Complete the table below. Write your values correct to two decimal places where necessary.
| x | 0 | 0.5 | 1 | ln(4) |
|:---------:|:---... show full transcript
Worked Solution & Example Answer:Complete the table below - Leaving Cert Mathematics - Question 3 - 2016
Step 1
Complete the table below. Write your values correct to two decimal places where necessary.
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Answer
To complete the table, we need to evaluate the functions f(x) and g(x) at the specified values of x:
For f(x) = 2/e^x:
When x = 0:
f(0)=2/e0=2/1=2.00
When x = 0.5:
f(0.5)=2/e0.5≈2/1.6487≈1.21
When x = 1:
f(1)=2/e1≈2/2.7183≈0.74
When x = ln(4):
f(ln4)=2/eln4=2/4=0.50
For g(x) = e^x - 1:
When x = 0:
g(0)=e0−1=1−1=0.00
When x = 0.5:
g(0.5)=e0.5−1≈1.6487−1≈0.65
When x = 1:
g(1)=e1−1≈2.7183−1≈1.72
When x = ln(4):
g(ln4)=eln4−1=4−1=3.00
Thus, the completed table is:
x
0
0.5
1
ln(4)
f(x)
2.00
1.21
0.74
0.50
g(x)
0.00
0.65
1.72
3.00
Step 2
In the grid on the right, use the table to draw the graphs of f(x) and g(x) in the domain 0 ≤ x ≤ ln(4). Label each graph clearly.
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Answer
To draw the graphs of f(x) and g(x):
Plotting Points:
For f(x): (0, 2.00), (0.5, 1.21), (1, 0.74), (ln(4), 0.50)
For g(x): (0, 0.00), (0.5, 0.65), (1, 1.72), (ln(4), 3.00)
Graph Lines:
Connect the points for f(x) with a smooth curve and label it.
Connect the points for g(x) with a smooth curve and label it.
Ensure both graphs are properly labeled and distinguishable on the grid.
Step 3
Use your graphs to estimate the value of x for which f(x) = g(x).
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Answer
From the graph, observe the intersection point of f(x) and g(x). By visual estimation, this intersection appears to occur at approximately x = 0.7.
Step 4
Solve f(x) = g(x) using algebra.
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Answer
To solve the equation algebraically:
Set the two functions equal to each other:
2/ex=ex−1
Multiply both sides by e^x to eliminate the denominator:
2=e2x−ex
Rearranging gives us:
e2x−ex−2=0
Letting y = e^x, we have:
y2−y−2=0
Factoring the quadratic yields:
(y−2)(y+1)=0
The solutions are:
y = 2 → e^x = 2 → x = ln(2)
y = -1 (not possible for e^x)
Thus, the only solution is:
x=ln(2)≈0.693
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