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Linearly Independent Vectors
Problem 133: Determine if the following vectors are linearly independent. (1) Solution: If the vectors in (1) are linearly independent, the equation (2) only has the trivial solution (i.e., ). On way we can identify this is by writing our vectors in an augmented matrix and reducing it to row reduced echelon form….
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Subspace of R^3
Problem 132: Determine if is a subspace of . Solution: If is a subspace, then it must meets the following three conditions. Indeed, is a subspace of .
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Extrema (Maximum and Minimum) II
Problem 130: Find the absolute maximum and minimum values of the function (1) over the interval = [-7,2]. Solution: First, we need to find the critical point by taking the derivative of the function and setting it equal to zero. That is, (2) The critical points are given by and . Sine only…
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Extrema (Maximum and Minimum)
Problem 130: Find the absolute maximum and minimum values of the function (1) over the interval = [-10,0]. Solution: First, we need to find the critical point by taking the derivative of the function and setting it equal to zero. That is, (2) The critical points are given by and . Sine only…
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Laplace Transform with Initial Value Problems (IVPs)
Problem 129: Consider the initial value problem (1) (a) Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of by . (b) Solve your equation for . Solution: First, let’s highlights some of the things we need to solve this problem….
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Second Order ODE with Repeated Roots
Problem 128: (a) Find the general solution to (b) Find the particular solution that satisfies and . Solution: The general solution of a second order ODE with repeated roots is given by (1) where and are the roots of the characteristic polynomial. (a) Let so that and . This means that (2) Hence,…
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Implicit Differentiation – Slope
Problem 127: Find the slope of the graph of (1) Solution: We need to find the derivative using implicit differentiation. After that, we will evaluate at our given point. That is, (2) That is, the slope of the given graph is given by .
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Integral Test for Series
Problem 126: Use the Integral Test to determine whether the infinite series is convergent. (1) Solution: In order to use the Integral Test, we need to convert the infinite series to its integral form, and then compute the integral. That is, (2) Hence, the infinite series is convergent since the integral is not…
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Derivative at A Point
Problem 125: Calculate of (1) Solution: We have two different options to find the derivative of (1). First, I will expand the logarithm using properties of logarithm. Then I will find the derivative and evaluate. That is, (2) Hence, (3) Therefore, .
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Second Order ODE with Basic Roots
Problem 124: Given the second order ODE (1) find the solutions and . Solution: The general solution of a second order ODE with basic 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)…
One Problem at a Time
