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10.1 Parametric Equations

10.1 Parametric Equations. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function. Remember:

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10.1 Parametric Equations

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  1. 10.1 Parametric Equations

  2. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function. • Remember: To graph, if there are trig functions in the parametric equations, plug in t values between 0 and 2Ө. If not, try different t-values between –10 and 10.

  3. 6 4 2 0 -6 -4 -2 2 4 6 -2 -4 -6 • Graph the parametric curve represented by x = t2 – 1 and y = 1 – t2 –2 3 –3 –1 0 0 0 –1 1 1 0 0 2 3 –3

  4. Write the Cartesian equation of the parametric curve represented by x = t2 – 1 and y = 1 – t2 x = t2 – 1 • Solve for t ±(x + 1)1/2 = t • Substitute t into the other expression y = 1 – (±(x + 1)1/2)2 y = 1 – (x + 1) y = –x

  5. If f and g have derivatives at t, then the parametrized curve also has a derivative at t. The formula for finding the slope of a parametrized curve is: This makes sense if we think about canceling dt.

  6. The formula for finding the slope of a parametrized curve is: We assume that the denominator is not zero.

  7. 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.

  8. Example:

  9. Example: • Find the first derivative (dy/dx).

  10. 2. Find the derivative of dy/dx with respect to t. Quotient Rule

  11. 3. Divide by dx/dt.

  12. 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.) (proof on pg. 533)

  13. Remember: • A particle is AT REST when both dy/dt and dx/dt are 0.

  14. Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:

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