|
|
Differential
calculus - derivatives
|
|
Derivative of
parametric functions example
|
Example: Write
equation of the line tangent to the curve x
= t
+
1 and
y
=
-
t2
+
4
at the point t
= 1.
|
Solution:
The equation
x
=
t + 1
solve for t
and plug into y
=
- t2 + 4,
thus |
t
=
x
- 1
=> y
=
- t2 + 4,
y
=
- (x
- 1)2 + 4
that is |
y
-
4
=
- (x
- 1)2
or y
=
- x2 + 2x + 3 translated parabola with the vertex V(x0,
y0),
so V(1,
4). |
When
plotting points of a parametric curve by increasing t, the
graph of the function is traced
out in the direction of motion. |
|
The
derivative of the given parametric equations at t
= 1 is the slope of the tangent
line, |
|
since
t
=
x
- 1
then m
=
y' (x)
=
-
2(x
- 1)
= -
2(2
- 1)
=
-
2,
m
=
-
2.
|
Therefore,
the equation of the line tangent to the given parametric curve at t
= 1 or the point P1(2,
3) is |
y
- y1
= m(x
- x1),
y
- 3
= -
2(x
- 2) =>
y
= -
2x
+ 7. |
|
Derivative
of vector-valued functions
|
If
the radius vector r
of a point in a plane depends on a parameter t,
say t
represents time, so that its magnitude
and direction change continuously while t
changes, then its arrow sweeping out a curve. |
Let
r(t)
denotes its value at the moment t
and r(t
+ h)
represents its value at t
+ h,
and P
and P1
are the corresponding
points of the curve, as is shown in the figure below. |
The
increment Dr
= r(t +
h)
-
r(t)
is the vector that falls in the direction of the secant line
PP1
and points from P
to P1.
The difference quotient |
|
obtained
by division with the scalar
h,
is the vector of the same
direction but of different length. |
The
limit of the difference quotient as
h
®
0 |
|
is
the
derivative vector of the
vector-valued
function that
falls in
the direction of the line tangent to the curve at P. |
|
|
|
If
x(t)
and y(t)
are the scalar components of the vector r
(t)
then, according to rules of
vector algebra, |
Dr
= [x(t +
h)
-
x(t)] i
+ [y(t +
h)
-
y(t)] j |
and
by use of the definition of the derivative |
|
is
the derivative vector,
where x' (t)
and
y' (t)
are its scalar components, and where |
|
|
is
its length or magnitude. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Contents
J
|
|
|
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
|
|
|