$\"\"$

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The vector is $\"\"$ where $\"\"$ .

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The vectors are $\"\"$ , $\"\"$ and $\"\"$ .

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(a)

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Graph:

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Graph the vectors $\"\"$ and $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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(b)

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Draw the vector $\"\"$ starting at $\"\"$ and compare it with the vector $\"\"$ .

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The vector is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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The vector is $\"\"$ .

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the derivative of the function is $\"\"$ .

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The vector is at the point $\"\"$ .

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$\"\"$

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Compare the vector $\"\"$ with $\"\"$ .

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Therefore, the derivative of the vector gives us the actual tangent vector at $\"\"$ .

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$\"\"$

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(a)

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Graph:

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Graph the vectors $\"\"$ and $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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(b)

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The derivative of the vector gives us the actual tangent vector at $\"\"$ .