Lesson 1

23.1.1 The values of t are restricted to nonnegative values and because x = t the values of x are also restricted to those values.  23.1.2   Swapping x and y produces the inverse relation whose graph is the original graph reflected across the line y = x.   23.1.3 [0, 2 ] x [-4, 4] x [-2, 2]

The first interval given in the window coordinates above represents t-values, the second and third intervals represent x and y-values, respectively.  23.1.4  x = 7sin t - sin 7t y = 7cos t - cos 7t x = 8sin t - sin 8t y = 8cos t - cos 8t

The number of petals is N - 1 in the epicycloid

x = N sin t - sin N t

y = N cos t - cos N t  Lesson 2

23.2.1 dydx(0.5) = 2.08583, which agrees with the value found using the "Derivative" feature of the Graph screen.  23.2.2   The slope of the tangent line when t = 2 is dydx(2) 0.642093  23.2.3 The window shown is [- , ] x [-3, 3] x [-5, 5].

The equation of the tangent to the curve when t = 2 is y 0.642093x + 0.715815.  Lesson 3

23.3.1 The arc length is 3/2 units.  Self Test 1/2

7/4 3.34122 units 