Leaving Cert Mathematics Graphs of Functions: The functions $f$ and $g$ are de

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ as $f : x \mapsto 2x^2 - 3x + 2$ and $g : x \mapsto x^3 + x + 7$ - Leaving Cert Mathematics - Question 5 - 2012

Question 5

$The-functions-$f$-and-$g$-are-defined-for-$x-\in-\mathbb{R}$-as-$f-:-x-\mapsto-2x^2---3x-+-2$---and--$g-:-x-\mapsto-x^3-+-x-+-7$-Leaving Cert Mathematics-Question 5-2012.png$

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ as $f : x \mapsto 2x^2 - 3x + 2$ and $g : x \mapsto x^3 + x + 7$. (a) Find the co-ordinates of the t... show full transcript

Worked Solution & Example Answer:The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ as $f : x \mapsto 2x^2 - 3x + 2$ and $g : x \mapsto x^3 + x + 7$ - Leaving Cert Mathematics - Question 5 - 2012

Step 1

Find the co-ordinates of the two points where the curves $y = f(x)$ and $y = g(x)$ intersect.

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Answer

To find the points of intersection, set the equations equal:

$2x^2 - 3x + 2 = x^3 + x + 7$

Rearranging gives:

$x^3 - 2x^2 + 4x + 5 = 0$

To find the roots of this polynomial, we can apply the factor theorem or try possible rational roots. After testing $x = -1$ and $x = 5$ , we arrive at:

For $x = -1$ :

ightarrow (-1, 7)$$

For $x = 5$ :

ightarrow (5, 37)$$

Thus, the co-ordinates where the curves intersect are the points $(-1, 7)$ and $(5, 37)$ .

Step 2

Find the area of the region enclosed between the two curves.

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Answer

To find the area between the curves, we integrate the difference:

$A = \int_{{x_1}}^{{x_2}} (g(x) - f(x))dx$ where $x_1 = -1$ and $x_2 = 5$ .

Calculating:

$A = \int_{{-1}}^{{5}} (x^3 + x + 7 - (2x^2 - 3x + 2))dx$

This simplifies to:

$= \int_{{-1}}^{{5}} (x^3 - 2x^2 + 5x + 5)dx$

Evaluating the integral gives:

$A = \left[ \frac{x^4}{4} - \frac{2x^3}{3} + \frac{5x^2}{2} + 5x \right]_{-1}^{5}$

Calculating at $x = 5$ and $x = -1$ , we obtain:

$= (\frac{625}{4} - \frac{250}{3} + \frac{125}{2} + 25) - (\frac{1}{4} + \frac{2}{3} - \frac{5}{2} - 5)$

This results in an area of 36 square units, hence the final result is:

$A = 36$ .

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ as $f : x \mapsto 2x^2 - 3x + 2$ and $g : x \mapsto x^3 + x + 7$ - Leaving Cert Mathematics - Question 5 - 2012

Worked Solution & Example Answer:The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ as $f : x \mapsto 2x^2 - 3x + 2$ and $g : x \mapsto x^3 + x + 7$ - Leaving Cert Mathematics - Question 5 - 2012

Find the co-ordinates of the two points where the curves $y = f(x)$ and $y = g(x)$ intersect.

Find the area of the region enclosed between the two curves.

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