Barrons AP Calculus

(Marvins-Underground-K-12) #1

(a)


(b)


(a)


(b)


BC  ONLY

Example 25 __

Find the equation of the tangent to the curve in Example 24 for .


SOLUTION:


When the slope of the tangent, , equals −2 sin . Since


    the equation    is

Example 26 __

Suppose two objects are moving in a plane during the time interval 0 ≤ t ≤ 4.
Their positions at time t are described by the parametric equations


x 1     =   2t, y 1     =   4t  −   t^2     and x 2     =   t + 1,  y 2     =   4   – t.
Find all collision points. Justify your answer.
Use a calculator to help you sketch the paths of the objects, indicating the
direction in which each object travels.

SOLUTION:


Equating    x 1     and x 2     yields  t = 1.  When    t = 1,  both    y 1     and y 2     equal   3.  So  t =     1
yields a true collision point (not just an intersection point) at (2,3). (An
intersection point is any point that is on both curves, but not necessarily at
the same time.)
Using parametric mode, we graph both curves with t in [0,4], in the window
[0,8] × [0,4] as shown in Figure N3–7.
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