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Clicker Question 1

Clicker Question 1. If x = e 2 t + 1 and y = 2 t 2 + t , then what is y as a function of x ? A. y = (1/2)(ln 2 (x – 1) + ln( x – 1)) B. y = ln 2 (x – 1) + (1/2)ln( x – 1)) C. y = (1/2)(ln 2 (x) + ln( x ) – 2) D. y = (1/2)(e 2(x – 1) + e ( x – 1) )

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Clicker Question 1

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  1. Clicker Question 1 • If x = e2t + 1 and y = 2t2 + t , then what is y as a function of x ? • A. y = (1/2)(ln2(x – 1) + ln(x – 1)) • B. y = ln2(x – 1) + (1/2)ln(x – 1)) • C. y = (1/2)(ln2(x) + ln(x) – 2) • D. y = (1/2)(e2(x – 1) + e(x – 1)) • E. y = e2(x – 1) + e(x – 1)

  2. Clicker Question 2 • What figure in the plane is given by the parametric equations x = 2 sin(t), y = 3 cos(t), 0  t   • A. A circle • B. The top half of a circle • C. An ellipse • D. The top half of an ellipse • E. The right half of an ellipse

  3. Calculus of Parametric Equations (11/1/13) • Most of the ideas and techniques of calculus carry over quite easily into the parametric setting (derivatives, integrals, areas, volumes, arc length, surface area, etc.) • The bottom line is that, by the ultra-important Chain Rule, (dy / dx )(dx / dt ) = dy / dt . • Hence we have dy / dx = (dy / dt ) / (dx / dt ).

  4. Example • Suppose x = e2t + 1 and y = t2 + t . • What is dy / dx ? • For what t is the tangent line to this curve horizontal? Vertical? • At what point in the plane does this curve have a horizontal tangent? • What is the equation of the tangent line to this curve when t = 0?

  5. Clicker Question 3 • If x = t cos(t) and y = t3, what is the slope of the tangent line to the curve when t = /2 ? • A. (-3/4) 2 • B. (3/4) 2 • C. (3/2)  • D. (-3/2)  • E. (-3/2)

  6. Areas • For example, the area enclosed by a curve and the x-axis is y dx (with appropriate endpoints on the integral). • Example: Find the area enclosed by the curve x = ln(t + 1), y = t – t 2 and the x-axis. Note that since the variable of integration is t, the limits of integration must be in terms of t.

  7. Arc Length • Recall that using the Pythagorean Theorem, we had ds = (dx 2 + dy2). This is easily computed in terms of t . • Example: Find or estimate the length of the curve x = ln(t + 1), y = t – t 2 from t = 0 to t = 1. • Similarly, surface area, volume, etc.

  8. Assignment for Monday • Read Section 10.2. • In that section, please do Exercises 3, 7, 19, 31, 41, and 61. This last is challenging, but see if you can get it. The answer book is right.

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