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Second Order ODE with Complex’s Roots
Problem 123: Given the second order ODE (1) find the solutions and . Solution: The general solution of a second order ODE with complex root is given by (2) where and are arbitrary constant, and are the complex roots of the characteristic polynomial. Finding the roots of the characteristic polynomial gives us (3)…
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Calculating Annuity Withdrawals
Problem 122: You have $875,000 saved in your 401k. Your account earns 6.5% interest compounded monthly. How much will you be able to withdrawal each month, if you want to be able to take withdrawals for the next 25 years? Solution: The present value of an annuity account can be calculate by (1) where…
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Limit at Infinity
Problem 121: Evaluate the following limit: (1) Solution: The first thing that comes to mind when calculating limit is to plug in the limiting value (value of the limit) into the function. If we do that in (1), we get which really does not make sense. That is, we need to do something more…
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Simplifying Trigonometry Expressions
Problem 120: Simplify the following expressions: (a) and (b) . Solution: (a) Here we can use the double angle formula: for an angle . That is, (1) (b) Here, we will use another double angle formula: . Now let so that . Since, this means that in a right triangle, the opposite side is…
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Trigonometry Equation
Problem 119: Verify the identity: (1) Solution: If we want to verify this equation, we need to pick a side in order to prove the other side. I will start with the left-hand side and show that it is equal to the right-hand side . That is, (2) Hence, as desired.
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Linearization
Problem 118: Find the linearization of the function at . (1) Solution: The linearization is given by . Thus we will find the pieces of the formula and then assemble them. That is, (2) We must now find the derivative evaluated at . (3) Hence, (4) One can now find the…
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Integration – Trigonometric Substitution
Problem 117: Solve the following integral using trigonometric substituition (1) Solution: Before solving the problem, let’s remember that . That is, (2) Now we can let . Thus (3) Going back to (2) and using our substitution we have (4) Note that the solution is given in term of , but…
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Logarithmic Differentiation
Problem 116: Find using the method of logarithmic differentiation when . Solution: We need to use property of logarithm to solve this problem. That is, (1) Hence, (2)
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Bayes’ Theorem
Problem 115: The New York Times of January 24, 1997, discussed the recommendation of a special panel concerning mammograms for women in their 40s. About 2% of women aged 40 to 49 years old develop breast cancer in their 40s. But the mammogram used for women in that age group has a high rate o…
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Independent Event
Problem 114: Roll a Die, and consider the following two events: and . Are the events and independent? Solution: If events and are independent, then and . 1. Calculate : (1) 2. Calculate : (2) 3. Calculate : (3) 4. Calculate : (4) Since and , the events and are independent.
One Problem at a Time
