An exponential function is a function one in which the a base value is
raised to the power of a variable.
The correct responses are;
- 1. Negative exponent; [tex]\displaystyle a^{-x} = \frac{1}{a^x}[/tex] Fractional exponent; [tex]a^{\frac{n}{m} } = \left(\sqrt[m]{x} \right)^n[/tex]
- 2. When saving for less than 12 years use the first account. When saving for more than 12 years use the second account.
- 3. An exponential growth function is; A = (1 + r)ⁿ. A real world situation of exponential growth function, is the amount in a compound interest account.
- An exponential decay function is A = (1 - r)ⁿ. An A real world situation of exponential decay is the price value of a device with time after purchase.
Reasons:
1. A negative exponent is the reciprocal of the given exponent with the sign of the power changed from negative to positive.
- [tex]\displaystyle a^{-x} = \frac{1}{a^x}[/tex]
A fractional exponent is an exponent of the root of the base to the denominator, which is then raised to the power of the numerator of the numerator of the fraction.
- [tex]a^{\frac{n}{m} } = \left(\sqrt[m]{x} \right)^n[/tex]
2. The interest paid in the first account with simple interest is given as follows;
[tex]\displaystyle I = \frac{P \times r \times t}{100}[/tex]
Where;
r = The interest rate = 5%
Which gives;
Interest from the first account I₁ = 0.05·P·t
The amount in the account, A₁ = P + 0.05·P·t = P·(1 + 0.05·t)
The compound interest in the second account is; [tex]A = \mathbf{P \times \left(1 + r\right)^t}[/tex]
The compound interest rate on the second account is; r = 4% = 0.04
Which gives;
[tex]A_2 = \mathbf{P \times \left(1 + 0.04\right)^t}[/tex]
In the first few years, we have;
The simple interest account gives a higher account balance than the compound interest account
On year t = 11, we have;
A₁ = P·(1 + 0.05 × 11) = 1.55·P
[tex]A_2 = P \times \left(1 + 0.04\right)^{11} \approx \mathbf{1.54 \cdot P}[/tex]
On year t = 12, we have;
A₁ = P × (1 + 0.05×12) = 1.6·P
[tex]A_2 = P \times \left(1 + 0.04\right)^{12} \approx \mathbf{1.601 \cdot P}[/tex]
Therefore, just before, year 12, the second account, (compound interest
account) yields more interest.
The account to use is therefore;
- For short term savings (less than 12 years) use the first account, having a simple rate.
- For long term savings (more than 12 years) use the second account having a compound interest rate.
3. An exponential growth function is; A = (1 + r)ⁿ
A real world situation that the function could represent is the amount A, in a compound interest account
r = The interest rate
n = The duration the interest is applied
An exponential decay function is; [tex]A = \mathbf{(1 - r)^t}[/tex]
A real world situation that the function could represent is the price value of a device with time
Where;
r = The rate at which the price changes;
t = The time
Learn more about exponential functions here:
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