The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below. (a) On the same diagram, draw the graph of $h(x) = e^{-x}$, $x \in \mathbb{R}$, in the domain $0 \leq x \leq 1$. (b) Find the area enclosed by $g(x) = e^x$, $h(x) = e^{-x}$, and the line $x = 0.75$. Give your answer correct to 4 decimal places.

Leaving Cert Mathematics Integration: The graph of the function $g(x)

The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below - Leaving Cert Mathematics - Question 6 - 2017

Question 6

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The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below. (a) On the same diagram, draw the graph of $h(x) = e^{... show full transcript

Worked Solution & Example Answer:The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below - Leaving Cert Mathematics - Question 6 - 2017

Step 1

Draw the graph of $h(x) = e^{-x}$

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Answer

To draw the graph of $h(x) = e^{-x}$ in the interval $0 \leq x \leq 1$ :

Calculate points:
- At $x = 0$ : $h(0) = e^{0} = 1$
- At $x = 0.2$ : $h(0.2) \approx e^{-0.2} \approx 0.8187$
- At $x = 0.4$ : $h(0.4) \approx e^{-0.4} \approx 0.6703$
- At $x = 0.6$ : $h(0.6) \approx e^{-0.6} \approx 0.5488$
- At $x = 0.8$ : $h(0.8) \approx e^{-0.8} \approx 0.4493$
- At $x = 1$ : $h(1) = e^{-1} \approx 0.3679$
Plot these points on the graph together with the graph of $g(x)$ .

Step 2

Find the area enclosed by the curves

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Answer

To find the area between the curves $g(x)$ and $h(x)$ from $0$ to $0.75$ :

Set up the integral for the area $A$ : $A = \int_{0}^{0.75} (g(x) - h(x)) \, dx$ $= \int_{0}^{0.75} (e^x - e^{-x}) \, dx$
Calculate the integral:
- The antiderivative of $e^x$ is $e^x$ .
- The antiderivative of $e^{-x}$ is $-e^{-x}$ .
Thus, $A = \left[ e^x + e^{-x} \right]_{0}^{0.75}$
Evaluate the definite integral: $A = \left[ e^{0.75} + e^{-0.75} \right] - \left[ e^{0} + e^{0} \right]$ $= e^{0.75} + e^{-0.75} - 2$ $\approx 2.1170 - 2 = 0.1170$
Thus, the area enclosed is approximately $0.1170$ .

The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below - Leaving Cert Mathematics - Question 6 - 2017

Worked Solution & Example Answer:The graph of the function $g(x) = e^x$, $x \in \mathbb{R}$, $0 \leq x \leq 1$, is shown on the diagram below - Leaving Cert Mathematics - Question 6 - 2017

Draw the graph of $h(x) = e^{-x}$

Find the area enclosed by the curves

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