Calculus: Analytic Geometry and Calculus, with Vectors

5.1 Graphs, slopes, and tangents 291

prove that the graph has exactly one tangent at Po and that the equation of this

tangent is

Y- Yo

Yo

(x-xo).

Prove that this line is perpendicular to the line joining the origin to Po and hence
that the definition of tangent given in this section is in agreement with ideas of
tangents employed in elementary plane geometry.

(^8) With or without more critical investigation of thematter, sketch a figure
which indicates that the graph of the preceding problem has exactlyone line of
support at Po.
(^9) If x = a cost and y = a sin t, it is easy to make the calculation
dy
dy =dt __ a cos t
dx dx - ssin t
dt
over each interval of values oft for which sin t > 0 or sin t < 0. Letting xo =
a cos to and yo = a sin to, we find that the equation of the tangent to the graph
at the point Po(xo,yo) is

_ a cos to(x - xo "co

Y Yo a sin to ) or y - yo = - (x - xo).

Yo

Sketch a graph which shows the geometric interpretations of these things.

(^10) Find the equation of the tangent to the graph of y = x3 at the origin.
Sketch the graph and show that it does not have a line of support at the origin.
11 Draw a graph of the equation y = Jxl. Show that this graph has no
tangent at the origin but does have many lines of support. Remark: Our word
"tangent" has its root in a Latin verb meaning "to touch," and a mathematician
from Mars can defend his contention that our lines of support are "touching
lines" and hence should be called tangents. We must, however, stick to our
guns and insist that, in languages used on earth, these lines are not tangents.
(^12) Sketch the graph of y = sin x and the normal to the graph at the point
(x, sin x). The normal intersects the x axis at the point (f (x), 0). Determine
whether f (x) increases as x increases. Hint: Borrow, from the next section, the
unsurprising fact that f(x) is increasing over an interval if f(x) > 0 over the
interval.
13 In connection with Problem 12, we note that problems in applied mathe-
matics sometimes involve extraneous material that may obscure their mathe-
matical aspects. A witch with a broom sweeps the x axis while walking along the
graph of y = sin x in such a way that x is always increasing. She keeps the
handle of her broom perpendicular to her path. Is the broom always pushing
dust to the right?
14 The two formulas
d. d
dx
sin x = cos x,
dx
cos x = - sin x