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PRACTICE PROBLEMS: Find the equations of the tangent line and the normal line
and the lengths of the tangent and the normal for each of the following curves
at the point indicated:
4.
Find the points of contact of the horizontal and the vertical tangents to the
curve
given
SUMMARY The following are the major topics covered in this
chapter: 1. Slope of a curve at a point:
where
equals the inclination of the tangent line.
If the line tangent to the curve is horizontal, then ^{
} If the line tangent to the curve is vertical, then
When the slope of a curve is zero, the curve may be at
either a maximum or a minimum. 2. Tangent at a given point on the standard parabola y^{2
}= 4ax:
where a is the same as in the standard equation for
parabolas, and y_{1} is the y coordinate of the given point
(x_{1},y_{1}). 3. Tangent at a given point on other curves: To find the
slope, m, of a given curve at point P_{1}(x_{1},y_{1}),
choose a second point, P', on the curve so that it has coordinates (x_{1}+
X,y_{1}
+
y); then
substitute each of the coordinates of P' and P_{1} in the equation of
the curve and simplify. Divide both sides by
x and
eliminate terms that contain powers of
y higher
than the first power. Solve for
Let
x
approach zero and
will approach the slope of the tangent line,
m, at point P_{1}. 4. Equation of the tangent line:
5. Equation of the normal line:
6.
Relationships between the slopes of the tangent and normal lines: The slope of
the normal line is the negative reciprocal of the slope of the tangent line. The
inclination of one line must be 90 greater than the other. 7.
Length of the tangent: The length of the tangent is defined as that portion of
the tangent line between the point P,(x,,y,) and the point where the tangent
line crosses the X axis. length
of the tangent =
8. Length
of the normal: The length of the normal is defined as that portion of the
normal line between the point P_{1}
(x_{1},y_{1}) and the X axis. length
of the normal =
9.
Parametric equations: If the variables x and y of the Cartesian coordinate
system are expressed in terms of a third variable, say t (or
), then
the variable t (or
) is
called a parameter. The two equations x = x(t) and y = y(t) [or x = x(
) and y =
y(
)] are
called parametric equations. 