Pre Calc Final

Compound Interest (Compounded n times per year): A = P(1 + r/n)^(nt).

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Use this formula when an initial investment (P) earns interest at an annual rate (r), compounded n times per year for t years to find the future value (A).

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For example, problem 6 involves finding the time it takes for an investment to double when compounded monthly, and problem 7 involves calculating the future value of an investment compounded quarterly.

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'n' represents the number of times interest is compounded per year. For example, if interest is compounded monthly, n = 12; if quarterly, n = 4.

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Continuous Compounding: A = Pe^(rt).

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Use this formula when interest is compounded continuously, where 'e' is the base of the natural logarithm.

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For example, problem 11 involves calculating the time it takes for an investment to double when interest is compounded continuously, and problem 8 involves finding the interest rate on an account that is compounded continuously.

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This formula is a specific case of exponential growth or decay, with 'e' as the base.

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Radioactive Decay Formula: A = Aoe^(-0.01155x).

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Use this formula to model the amount of radioactive material remaining after a period of time.

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In this formula, Ao is the initial amount of the substance, x is the number of years, and A is the amount remaining after x years.

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Half-Life Formula: This is not explicitly stated as a formula in the sources, but it is implied in problem 14.

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You'll use this concept when dealing with radioactive decay, specifically with carbon-14.

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The problem requires understanding that the half-life is the time it takes for half of the substance to decay.

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The specific formula used to solve problem 14 is not provided in the sources. However, you can use the general exponential decay formula, substituting half the initial amount for the final amount after the half-life period.

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Solving Exponential Equations: You will have to manipulate equations involving exponents and logarithms to isolate the variable.

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For example, in problem 4, you need to solve for x in an equation like 4x + 6 = 5.

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In problem 16, you need to solve an equation of the form ex + 2 = 4.

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You will often use logarithms to solve exponential equations. Remember that when solving for a variable in the exponent, taking the natural logarithm (ln) or common logarithm (log) of both sides can simplify the equation, and that when the base is e you will typically use ln.

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Logarithm Properties:

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The sources do not explicitly state any log properties but demonstrate the use of properties implicitly in problem 11.

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When using a calculator to evaluate logarithms that have a base other than 10 or e, you will likely need to use the change of base formula, which is not given in the sources. This can be done by using the log function or the ln function because logb(x) = log(x)/log(b) or ln(x)/ln(b).

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Solving for Intersections:

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To find where two functions intersect, set them equal to each other and solve for x. For instance, if you have f(x) and g(x), solve f(x) = g(x).

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Problem 5 is an example of finding the intersection of two functions.

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Problem 18 involves solving for the intersection of two logarithmic functions.

When to Use "ln" vs. "log":

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Use "ln" (natural logarithm) when you have a base of e in the equation, especially with continuous compounding or radioactive decay problems.

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Use "log" (common logarithm) when the base of the logarithm is 10, or when the base of the logarithm is not explicitly stated in the problem. The sources use "log" to represent the base-10 logarithm.

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When using a calculator, remember that the "log" button typically refers to the common logarithm (base 10), while "ln" refers to the natural logarithm (base e).