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Squeeze Theorem
Problem 143: Calculate the following limit: (1) Solution: A little trick to do this problem is to remember that the absolute value of cosine is bounded by 1. That is, (2) Since is bounded by two functions, says and , we will use the Squeeze Theorem to find our limit. The Squeeze Theorem:…
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The Sandwich Theorem II
Problem 142: If for all , find (1) Solution: Since is bounded by two functions, says and , we will use the Sandwich Theorem to find our limit. The Sandwich Theorem: Suppose that for all in some open interval containing , except possibly at itself. Suppose also that . Then . That is, since…
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The Sandwich Theorem
Problem 141: If for , find . Solution: Since is bounded by two functions, says and , we will use the Sandwich Theorem to find our limit. The Sandwich Theorem: Suppose that for all in some open interval containing , except possibly at itself. Suppose also that . Then . That is, since we are…
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Average Rate of Change
Problem 140: Find the average rate of change of (1) Solution: The average rate of change of a function is given by (2) That is, starting at we need to get t so that . Hence, (3)
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Average Speed
Problem 139: A rock breaks loose from the top of a tall cliff. What is its average speed (a) during its first 2 sec of fall? (b) during the 1-sec interval between second 1 and second 2? Solution: The average speed (S) of a function over the interval is given by (1) (a) Using…
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Complex Integral II
Problem 138: Evaluate the following complex integral (1) Solution: When evaluating complex integral, we consider the real part and imaginary part of the integral separately. That is, (2) That is, (3)
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Definite Complex Integral
Problem 137: Evaluate the following complex integral (1) Solution: When evaluating complex integral, we consider the real part and imaginary part of the integral separately. That is, (2) That is, (3)
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Law of Sine
Problem 136: If , , and , then find the length of the missing side . Solution: In order to solve this problem, we will use the Law of Sine. That is, (1) Solving for (2) Hence, the other side of the triangle measure 5.144.
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Law of Cosine
Problem 135: If , ,and , then find the length of the missing side . Solution: In order to solve this problem, we will use the Law of Cosine. That is, (1) Solving for (2) Hence, the other side of the triangle measure 10.652.
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Basis of Null Space
Problem 134: Find the basis for the null space of (1) Solution: In order to find the null space of the given matrix, we first need to make the matrix in row reduced echelon form. That is, when we row reduced the matrix , we get the following matrix (2) Since we are…
One Problem at a Time
