Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Suppose we wished to estimate this length with two line segments where $\mathrm{\Delta }x=2\text{.}$

Which of the following expressions represents the sum of the lengths of the line segments with endpoints $(0,0)\text{,}$ $(2,8)$ and $(4,8)\text{?}$

1. ${\displaystyle \sqrt{4+8}}$

2. ${\displaystyle \sqrt{2^2+8^2}+\sqrt{(4-2{)}^2+(8-8{)}^2}}$

3. ${\displaystyle \sqrt{4^2+8^2}}$

4. ${\displaystyle \sqrt{2^2+8^2}+\sqrt{4^2+8^2}}$

(b)

Suppose we wished to estimate this length with four line segments where $\mathrm{\Delta }x=1\text{.}$

Which of the following expressions represents the sum of the lengths of the line segments with endpoints $(0,0)\text{,}$ $(1,5)\text{,}$ $(2,8)\text{,}$ $(3,9)$ and $(4,8)\text{?}$

1. ${\displaystyle \sqrt{4^2+8^2}}$

2. ${\displaystyle \sqrt{1^2+(5-0{)}^2}+\sqrt{1^2+(8-5{)}^2}+\sqrt{1^2+(9-8{)}^2}+\sqrt{1^2+(8-9{)}^2}}$

3. ${\displaystyle \sqrt{1^2+5^2}+\sqrt{2^2+8^2}+\sqrt{3^2+9^2}+\sqrt{4^2+8^2}}$

(c)

Suppose we wished to estimate this length with $n$ line segments where ${\displaystyle \mathrm{\Delta }x=\frac{4}{n}\text{.}}$ Let $f(x)=-x^2+6x\text{.}$

Which of the following expressions represents the length of the line segment from $(x_0,f(x_0))$ to $(x_0+\mathrm{\Delta }x,f(x_0+\mathrm{\Delta }x))\text{?}$

1. ${\displaystyle \sqrt{x_0^2+f(x_0{)}^2}}$

2. ${\displaystyle \sqrt{(x_0+\mathrm{\Delta }x{)}^2+f(x_0+\mathrm{\Delta }x{)}^2}}$

3. ${\displaystyle \sqrt{(\mathrm{\Delta }x{)}^2+f(\mathrm{\Delta }x{)}^2}}$

4. ${\displaystyle \sqrt{(\mathrm{\Delta }x{)}^2+(f(x_0+\mathrm{\Delta }x)-f(x_0){)}^2}}$

(d)

Which of the following Riemann sums best estimates the arclength of the parabola $y=-x^2+6x$ over the interval $[0,4]\text{?}$ Let $f(x)=-x^2+6x\text{.}$

1. ${\displaystyle {\displaystyle \sum \sqrt{(\mathrm{\Delta }x{)}^2+f(\mathrm{\Delta }x{)}^2}}}$

2. ${\displaystyle {\displaystyle \sum \sqrt{(x_i+\mathrm{\Delta }x{)}^2+f(x_i+\mathrm{\Delta }x{)}^2}}}$

3. ${\displaystyle {\displaystyle \sum \sqrt{x_i^2+f(x_i{)}^2}}}$

4. ${\displaystyle {\displaystyle \sum \sqrt{(\mathrm{\Delta }x{)}^2+(f(x_i+\mathrm{\Delta }x)-f(x_i){)}^2}}}$

(e)

Note that

Which of the following best describes ${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x_i+\mathrm{\Delta }x)-f(x_i)}{\mathrm{\Delta }x}\text{?}}$

1. ${\displaystyle 0}$

2. ${\displaystyle 1}$

3. ${\displaystyle f^{\prime }(x_i)}$

4. This limit is undefined.

(a)

Estimate the arclength of this curve with two line segments where $\mathrm{\Delta }x=2\text{.}$

(b)

Estimate the arclength of this curve with four line segments where $\mathrm{\Delta }x=1\text{.}$

(c)

Find an integral which computes the arclength of the curve $y=2^x-1$ defined on $[1,5]\text{.}$

(a)

Estimate this arclength with 5 line segments where $\mathrm{\Delta }x=1\text{.}$

(b)

Find an integral which computes this arclength.