(^292) Functions, graphs, and numbers
say something very specific about the slopes of graphs of y = sin x and y = cos X.
Sketch these graphs and observe that the formulas seem to be correct.
15 Let x = L cos3 0, y = L sin3 0, where L is a given positive constant.
Find the equation of the tangent to the graph
Y at the point for which 0 = Bo and show that this
tangent intersects the x and y axes at the points
Q
.1(L cos Bo, 0) and B(0, L sin Oo). Show that
ABI = L.
16 Let x = L cos3 0, y = L sin3 0 as in the
x
x3f + Y3i = L%
and use this to find equations of tangents and to
i find the final result IABI = L of the preceding
problem. Remark: The graph of these equations
Figure 5.191 is, as we shall see later, a hypocycloid of four cusps.
It appears in Figure 5.191.
17 We now solve a problem that is similar to Problem 11 of Section 3.6.
It is a rather tedious task to draw a graph of the equation
(1) x5-x'y-2x-7x3+ys=721
unless we have an electronic computer to help us do the chores. The graph does
contain the point P0(2,3), the constant 721 having been so determined that this
is so. Our problem is to find the equation of the tangent (if any) to the graph
at Po. Without being sure about the facts, we assume that there is a function
4, defined over some interval 2 - S < x < 2 + S, such that the part of the
graph near Po has the equation y = 4,(x) and, moreover, 4, is differentiable.
Then (1) holds when y = 4,(x) and, with the aid of our formula for differentiating
products of differentiable functions of x, we differentiate the members of (1) and
equate the results to obtain
(2) 6x5-x2dx-2xy-2-21x2-} 6y5dx=0
or(3) d y 6x5 - 2xy - 2 - 21x2
dx 6y5 - x2At the point (2,3) this has the value-. The required equation of the tangent
line is(4) Y - 3 = i- fr(x - 2),
provided, of course, that our assumption is correct.
18 Apply the method of the preceding problem to find the slope of the graph
of the equation
x2 + y2 = 25
at the point (3,4).