I need help with this picture.
1 answer:
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See the graph attached.
The midpoint rule states that you can calculate the area under a curve by using the formula:
![M_{n} = \frac{b - a}{2} [ f(\frac{x_{0} + x_{1} }{2}) + f(\frac{x_{1} + x_{2} }{2}) + ... + f(\frac{x_{n-1} + x_{n} }{2})]](https://tex.z-dn.net/?f=M_%7Bn%7D%20%3D%20%5Cfrac%7Bb%20-%20a%7D%7B2%7D%20%5B%20f%28%5Cfrac%7Bx_%7B0%7D%20%2B%20x_%7B1%7D%20%7D%7B2%7D%29%20%2B%20%20f%28%5Cfrac%7Bx_%7B1%7D%20%2B%20x_%7B2%7D%20%7D%7B2%7D%29%20%2B%20...%20%2B%20%20f%28%5Cfrac%7Bx_%7Bn-1%7D%20%2B%20x_%7Bn%7D%20%7D%7B2%7D%29%5D)
In your case:
a = 0
b = 1
n = 4
x₀ = 0
x₁ = 1/4
x₂ = 1/2
x₃ = 3/4
x₄ = 1
Therefore, you'll have:
![M_{4} = \frac{1 - 0}{4} [ f(\frac{0 + \frac{1}{4} }{2}) + f(\frac{ \frac{1}{4} + \frac{1}{2} }{2}) + f(\frac{\frac{1}{2} + \frac{3}{4} }{2}) + f(\frac{\frac{3}{4} + 1} {2})]](https://tex.z-dn.net/?f=M_%7B4%7D%20%3D%20%5Cfrac%7B1%20-%200%7D%7B4%7D%20%5B%20f%28%5Cfrac%7B0%20%2B%20%20%5Cfrac%7B1%7D%7B4%7D%20%7D%7B2%7D%29%20%2B%20%20f%28%5Cfrac%7B%20%5Cfrac%7B1%7D%7B4%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%7B2%7D%29%20%2B%20%20f%28%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%7D%7B2%7D%29%20%2B%20f%28%5Cfrac%7B%5Cfrac%7B3%7D%7B4%7D%20%2B%201%7D%20%7B2%7D%29%5D)
![M_{4} = \frac{1}{4} [ f(\frac{1}{8}) + f(\frac{3}{8}) + f(\frac{5}{8}) + f(\frac{7}{8})]](https://tex.z-dn.net/?f=M_%7B4%7D%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5B%20f%28%5Cfrac%7B1%7D%7B8%7D%29%20%2B%20%20f%28%5Cfrac%7B3%7D%7B8%7D%29%20%2B%20%20f%28%5Cfrac%7B5%7D%7B8%7D%29%20%2B%20f%28%5Cfrac%7B7%7D%7B8%7D%29%5D)
Now, to evaluate your f(x), you need to look at the graph and notice that:
f(x) = x - x³
Therefore:
![M_{4} = \frac{1}{4} [(\frac{1}{8} - (\frac{1}{8})^{3}) + (\frac{3}{8} - (\frac{3}{8})^{3}) + (\frac{5}{8} - (\frac{5}{8})^{3}) + (\frac{7}{8} - (\frac{7}{8})^{3})]](https://tex.z-dn.net/?f=M_%7B4%7D%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5B%28%5Cfrac%7B1%7D%7B8%7D%20-%20%28%5Cfrac%7B1%7D%7B8%7D%29%5E%7B3%7D%29%20%2B%20%28%5Cfrac%7B3%7D%7B8%7D%20-%20%28%5Cfrac%7B3%7D%7B8%7D%29%5E%7B3%7D%29%20%2B%20%28%5Cfrac%7B5%7D%7B8%7D%20-%20%28%5Cfrac%7B5%7D%7B8%7D%29%5E%7B3%7D%29%20%2B%20%28%5Cfrac%7B7%7D%7B8%7D%20-%20%28%5Cfrac%7B7%7D%7B8%7D%29%5E%7B3%7D%29%5D)
![M_{4} = \frac{1}{4} [(\frac{1}{8} - \frac{1}{512}) + (\frac{3}{8} - \frac{27}{512}) + (\frac{5}{8} - \frac{125}{512}) + (\frac{7}{8} - \frac{343}{512})]](https://tex.z-dn.net/?f=M_%7B4%7D%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5B%28%5Cfrac%7B1%7D%7B8%7D%20-%20%5Cfrac%7B1%7D%7B512%7D%29%20%2B%20%28%5Cfrac%7B3%7D%7B8%7D%20-%20%5Cfrac%7B27%7D%7B512%7D%29%20%2B%20%28%5Cfrac%7B5%7D%7B8%7D%20-%20%5Cfrac%7B125%7D%7B512%7D%29%20%2B%20%28%5Cfrac%7B7%7D%7B8%7D%20-%20%5Cfrac%7B343%7D%7B512%7D%29%5D)
M₄ = 1/4 · (2 - 478/512)
= 0.2666
Hence, the <span>area of the region bounded by y = x³ and y = x</span> is approximately 0.267 square units.
The midpoint rule states that you can calculate the area under a curve by using the formula:
In your case:
a = 0
b = 1
n = 4
x₀ = 0
x₁ = 1/4
x₂ = 1/2
x₃ = 3/4
x₄ = 1
Therefore, you'll have:
Now, to evaluate your f(x), you need to look at the graph and notice that:
f(x) = x - x³
Therefore:
M₄ = 1/4 · (2 - 478/512)
= 0.2666
Hence, the <span>area of the region bounded by y = x³ and y = x</span> is approximately 0.267 square units.
Compound Interest
A total of $20,000 is invested in different assets.
45% is invested in a Treasury bond for 3 years at 4.35 APR compounded annually.
For this investment, the principal is P = 0.45*$20,000 = $9,000.
The compounding period is yearly, thus the interest rate is:
i = 4.35 / 100 = 0.0435
The duration (in periods) is n = 3
Calculate the final value with the formula:
Substituting:
The second investment is a CD at 3.75% APR for 3 years compounded annually. The parameters for the calculations are as follows:
P = 15% of $20,000 = $3,000
i = 3.75 / 100 = 0.0375
n = 3
Calculating:
The third investment is in a stock plan. The initial value of the investment is
P = 20% of $20,000 = $4,000
By the end of the first year, the stock plan increased by 8%, thus its value is:
M1 = $4000 * 1.2 = $4,800
By the end of the second year, the stock plan decreased by 4$, thus the value is:
M2 = $4,800 * 0.96 = $4,608
Finally, the stock plan increases by 6%, resulting in a final balance of:
M3 = $4,608 * 1.06 = $4,884.48
Finally, the last investment is in a savings account at 2.90% APR compounded annually for 3 years (not mentioned, but assumed).
P = $20,000 - $9,000- $3,000 - $4,000 = $4,000
i = 2.90 / 100 = 0.029
n = 3
Calculating:
To summarize, the final balances for each type of investment at the end of the third year are:
Investment 1; $10,226.33
Investment 2: $3,350.31
Investment 3: $4,884.48
Investment 4: $4,358.19
Total balance: $22,819.32