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Continuity at a Point II
Problem 233: Determine whether the following functions are continuous at . (1) Solution: Before we determine whether the given function is continuous or not, we need to know when is a function continuous. That is, Theorem 1: A function is continuous at if (2) Using Theorem 1, one can now determine whether (1)…
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Continuity at a Point
Problem 232: Determine whether the following functions are continuous at . (1) Solution: Before we determine whether the given function is continuous or not, we need to know when is a function continuous. That is, Theorem 1: A function is continuous at if (2) Using Theorem 1, one can now determine whether (1)…
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Calculating Annuity (Account)
Problem 231: A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years? Solution: This…
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Application: Optimization II
Problem 230: A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $40/ft and on the other three sides by a metal fence costing $20/ft. If the area of the garden is 182 square feet, find the dimensions of the garden that minimize the cost. Solution: Let be…
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Application of Derivative: Optimization
Problem 229: If you have 60 meters of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose? Solution: The perimeter of a rectangle is given by (1) where is the length of the rectangle and is the width of the rectangle….
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Geometric Series
Problem 228: Find the requested sums: a. The first term appearing in this sum is? b. The common ratio for our sequence is? c. The sum is? Solution: The general form of a geometric sequence is given by where is the first term of the sequence and is the common ratio of the…
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Sequence – Geometric
Problem 227: Consider the sequence . a. What is the first term of the sequence? b. What is the common ration? c. What is the closed form for the term of the sequence? d. Use the closed form to determine the value of . Solution: a. The first term is given by the first number…
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Derivatives and Tangent Lines
Problem 226: Let . Compute , the derivative of at , and use the result to find an equation of the line tangent to the graph of at . Solution: We will start by finding the derivative of and then evaluate it at . (1) One can now find the tangent line using the…
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End Behavior of Rational Functions
Problem 225: Use the limits at infinity to determine the end behavior of (1) Solution: To determine the end behavior of a rational expression, we must take the limit at infinity of the function. That is, (2) Thus . Likewise, . This means that the function (1) has a horizontal asymptote at ….
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Geometric Sequences II
Problem 224: Consider the sequence a. What is the common ration? b. What are the next five terms in the sequence? Solution: A geometric sequence is different from an arithmetic sequence. In an arithmetic sequence, we look for a common difference between the terms in the sequence whereas in the geometric sequence we look for…
One Problem at a Time
