Tangents and Normals
SEMESTER- I (BMG1CC1A)
BY
DR. AMRITA DAS
TANGENT LINES
P0 Let= (x0,y0,z0) be a point on a surface S, and let C be any curve passing through P0 and lying entirely in S. If the tangent lines to all such curves C at P0 lie in the same plane, then this plane is called the tangent plane to S at P0.
Definition
Tangent Plane at P0{x0,y0,z0}
Curve C
passes through P0{x0,y0,z0}
In this figure The tangent plane to a surface S at a point P0 contains all the tangent lines to curves in S that pass through P0.
TANGENT PLANES
Let S be a surface defined by a differentiable function z = f(x, y), and let P0 = (x0, y0) be a point in the domain of f. Then, the equation of the tangent plane to S at P0 is given by
z = f(x0, y0) + fx (x0, y0) (x- x0) + fy (x0, y0) (y- y0)
To see why this formula is correct , let’s first find two tangent lines to the surface S. The equation of the tangent line to the curve that is represented by the intersection of S with the vertical trace given by x = x0 is
z = f (x0, y0) + fy (x0, y0) (y- y0)
Similarly , the equation of the tangent line to the curve that is represented by the intersection of S with the vertical trace given by y = y0 is
z = f(x0, y0) + fx (x0, y0) (x- x0)
A parallel vector to the first tangent line is a= j + fy (x0, y0) k
A parallel vector to the second tangent line is b = i + fx (x0, y0) k
We can take the cross product of these two vectors:
Example:-
Find the equation of the tangent plane to the surface defined by the function
f(x, y) = 2x2 − 3xy + 8y2 + 2x − 4y + 4 at point (2, −1).
Solution:-
First, we must calculate fx (x, y) and fy (x, y) then use Equation with x0 = 2 and y0 = -1:
fx (x, y) = 4x − 3y + 2
fy (x, y) = −3x + 16y - 4
f (2, −1) = 2*(2)2 − 3*(2)*(−1) + 8*(−1)2 +2*(2) − 4*(−1) + 4 = 34
fx (2, −1) = 4*(2) − 3*(−1) + 2 = 13
fy (2, −1) = −3*(2) + 16*(−1) − 4 = −26
Then Equation becomes
z = f(x0, y0) + fx (x0, y0) (x- x0) + fy (x0, y0) (y- y0)
z = 34 + 13 (x − 2) − 26 (y − (−1))
z = 34 + 13x − 26 − 26y − 26
z = 13x − 26y − 18.
A tangent plane to a surface does not always exist at every point on the surface. Consider the piecewise function
The graph of this function follows
LINEAR APPROXIMATION
Linear approximation is a mathematical technique used to approximate the behavior of a function near a specific point by using a linear function (or a linear equation) that closely matches the function's behavior at that point. It is based on the idea that for sufficiently small changes in the input variables around a particular point, the function can be approximated by a linear function. Linear approximation is also known as tangent line approximation or linearization.
Definition
Method of Linearly Approximating a Function with respect to a Point
Linear approximation is closely linked with tangent planes in multivariable calculus. In the case of functions of several variables, such as f (x, y), linear approximation is extended to linear approximation of functions of multiple variables at a specific point (x0 , y0). This is achieved by using the equation of the tangent plane to the graph of the function at the point (x0, y0, f(x0, y0)).
The equation of the tangent plane to the graph of f(x,y) at the point (x0, y0,f(x0, y0)) can be expressed as:
z = f(x0, y0) + fx (x0, y0) (x− x0) + fy (x0, y0) (y− y0)
where:
z is the variable representing the function value
fx(x0, y0) is the partial derivative of f(x, y) with respect to x, evaluated at (x0, y0).
fy(x0, y0) is the partial derivative of f(x, y) with respect to y, evaluated at (x0, y0).
(x − x0) and (y − y0) represent the deviations from the point(x0, y0).
By doing this we obtain a linear function,
L (x,y) = f(x0, y0) + fx (x0, y0) (x− x0) + fy (x0, y0) (y− y0)
which is good approximation of f(x, y) when (x, y) is near (x0, y0).
The function L is called the Linearization of f at (x0, y0) and the approximation
f (x, y) ≈ f(x0, y0) + fx (x0, y0) (x− x0) + fy (x0, y0) (y− y0)
is called the Linear Approximation or Tangent Plane Approximation of f at (x0, y0).
Let us linearly approximate f(x,y)= x2 - y2 at the point (-2,1).
Example:-
Here,
f(-2,1) = (-2)2 - (1)2 = 3
fx(-2,1) = 2*(-2) = -4
fy(-2,1) = -2*(1) = -2
Now we fit these values in
L (x, y) = f (-2, 1) + fx (-2, 1) (x + 2) + fy (-2, 1) (y − 1)
We get,
L (x, y) = 3 + (-4)*(x+ 2) + (-2)*(y- 1)
= 3 – 4x – 8 – 2y + 2
= – 4x – 2y – 3
At (-2, 1), f(x, y) = 3 and L(x, y) = 3 ⇒ f(x, y) = L(x, y)
At (-1.998, 1.011), f(x, y) = 2.969883 and L(x, y) = 2.97 ⇒ f(x, y) ≈ L(x, y)
But, At (0, 0), f(x, y) = 0 and L(x, y)= -3 ⇒ f(x, y) ≠ L(x, y)
3D-Plot
x2- y2
- 4x - 2y - 3
APPLICATIONS OF LINEAR APPROXIMATION
PLANETARY ORBIT
In celestial mechanics, the motion of planets and other celestial bodies is often described using Newton's laws of motion and gravitation. When a planet orbits a star or a moon orbits a planet, the gravitational force between the two bodies determines the shape and characteristics of the orbit.
Linear approximation comes into play when we want to understand how the motion of the celestial body changes over time or in response to external forces. By considering small deviations from a particular point on the orbit, we can use linear approximation to approximate the change in the body's position, velocity, and acceleration.
In summary, the tangent plane to a planetary orbit provides a local approximation of the motion of the celestial body at a specific point along its path. Linear approximation allows us to use this tangent plane to predict how the body's motion will evolve over time and understand its behavior in curved spaces such as planetary orbits.
OTHER APPLICATIONS
Engineering: Engineers often encounter curved surfaces in various applications such as aerodynamics, structural analysis, and design of complex mechanical systems. The tangent plane is essential in engineering for designing and analyzing structures that interact with curved surfaces.
Computer Graphics: In computer graphics and animation, creating realistic images of three-dimensional objects often involves rendering curved surfaces. The tangent plane is crucial in this context for techniques like texture mapping, shading, and surface deformation.
Robotics and Automation: In robotics and automation, understanding the local geometry of surfaces is essential for tasks such as robot navigation, grasping, and manipulation of objects. Robots equipped with sensors or vision systems use the tangent plane information to detect and navigate along curved surfaces, avoiding obstacles or following predefined paths accurately.
MAXIMUM AND MINIMUM VALUES FOR FUNCTIONS OF TWO VARIABLES
Example-1
Example-2
Critical Points
Local Minimum
Saddle Point
Local Maximum
SADDLE POINT
Example-1
Second Derivative Test
Absolute Maximum and Minimum Values
Extreme Value Theorem for Functions of Two Variables
Finding absolute maximum and minimum values of a function of two variables
To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D:
Step 1: Find the values of f at the critical points of f in D.
Step 2: Find the extreme values of f on the boundary of D.
Step 3: The largest of the values from steps 1 and 2 is the absolute maximum value; the . . . smallest of these values is the absolute minimum value.
Reference
1. Multivariable Calculus 7th Edition By James Stewart
2. An Introduction To Analysis Differential Calculus Part-I by Ram Krishna Ghosh and Kantish Chandra Maity
3. http://math.libretexts.org
4. https://chat.openai.com/