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CHAPTER 3 TANGENTS, NORMALS, AND SLOPES OF CURVES LEARNING OBJECTIVES Upon completion of this chapter, you should be able to do
the following: 1. Find the slope and equation of the tangent line for a
standard parabola and for other curves at a given point. 2. Find equations of the tangent line and the normal line
and lengths of the tangent and the normal between a point on a curve and the X
axis for various curves. 3. Apply parametric equations to motion in a straight line
and in a circle; and also, find equations and lengths of tangents and normals
of curves using parametric equations. INTRODUCTION In chapter 1, the notation
was introduced to represent the slope of a line. The straight line discussed had a
constant slope and the symbol Ay was defined as (y_{2}  y_{1})
and Ax was defined as (x_{2}  x_{1}). In this chapter we will
discuss the slope of curves at specific points on the curves. We will do this
with as little calculus as possible, but our discussion will be directed toward
the study of calculus. SLOPE
OF A CURVE AT A POINT In
figure 31 the slope of the curve is represented at two different places by
. The
value of
Ay
on the lower part of the curve is extremely close
to the actual slope at P_{1}
because P_{1} lies on a nearly
straight portion of the curve. The value of the slope at P_{2} is less accurate than the slope near
P_{1} because P_{2} lies on a portion of the curve that
has more curvature than the portion of the curve near P_{1}. To obtain an accurate measure of
the slope of the curve at each point, as small a portion of the curve as possible
should be used. When the curve is nearly a straight line, a very small error
will occur when you are finding the slope, regardless of the value of the
increments
y and
x. If the
curvature is great and large increments are used when you are finding the slope
of a curve, the error will become very large. Thus,
the error can be reduced to as small an amount as you want provided you choose
your increments to be sufficiently small. Whenever the slope of a curve at a
given point is desired, the increments
y and
x should be extremely small. Consequently, the arc of the curve can be
replaced by a straight line, which determines the slope of the line tangent to
the curve at that point. You
must understand that when we speak of the line tangent to a curve at a specific
point, we are really considering the secant line, which cuts a curve in at
least two points. Refer to figure 32. As point P_{2} moves closer to point P_{1}, the slope of the secant line varies by smaller and
smaller amounts and causes the secant line between P_{1} and P_{2} to approach P_{1}
. P_{1} is then extended to
form the line tangent to the curve at that specific point. If
we allow to
represent
the equation of a curve, then ~ is the slope of the line tangent to the curve
at P(x,y).
Figure 31.Curve with increments Ay and Ax.
Figure 32.Curve with secant line and tangent line. The direction of a curve is defined as the direction of
the tangent line at any point on the curve. Let
equal the inclination of the tangent line;
then the slope equals tan
and
is the slope of the curve at any point P(x,y). The angle
, is the
inclination of the tangent line at P, in figure 33. This angle is acute and
the value of tan 0, is positive. Hence, the slope of the tangent line is
positive at point P_{1}. The angle
_{2}
is an obtuse angle, tan
_{2}
is negative, and the slope of the tangent line at point P_{2} is
negative. All lines leaning to the right have positive slopes, and all lines
leaning to the left have negative slopes. At point P_{3} the tangent
line to the curve is horizontal and
equals 0. This means that
Figure 33.Curve with tangent lines.
The fact that the slope of a curve is zero when the
tangent line to the curve at that point is horizontal is of great importance in
calculus when you are determining the maximum or minimum points of a curve. Whenever the slope of a
curve is zero, the curve may be at either a maximum or a minimum. Whenever the inclination of the tangent line to a curve at
a point is 90', the tangent line is vertical and parallel to the Y axis. This
results in an infinitely large slope where
