Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A. (a) Using calculus, show that the x coordinate of A is 1. The curve crosses the x-axis at the points B(2, 0) and C(−rac{1}{4}, 0). The finite region R, shown shaded in Figure 2, is bounded by the curve, the line AB, and the x-axis. (b) Use integration to find the area of the finite region R, giving your answer to 2 decimal places.

A-Level Edexcel Maths Pure Binomial Expansion: Figure 2 shows a sketch of part

Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 2

Question 1

Figure-2-shows-a-sketch-of-part-of-the-curve-with-equation--y-=-4x³-+-9x²---30x---8,---0.5-≤-x-≤-2.2--The-curve-has-a-turning-point-at-the-point-A-Edexcel-A-Level Maths Pure-Question 1-2015-Paper 2.png

Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A. (a) Using calcul... show full transcript

Worked Solution & Example Answer:Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 2

Step 1

Using calculus, show that the x coordinate of A is 1.

96%

114 rated

Answer

To find the x-coordinate of the turning point A, we first need to calculate the derivative of the function.

Starting with the given function:

$y = 4x^3 + 9x^2 - 30x - 8$

The first derivative is:

$\frac{dy}{dx} = 12x^2 + 18x - 30$

Now, we set the derivative equal to zero to find the turning points:

$12x^2 + 18x - 30 = 0$

Dividing the entire equation by 6 gives:

$2x^2 + 3x - 5 = 0$

Using the quadratic formula, where (a = 2), (b = 3), and (c = -5):

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)}$

Calculating the discriminant:

$b^2 - 4ac = 9 + 40 = 49$

Thus, the solutions for x are:

$x = \frac{-3 \pm 7}{4}$

This produces two potential solutions:

$x = \frac{4}{4} = 1$
$x = \frac{-10}{4} = -2.5$

Since we're considering the range (-0.5 ≤ x ≤ 2.2), we conclude that the x-coordinate of point A is indeed 1.

Step 2

Use integration to find the area of the finite region R, giving your answer to 2 decimal places.

99%

104 rated

Answer

To find the area of region R enclosed by the curve, line AB, and the x-axis, we first need to establish the integration limits which are between C(−\frac{1}{4}, 0) and B(2, 0).

Thus, the area A can be defined as:

$A = \int_{-\frac{1}{4}}^{2} (f(x) - 0) \, dx$

where f(x) is the given curve equation. Substituting the curve's equation into the integral gives:

$A = \int_{-\frac{1}{4}}^{2} (4x^3 + 9x^2 - 30x - 8) \, dx$

Calculating the integral, we have:

Find the antiderivative:
- $\int 4x^3 dx = x^4$
- $\int 9x^2 dx = 3x^3$
- $\int -30x dx = -15x^2$
- $\int -8 dx = -8x$

So the antiderivative is:

$F(x) = x^4 + 3x^3 - 15x^2 - 8x$

Evaluating from (-\frac{1}{4}) to 2:
- $F(2) = 2^4 + 3(2^3) - 15(2^2) - 8(2)$
- $= 16 + 24 - 60 - 16 = -36$
- $F(-\frac{1}{4}) = \left(-\frac{1}{4}\right)^4 + 3\left(-\frac{1}{4}\right)^3 - 15\left(-\frac{1}{4}\right)^2 - 8\left(-\frac{1}{4}\right)$
- This simplifies to:
- $= \frac{1}{256} - \frac{3}{64} - \frac{15}{16} + 2 = \frac{1 - 12 - 240 + 512}{256} = \frac{261}{256}$

Finally, we can calculate the area:

$A = F(2) - F(-\frac{1}{4}) = -36 - \frac{261}{256}$

Calculating this gives us: $A = -36 - 1.019 = -37.019$

Since we're interested in the absolute area, we can express:

$|A| = 37.019.$

Thus, giving the final answer to two decimal places results in:

$A \approx 37.02.$

Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 2

Worked Solution & Example Answer:Figure 2 shows a sketch of part of the curve with equation y = 4x³ + 9x² - 30x - 8, -0.5 ≤ x ≤ 2.2 The curve has a turning point at the point A - Edexcel - A-Level Maths Pure - Question 1 - 2015 - Paper 2

Using calculus, show that the x coordinate of A is 1.

Use integration to find the area of the finite region R, giving your answer to 2 decimal places.

Join the A-Level students using SimpleStudy...