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(a) Sketch the graph of y = 3^(-x), x ∈ ℝ, showing the coordinates of the point at which the graph meets the y-axis - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 2
Question 7
(a) Sketch the graph of y = 3^(-x), x ∈ ℝ, showing the coordinates of the point at which the graph meets the y-axis. (b) Copy and complete the table, giving the val... show full transcript
Worked Solution & Example Answer:(a) Sketch the graph of y = 3^(-x), x ∈ ℝ, showing the coordinates of the point at which the graph meets the y-axis - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 2
Step 1
Sketch the graph of y = 3^(-x) and y-intercept
Answer
To sketch the graph of the function ( y = 3^{-x} ), note that this is an exponential decay function. It intersects the y-axis when ( x = 0 ). Therefore, the coordinates of the point at which the graph meets the y-axis are (0, 1), since ( y = 3^{0} = 1 ). The graph will asymptotically approach the x-axis as ( x ) increases.
Step 2
Copy and complete the table, giving the values of 3^x to 3 decimal places
Answer
To complete the table with the values of ( 3^x ), we calculate the missing entries:
- ( 3^{0.2} \approx 1.246 )
- ( 3^{0.4} \approx 1.552 )
- ( 3^{0.6} \approx 1.933 )
- ( 3^{0.8} \approx 2.408 )
- ( 3^{1} = 3 )
The completed table is:
| x | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 |
|---|---|---|---|---|---|---|
| 3^x | 1 | 1.246 | 1.552 | 1.933 | 2.408 | 3 |
Step 3
Use the trapezium rule to approximate the value of \( \int_0^1 3^x \, dr \)
Answer
To apply the trapezium rule, we use the trapezoidal sum:
[ A \approx \frac{1}{2} \Delta x \left( f(x_0) + f(x_1) + 2 \sum_{i=1}^{n-1} f(x_i) \right) ]
For this problem, ( \Delta x = 0.2 ) and we need to evaluate:
[ A \approx \frac{1}{2} \times 0.2 \left( 1 + 3 + 2(1.246 + 1.552 + 1.933 + 2.408) \right) ]
Calculating the sum gives: [ A \approx 0.1 \left( 4 + 2(7.139) \right) \approx 0.1 \times 18.278 \approx 1.8278 ]
Thus, the approximate value of ( \int_0^1 3^x , dr ) is 1.8278.