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Fundamental Theorem of Calculus Part I
Problem 183: Let (1) Find . Solution: We will use the Fundamental Theorem of Calculus Part I to find the derivative of the integral. That is, (2) That is, (3)
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Complex Number Arithmetic
Problem 182: Show that where . Solution: Using the properties of complex numbers we will solve our problem. That is, if and are two complex numbers, then (1) Likewise, (2) That is, (3)
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Complex Number
Problem 181: Show that (a) Re = -Im (b) Im = Re where . Solution: It is important to note that is a complex number. Also, Re = and Im = . That is, (a) (1) Hence, Re -Im . (b) Similarly, from above, Im Re .
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Taking Limit of Riemann Sums
Problem 180: Find a formula for the Riemann sum obtained by dividing the interval into equal subintervals and using the right-hand endpoint for each . Then take a limit of these sums as to calculate the area under the curve over . Solution: We will divide the interval into subintervals. Since the subintervals…
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Limit of Riemann Sums
Problem 179: Find a formula for the Riemann sum obtained by dividing the interval into equal subintervals and using the right-hand endpoint for each . Then take a limit of these sums as to calculate the area under the curve over . Solution: We will divide the interval into subintervals. Since the subintervals…
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Riemann Sums Limit
Problem 178: Find a formula for the Riemann sum obtained by dividing the interval into equal subintervals and using the right-hand endpoint for each . Then take a limit of these sums as to calculate the area under the curve over . Solution: We will divide the interval into subintervals. Since the subintervals…
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Displacement of an Object
Problem 177: A rock is dropped from the top of a 400 foot cliff. Its velocity at time seconds is measured by per second. Find the displacement of the rock during the time interval . Solution: The displacement of the rock is given by the integral of the velocity function. That is, (1) Therefore,…
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Newton’s Method
Problem 176: Use Newton’s Method to approximate the positive root of the equation (1) Solution: We know that the actual positive root of to 5-decimal places. We will use (2) which is the Newton’s Method applied to our function . We start with the guess . Then (3) Note that we get…
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Concavity of A Function
Problem 175: Determine the concavity of the function on its domain. Solution: The domain is given by . In order to find the concavity of a differentiable function, we use the second derivative. That is, (1) We have a change of concavity at the inflection point (that is, ). Then we have …
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Concavity of A Function
Problem 174: Determine the concavity of on . Solution: In order to find the concavity of a differentiable function, we use the second derivative. That is, (1) We have a change of concavity at the inflection point (that is, ). Then we have Since on the interval , the function is concave…
One Problem at a Time
