Calculus for Team-Based Inquiry Learning: 2024 Edition

(a)

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

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Plot of \(y=-x^2+6x\) over \([0,4]\text{.}\)

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 \(\Delta x=1\text{.}\)

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Plot of \(y=-x^2+6x\) over \([0,4]\text{.}\)

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 \(\displaystyle \Delta x=\frac{4}{n}\text{.}\) Let \(f(x)=-x^2+6x\text{.}\)

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Plot of \(y=-x^2+6x\) over \([0,4]\text{.}\)

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

(d)

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

(e)

(a)

(b)

(c)

(a)

(b)