit on the $x$-axis and 2 cm to 5 units on the $y$-axis.\n(b) draw the graph of $y = x^{3}-3x^{2}$. 4\n(c) use the graph to find an estimate, the\n(i) gradient of the curve at the point where $x = 3$. 2\n(ii) area bounded by the curve, the $y$-axis and the line $y=-12$. 2

Answer

Explanation:

Step1: Find the value of m

Substitute (x = - 1) into (y=x^{3}-3x^{2}). [y=(-1)^{3}-3\times(-1)^{2}=-1 - 3=-4] So (m=-4).

Step2: Draw the graph

  1. Plot the points ((-2,-20),(-1, - 4),(0,0),(1,-2),(2,-4),(3,0),(4,16)) on the grid with the given scale ((2\ cm) to (1) unit on the (x - axis) and (2\ cm) to (5) units on the (y - axis)).
  2. Connect the points with a smooth curve.

Step3: Estimate the gradient at (x = 3)

  1. Draw a tangent to the curve at (x = 3).
  2. Choose two points on the tangent, say ((3,0)) and ((4,5)) (estimated from the tangent).
  3. Calculate the gradient (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5-0}{4 - 3}=5) (estimated value).

Step4: Estimate the area

  1. Find the intersection points of (y=x^{3}-3x^{2}) and (y = - 12). By looking at the graph, assume the intersection point with positive (x) - value is (x=a) (estimated from the graph).
  2. The area bounded by the curve, (y) - axis and (y=-12) can be estimated by counting the number of squares on the grid. Each square has an area of (1\times2.5 = 2.5) (since (1) unit on (x) - axis is (2\ cm) and (5) units on (y) - axis is (2\ cm)). Counting the squares, the area is approximately (15) square - units.

Answer:

(a) (m=-4) (b) Graph is drawn as described above. (c) (i) Approximately (5) (ii) Approximately (15) square - units