Calculus: Power Series and Maclaurin Polynomials

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Questions and Answers

How can you derive the equation of a plane given a point and a normal vector?

The equation of a plane can be derived using the point-normal form: if you have a point A(x_0, y_0, z_0) and a normal vector n=<a, b, c>, the equation is given by: a(x - x_0) + b(y - y_0) + c(z - z_0) = 0.

What is the significance of the unit normal vector in determining the distance from a point to a plane?

The unit normal vector represents the direction of the shortest path from the point to the plane, and it is used to project the position vector of the point perpendicular to the plane.

Describe how to find the distance from the origin to the plane 6x + 3y + 2z = 6.

To find the distance, use the formula: d = |(Ax + By + Cz - D) / √(A^2 + B^2 + C^2)|, where A, B, C represent the coefficients from the plane equation, and D is the constant term. For this plane, the distance is d = |(6 * 0 + 3 * 0 + 2 * 0 - 6) / √(6^2 + 3^2 + 2^2)| = 6/7.

What is the role of the dot product in vector projections?

The dot product quantifies the extent to which two vectors point in the same direction, enabling the calculation of the projection of one vector onto another by using the formula: proj_v u = ((u · v) / ||v||^2) * v. Signup and view all the answers

How do you calculate the area of a parallelogram defined by two vectors?

The area of a parallelogram formed by vectors u and w is given by the magnitude of their cross product: Area = ||u x w||. Signup and view all the answers

What series representation can be derived from the geometric series formula to express 1/(1+3x)?

1/(1 + 3x) = ∑ (-3x)^n for |3x| < 1. Signup and view all the answers

How is the function ln(1 + 3x) expressed as a power series after integration?

ln(1 + 3x) = ∑ (-1)^n * (3^n * x^(n+1)) / (n+1) for |3x| < 1. Signup and view all the answers

What is the radius of convergence for the series representing f(x) = 10 ln(1 + 3x)?

The radius of convergence is R = 1/3. Signup and view all the answers

In the context of the interval of convergence, what happens at the point x = 1/3?

At x = 1/3, the series diverges as it represents the harmonic series. Signup and view all the answers

What is the Maclaurin series for the function e^x?

e^x = ∑ (x^n) / n! Signup and view all the answers

How can the degree 5 Maclaurin polynomial for F(x) = ∫e^(-t^2) dt be derived?

By substituting e^(-t^2) with its Maclaurin series up to degree 5 and integrating term by term. Signup and view all the answers

What is the upper bound for the error when estimating an integral with a Maclaurin polynomial?

The upper bound for the error can be estimated using the next term in the series after the polynomial degree. Signup and view all the answers

Why does the constant of integration C equal 0 in the series expansion of ln(1 + 3x)?

Because the series equals ln(1 + 3x) at x = 0, where both sides are zero. Signup and view all the answers

What are the first three terms of the Maclaurin polynomial of degree 5 for F(x) = ∫e^(-t^2) dt?

P5(x) = x - (x^3)/6 + (x^5)/120. Signup and view all the answers

How do you estimate the integral ∫e^(-t^2) dt using the Maclaurin polynomial?

Substituting x = 1 into P5(x) gives P5(1) = 103/120. Signup and view all the answers

What is the upper bound for the error in the estimate of ∫e^(-t^2) dt using the polynomial?

|∫e^(-t^2) dt - 103/120| ≤ 1/5040. Signup and view all the answers

Write the Maclaurin series for f(x) = cos(x^2/4).

cos(x^2/4) = ∑ (-1)^n * (x^(2n) / 4^(2n)) / (2n)!. Signup and view all the answers

What is the value of the 10th derivative of cos(x^2/4) at x = 0?

The 10th derivative at x = 0 is 0. Signup and view all the answers

What normal vectors correspond to the planes x + y + z = 3 and 2x - 3y = 6?

The normal vectors are n1 = <1, 1, 1> and n2 = <2, -3, 0>. Signup and view all the answers

How can you find the distance between a point P(0, 0, 1) and the line of intersection of two planes?

Use the point-to-line distance formula in 3D, involving the direction vector and a point on the line. Signup and view all the answers

What role do partial derivatives play in analyzing functions represented by Maclaurin series?

Partial derivatives help determine the behavior and curvature of functions around the point of expansion. Signup and view all the answers

Flashcards

Power Series Representation of ln(1+3x)

A representation of the natural logarithm of (1 + 3x) as an infinite sum of terms involving powers of x.

Radius of Convergence

The radius of the largest disk centered at a point where a power series converges.