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1. Find the area bounded by r = cos 0 for -π/2 ≤0 ≤π/2. 2. Find the length of the r(t)= <9t, 4t³/2, t²>, 1 ≤ t ≤ 5 3. Find parametric equations of the tangent line to the curve given by r(t) =< et, t4,t + 3> at the point with t = 0. * ()", does it converge and if so what does it converge to? n² 5. Determine whether the series E-1(−1)" * - is conditionally convergent, n*+n+1 absolutely convergent, or divergent and explain why. 6. Find the radius of convergence and the interval of convergence for Σn-on (x-2). 4. For the series Σn= n=1; 7. Find the Taylor Series for cos x centered at π/2. What is the radius of convergence? (x)n n! 100 8. The Maclaurin Series for ex is ex = Σn=0 a. Find T3(x), the 4th degree Taylor Polynomial approximation of ex. b. Use T3(x) to find an approximate value for e¹/². c. Find an upper bound for the error in this approximation. 9. Find parametric equations for the line through (1, -2, 2) and perpendicular to <1, 0, 1> and <1, 1, 0>. 10. Find an equation for the plane through (1, 3, 1), (2, 1, 1) and (-1, 4, 2). 11. Find the position r(t) of an object if the acceleration is a(t) = <6t, 4, -32> and the initial position is r(0) = <0, 0, 1> and the initial velocity is v(0) = <50, 0, 128>.. 12. Find the curvature as a function of t of the curve r(t) = .

1. Find the area bounded by r = cos 0 for -π/2 ≤0 ≤π/2. 2. Find the length of the r(t)= <9t, 4t³/2, t²>, 1 ≤ t ≤ 5 3. Find parametric equations of the tangent line to the curve given by r(t) =< et, t4,t + 3> at the point with t = 0. * ()", does it converge and if so what does it converge to? n² 5. Determine whether the series E-1(−1)" * - is conditionally convergent, n*+n+1 absolutely convergent, or divergent and explain why. 6. Find the radius of convergence and the interval of convergence for Σn-on (x-2). 4. For the series Σn= n=1; 7. Find the Taylor Series for cos x centered at π/2. What is the radius of convergence? (x)n n! 100 8. The Maclaurin Series for ex is ex = Σn=0 a. Find T3(x), the 4th degree Taylor Polynomial approximation of ex. b. Use T3(x) to find an approximate value for e¹/². c. Find an upper bound for the error in this approximation. 9. Find parametric equations for the line through (1, -2, 2) and perpendicular to <1, 0, 1> and <1, 1, 0>. 10. Find an equation for the plane through (1, 3, 1), (2, 1, 1) and (-1, 4, 2). 11. Find the position r(t) of an object if the acceleration is a(t) = <6t, 4, -32> and the initial position is r(0) = <0, 0, 1> and the initial velocity is v(0) = <50, 0, 128>.. 12. Find the curvature as a function of t of the curve r(t) = .

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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### Advanced Calculus Problem Set 1. **Find the area bounded by \( r = \cos \theta \) for \(-\pi/2 \leq \theta \leq \pi/2 \).** 2. **Find the length of the \( r(t) = <9t, 4t^3/2, t^2> \), \( 1 \leq t \leq 5 \).** 3. **Find parametric equations of the tangent line to the curve given by \( r(t) = < e^t, t^4, t + 3 > \) at the point with \( t = 0 \).** 4. **For the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} * \left( \frac{2}{5} \right)^n \), does it converge and if so what does it converge to?** 5. **Determine whether the series \( \sum_{n=1}^{\infty}(-1)^n * \frac{n^2}{n^4 + n + 1} \) is conditionally convergent, absolutely convergent, or divergent and explain why.** 6. **Find the radius of convergence and the interval of convergence for \( \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n (x - 2)^n \).** 7. **Find the Taylor Series for \( \cos x \) centered at \( \pi/2 \). What is the radius of convergence?** 8. **The Maclaurin Series for \( e^x \) is \( e^x = \sum_{n=0}^{\infty} \frac{(x)^n}{n!} \).** - a. Find \( T_3(x) \), the 4th degree Taylor Polynomial approximation of \( e^x \). - b. Use \( T_3(x) \) to find an approximate value for \( e^{1/2} \). - c. Find an upper bound for the error in this approximation. 9. **Find parametric equations for the line through (1, -2, 2) and perpendicular to \( <1, 0, 1> \) and \( <1, 1, 0> \
Transcribed Image Text:### Advanced Calculus Problem Set 1. **Find the area bounded by \( r = \cos \theta \) for \(-\pi/2 \leq \theta \leq \pi/2 \).** 2. **Find the length of the \( r(t) = <9t, 4t^3/2, t^2> \), \( 1 \leq t \leq 5 \).** 3. **Find parametric equations of the tangent line to the curve given by \( r(t) = < e^t, t^4, t + 3 > \) at the point with \( t = 0 \).** 4. **For the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} * \left( \frac{2}{5} \right)^n \), does it converge and if so what does it converge to?** 5. **Determine whether the series \( \sum_{n=1}^{\infty}(-1)^n * \frac{n^2}{n^4 + n + 1} \) is conditionally convergent, absolutely convergent, or divergent and explain why.** 6. **Find the radius of convergence and the interval of convergence for \( \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n (x - 2)^n \).** 7. **Find the Taylor Series for \( \cos x \) centered at \( \pi/2 \). What is the radius of convergence?** 8. **The Maclaurin Series for \( e^x \) is \( e^x = \sum_{n=0}^{\infty} \frac{(x)^n}{n!} \).** - a. Find \( T_3(x) \), the 4th degree Taylor Polynomial approximation of \( e^x \). - b. Use \( T_3(x) \) to find an approximate value for \( e^{1/2} \). - c. Find an upper bound for the error in this approximation. 9. **Find parametric equations for the line through (1, -2, 2) and perpendicular to \( <1, 0, 1> \) and \( <1, 1, 0> \
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