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|…

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.\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.\n(c) use the graph to find an estimate,\n(i) the gradient of the curve at the point where x = 3.\n(ii) area bounded by the curve, the y - axis and the line y = -12.\n(d) write down, from the graph, the range of values of x when the curve has a negative gradient.\n(e) use the graph to solve the equation x^3 - 3x^2 = 2 for positive values of x.

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.\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.\n(c) use the graph to find an estimate,\n(i) the gradient of the curve at the point where x = 3.\n(ii) area bounded by the curve, the y - axis and the line y = -12.\n(d) write down, from the graph, the range of values of x when the curve has a negative gradient.\n(e) use the graph to solve the equation x^3 - 3x^2 = 2 for positive values of x.

Answer

Explanation:

Step1: Find the value of m

Substitute $x=-1$ into $y = x^{3}-3x^{2}$. $y=(-1)^{3}-3\times(-1)^{2}=-1 - 3=-4$

Step2: (b) Drawing the graph

This requires plotting the points from the table ($(-2,-20),(-1, - 4),(0,0),(1,-2),(2,-4),(3,0),(4,16)$) on the given scale and connecting them with a smooth - curve.

Step3: (c)(i) Estimate the gradient at $x = 3$

Draw a tangent to the curve at $x = 3$. Then, estimate the change in $y$ over the change in $x$ for the tangent. The tangent at $x = 3$ appears to be horizontal, so the gradient is approximately $0$.

Step4: (c)(ii) Estimate the area

Find the $x$ - values where $y=-12$. Then, use the trapezium rule or count the squares under the curve between the $y$ - axis and the relevant $x$ - values to estimate the area. First, find the $x$ - value when $y=-12$. Let $x^{3}-3x^{2}=-12$. By trial and error or looking at the graph, we find the $x$ - value. Then, estimate the area between the curve, $y$ - axis and $y = - 12$.

Step5: (d) Find the range of $x$ for negative gradient

The gradient of the curve $y=x^{3}-3x^{2}$ is given by $y'=3x^{2}-6x$. Set $y'<0$, $3x^{2}-6x<0$, $3x(x - 2)<0$. The solution is $0<x<2$. From the graph, we can also observe the intervals where the curve is decreasing.

Step6: (e) Solve the equation for positive $x$

Draw the line $y = 2$ on the graph. The positive $x$ - values where the line $y = 2$ intersects the curve $y=x^{3}-3x^{2}$ are the solutions. By looking at the graph, we can estimate these $x$ - values.

Answer:

(a) $m=-4$ (b) Graph is drawn as described above. (c)(i) $0$ (c)(ii) Estimated value based on graph - reading (d) $0<x<2$ (e) Estimated positive $x$ - values based on graph - reading (where $y = 2$ intersects $y=x^{3}-3x^{2}$)