Calculus 10.1

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10.1 Parametric functions
Mark Twains Boyhood Home Hannibal, Missouri
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Mark Twains Home Hartford, Connecticut
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In chapter 1, we talked about parametric
equations. Parametric equations can be used to
describe motion that is not a function.
If f and g have derivatives at t, then the
parametrized curve also has a derivative at t.
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The formula for finding the slope of a
parametrized curve is
This makes sense if we think about canceling dt.
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The formula for finding the slope of a
parametrized curve is
We assume that the denominator is not zero.
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To find the second derivative of a parametrized
curve, we find the derivative of the first
derivative

• Find the first derivative (dy/dx).

2. Find the derivative of dy/dx with respect to
t.
3. Divide by dx/dt.
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Example
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Example

• Find the first derivative (dy/dx).

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2. Find the derivative of dy/dx with respect to
t.
Quotient Rule
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3. Divide by dx/dt.
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The equation for the length of a parametrized
curve is similar to our previous length of
curve equation
(Notice the use of the Pythagorean Theorem.)
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Likewise, the equations for the surface area of a
parametrized curve are similar to our previous
surface area equations
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This curve is
p