Calculus: Analytic Geometry and Calculus, with Vectors

290 Functions, graphs, and numbers

the unqualified assertion that one of r and p is a differentiable function of

the other is quite dubious.
Returning to simpler considerations, we note that if the graph of a
function has a tangent line at a point P on the graph, then the line through
P perpendicular to the tangent is called the normal to the graph at P.

Problems 5.19

1 Find the equation of the line tangent to the graph of the given equation

at the given point

(a) y=x2,(1,1) Ans.:y-1 =2(x-1)

(b) y = sin 2x, (0,0) tins.: y = 2x

(c) y = x log x, (1,0) Ans.: y = x - 1

(d) Y = e x, (0,1) tins.: y = ax + 1

(e) y = sin x2, (0,0) Ans.: y = 0

(f) y = x cos x, (27r,21r) Ans.: y = x

(g) y = (x + x2)6, (1,32) Ans.: y = 240x - 208

2 Find the equation of the tangent to the graph of the equation y = x"

at the point (xi,x1). Ans.:

y = nx1-lx - (n - 1)x1.

3 First find the slopes of the graph of the equation y = x3 at the points for

which x = -1, x = -, x = 0, x = -r, and x = 1. Use this information to

help construct a figure showing the graph and five tangents.

4 Find the area of the region bounded by the graph of y = x3 and the tangent

to this graph at the point (1,1). Ans.: N'_-

5 Even a crude graph suggests that at least one line can be drawn through

the point (-2,-3) tangent to the graph of the equation y = x2 + 2. Investi-

gate this matter.

6 Sketch reasonably accurate graphs of y = sin x, y = x, and y = -x over

the interval -27r 5 x < 4ir. Let

f(x) = x sin x

and, after observing that f(x) = 0 when sin x = 0, f(x) = x when sin x = 1,
and f(x) = -x when sin x = -1, sketch a graph of f(x). It is easy to guess
that the graph of y = f(x) is tangent to the graph of y = x wherever sin x = 1
and that the graph of y = f(x) is tangent to the graph of y = -x wherever
sin x = -1. Prove that it is so. Hint: Calculate f' (x) and observe that cos x =
0 wherever sin x is 1 or -1.
7 As we know, the part of the graph of the equation

y = Va2-x2=(a'-x2)''

for which -a < x < a is an" upper semicircle" with center at the origin. Let
Po(xo,yo) be a point on this graph. Use definitions or theorems of this section to