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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. 