Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Which of the following represents the power series for \(\exp(a)=e^a\text{?}\)

1. \(\exp(a)=\displaystyle\sum_{n=0}^\infty a\frac{x^n}{n!}\text{.}\)

2. \(\exp(a)=\displaystyle\sum_{n=0}^\infty \frac{a^n}{n!}\text{.}\)

3. \(\exp(a)=\displaystyle\sum_{n=0}^\infty \frac{x^n}{a!}\text{.}\)

4. \(\exp(a)=\displaystyle\sum_{a=0}^\infty \frac{x^n}{n!}\text{.}\)

(b)

What is the interval of convergence for \(a\) in the series you found in part (a)?

1. \(\left(0,1\right)\text{.}\)

2. \(\left(-1,1\right)\text{.}\)

3. \(\left[-1,1\right]\text{.}\)

4. \(\left[0,1\right]\text{.}\)

5. \(\left(-\infty,\infty\right)\text{.}\)

(c)

Evaluating \(a=2x\text{,}\) what is the power series for \(\exp(a)=\exp(2x)=f(x)\text{?}\)

1. \(\exp(2x)=\displaystyle\sum_{n=0}^\infty \frac{2x^n}{n!}\text{.}\)

2. \(\exp(2x)=\displaystyle\sum_{2x=0}^\infty \frac{x^n}{n!}\text{.}\)

3. \(\exp(2x)=\displaystyle\sum_{n=0}^\infty 2x\frac{x^n}{n!}\text{.}\)

4. \(\exp(2x)=\displaystyle\sum_{n=0}^\infty \frac{(2x)^n}{n!}\text{.}\)

5. \(\exp(2x)=\displaystyle\sum_{n=0}^\infty \frac{x^n}{(2x)!}\text{.}\)

(d)

What is the interval of convergence for \(x\) in the series you found in part (c)?

1. \(\left(-\infty,\infty\right)\text{.}\)

2. \(\displaystyle \left(-\frac{1}{2},\frac{1}{2}\right)\text{.}\)

3. \(\displaystyle \left(0,\frac{1}{2}\right)\text{.}\)

4. \(\displaystyle \left[0,\frac{1}{2}\right]\text{.}\)

5. \(\displaystyle \left(-\frac{1}{2},\frac{1}{2}\right]\text{.}\)

(a)

Which of the following represents the power series for \(\frac{1}{1-a}\text{?}\)

1. \(g(a)=\displaystyle\sum_{n=0}^\infty ax^n\text{.}\)

2. \(g(a)=\displaystyle\sum_{n=0}^\infty (ax)^n\text{.}\)

3. \(g(a)=\displaystyle\sum_{n=0}^\infty a^n\text{.}\)

4. \(g(a)=\displaystyle\sum_{a=0}^\infty x^a\text{.}\)

(b)

What is the radius of convergence for \(a\) in the series you found in part (a)?

1. \(\left(0,1\right)\text{.}\)

2. \(\left(-1,1\right)\text{.}\)

3. \(\left[-1,1\right]\text{.}\)

4. \(\left[0,1\right]\text{.}\)

5. \(\left(-\infty,\infty\right)\text{.}\)

(c)

For what value of \(a\) is \(\frac{1}{1-a}=\frac{1}{x}\text{?}\)

1. \(a=x-1\text{.}\)

2. \(a=1-x\text{.}\)

3. \(a=x+1\text{.}\)

4. \(a=-x\text{.}\)

5. \(a=x\text{.}\)

(d)

Evaluating \(a\) at the previously found value, which of the following is the power series of \(f(x)=\frac{1}{x}\text{?}\)

1. \(f(x)=\displaystyle\sum_{n=0}^\infty x^n\text{.}\)

2. \(f(x)=\displaystyle\sum_{n=0}^\infty (-x)^n\text{.}\)

3. \(f(x)=\displaystyle\sum_{n=0}^\infty (1-x)^n\text{.}\)

4. \(f(x)=\displaystyle\sum_{n=0}^\infty (x-1)^n\text{.}\)

5. \(f(x)=\displaystyle\sum_{n=0}^\infty (1+x)^n\text{.}\)

(e)

Given that the interval of convergence for \(a\) is \(-1 < a < 1\text{,}\) what is the interval of convergence for \(x\text{?}\)

1. \(-1 < x < 1\text{.}\)

2. \(-2 < x < 0\text{.}\)

3. \(-2 < x < 2\text{.}\)

4. \(0 < x < 2\text{.}\)

5. \(2 < x < 0\text{.}\)

(a)

For what value of \(a\) is \(\frac{1}{1-a}=\frac{1}{3-2x}\text{?}\)

1. \(a=2x-2\text{.}\)

2. \(a=2-2x\text{.}\)

3. \(a=2x-3\text{.}\)

4. \(a=x\text{.}\)

5. \(a=3-2x\text{.}\)

(b)

Evaluating \(a\) at the previously found value, which of the following is the power series of \(f(x)=\frac{1}{3-2x}\text{?}\)

1. \(f(x)=\displaystyle\sum_{n=0}^\infty (3-2x)^n\text{.}\)

2. \(f(x)=\displaystyle\sum_{n=0}^\infty (2x-3)^n\text{.}\)

3. \(f(x)=\displaystyle\sum_{n=0}^\infty (2-2x)^n\text{.}\)

4. \(f(x)=\displaystyle\sum_{n=0}^\infty (2x-2)^n\text{.}\)

5. \(f(x)=\displaystyle\sum_{n=0}^\infty x^n\text{.}\)

(c)

Given that the interval of convergence for \(a\) is \(-1 < a < 1\text{,}\) what is the interval of convergence for \(x\text{?}\)

1. \(-1 < x < 1\text{.}\)

2. \(-1 < x < \frac{3}{2}\text{.}\)

3. \(-\frac{1}{2} < x < 1\text{.}\)

4. \(\frac{1}{2} < x < \frac{3}{2}\text{.}\)

5. \(-\frac{1}{2} < x < \frac{3}{2}\text{.}\)

(a)

For what value of \(a\) is \(\frac{1}{1-a}=\frac{1}{1+x^2}\text{?}\)

1. \(a=x^2\text{.}\)

2. \(a=-x^2\text{.}\)

3. \(a=1+x^2\text{.}\)

4. \(a=1-x^2\text{.}\)

5. \(a=x^2-1\text{.}\)

(b)

Evaluating \(a\) at the previously found value, which of the following is the power series of \(f(x)=\frac{1}{1+x^2}\text{?}\)

1. \(f(x)=\displaystyle\sum_{n=0}^\infty (-x^2)^{n}=\sum_{n=0}^\infty (-1)^nx^{2n}\text{.}\)

2. \(f(x)=\displaystyle\sum_{n=0}^\infty (1+x^2)^n\text{.}\)

3. \(f(x)=\displaystyle\sum_{n=0}^\infty (1-x^2)^n\text{.}\)

4. \(f(x)=\displaystyle\sum_{n=0}^\infty (x^2)^{n}=\sum_{n=0}^\infty x^{2n}\text{.}\)

5. \(f(x)=\displaystyle\sum_{n=0}^\infty (x^2-1)^n\text{.}\)

(c)

Given that the interval of convergence for \(a\) is \(-1 < a < 1\text{,}\) what is the interval of convergence for \(x\text{?}\)

1. \(-1 < x < 1\text{.}\)

2. \(-1 < x < 0\text{.}\)

3. \(0 < x < 1\text{.}\)

4. \(0 < x < 2\text{.}\)

5. \(-2 < x < 0\text{.}\)

6. \(-2 < x < 2\text{.}\)

(a)

Which of the following best represents the power series for \(n(x)=e^{-x^2}\text{?}\)

1. \(n(x)=\displaystyle -x^2\sum_{n=0}^\infty \frac{1}{n!}x^n=\displaystyle \sum_{n=0}^\infty -\frac{1}{n!}x^{n+2}\text{.}\)

2. \(n(x)=\displaystyle \sum_{n=0}^\infty \frac{1}{n!}(-x^2)^n=\displaystyle \sum_{n=0}^\infty \frac{1}{n!}(-1)^{n}x^{2n}\text{.}\)

3. \(n(x)=\displaystyle x^{-2}\sum_{n=0}^\infty \frac{1}{n!}(-x)^n=\displaystyle \sum_{n=0}^\infty \frac{1}{n!}(-1)^{n+2}x^{n+2}\text{.}\)

(b)

Which of the following best represents a degree 10 polynomial that approximates \(n(x)\text{?}\)

1. \(n_{10}(x)=\displaystyle -x^2-x^3-\frac{1}{2}x^4-\frac{1}{6}x^5-\frac{1}{24}x^6-\frac{1}{120}x^7-\frac{1}{720}x^8-\frac{1}{5040}x^9-\frac{1}{40320}x^{10}\text{.}\)

2. \(n_{10}(x)=\displaystyle x^2-x^3+\frac{1}{2}x^4-\frac{1}{6}x^5+\frac{1}{24}x^6-\frac{1}{120}x^7+\frac{1}{720}x^8-\frac{1}{5040}x^9+\frac{1}{40320}x^{10}\text{.}\)

3. \(n_{10}(x)=1-x^2+\frac{1}{2}x^4-\frac{1}{6}x^6+\frac{1}{24}x^8-\frac{1}{120}x^{10}\text{.}\)

(c)

Use your choice of \(n_{10}(x)\) to estimate \(\displaystyle \int_0^1 n(x)dx\) by computing \(\displaystyle \int_0^1 n_{10}(x)dx\text{.}\)

(a)

Which of the following represents an antiderivative of \(g(x)=\displaystyle \frac{1}{1-x}\text{?}\)

1. \(G(x)=C+\displaystyle\sum_{n=0}^\infty x^{n+1}\text{.}\)

2. \(G(x)=C+\displaystyle\sum_{n=1}^\infty \frac{1}{n}x^{n+1}\text{.}\)

3. \(G(x)=C+\displaystyle\sum_{n=0}^\infty \frac{1}{n+1}x^{n+1}\text{.}\)

4. \(G(x)=C+\displaystyle\sum_{n=1}^\infty \frac{1}{n+1}x^{n}\text{.}\)

(b)

Find the interval of convergence for this series.

(c)

Recall that \(\tilde{G}(x)=\ln|1-x|\) is an antiderivative of \(g(x)=\displaystyle \frac{1}{1-x}\text{.}\) For which \(C\) is your chosen \(G(x)=\ln|1-x|\text{?}\)

(d)

Use \(G_4(x)\) to estimate \(\displaystyle \int_2^4 \ln|1-x|dx\text{.}\)

(a)

Find a power series for \(\sin\left(-5 \, x^{2}\right)\text{.}\)

(b)

Find a power series for \(x^{4} \sin\left(x\right)\text{.}\)

(c)

Find a power series for \(F(x)\text{,}\) an antiderivative of \(f(x)\) such that \(F(0)=4\text{.}\)

(a)

Find a power series for \(\frac{1}{x^{4} + 1}\text{.}\)

(b)

Find a power series for \(-\frac{x^{5}}{x - 1}\text{.}\)

(c)

Find a power series for \(f'(x)\text{.}\)

(a)

Anti differentiate this power series and find \(C\) to find a power series for \(H(x)=\arctan(x)\text{.}\) Recall that \(\arctan(0)=0\text{.}\)

(b)

Find the interval of convergence for this series.

(a)

Find the power series for \(\alpha(x)=\ln|x|\text{.}\)

(b)

Find the interval of convergence for this series.

(a)

Find the power series for \(\beta(x)=\arctan(-3x^2)\text{.}\)

(b)

Find the interval of convergence for this series.