Tangent Approximation

Tangent Line Approximation

The origin of many approximation methods is Newton's Quotient:

Newton realized that you can use this practically. If is small,

Or,

Setting :

Equation

A linear approximation is defined by:

Intuition

We can intuitively make sense of this if we want. We have the point of the function we know, and then add the rate of change of the function at that point, and follow that rate until we reach the x-coordinate we want.

Visualization

Example

Approximate

solution
Knowing , let . We want .
We will try a linear approximation. Let , which is close to 30. and are simple to calculate.

So,

Compare this to the actual value of

Equation

If we take the approximation, we can find the differential to use:

So

Example

Use differentials to approximate

solution
Knowing , we identify , and .
We then calculate

When decreases by , decreases by approximately . Thus

Compare this to the actual value of

Remark

Interesting note: L'Hôpital's rule comes from the linear approximation

Tangent Planes

We wish to generalize the tangent line approximation for multivariable functions.

If we have a smooth surface , then at each point there should be a tangent plane rather than a tangent line.

Equations of Planes

Scalar Equation of a Plane

Definition

let be a normal vector to the plane containing
let be any point on the plane

Intuition

Consider a non-vertical plane passing through the point , with . These are vertical extensions of lines in the xy-plane, so their equations are .

  • Let be another plane, and consider the vector . It will lie in the plane as well.
  • Let but a normal vector to the plane.

These two vectors must be perpendicular, that is, (see dot product), which gives

Visualization

Equation

If we take , and then set and , we obtain:

so an alternate equation to our plane is

Lets consider the "alternate" equation (), and suppose that this is the tangent plane to a surface with equation .
If we set , we obtain the equation of a line in the plane, (since we're not moving along the y axis, and ). Observe that this is just the linear approximation formula with
Recall partial derivatives, that the slope of this line is , which is the value of .
Similarly, if we set instead, we see that .

Formula

The equation of a tangent plane of is:

Example

Find the tangent plane and normal vector to at .

solution

so we have

and

so our tangent plane is

with normal vector

Tangent Plane Approximation

Equation

Recall the differential approximation for single variables. For multivariable calculus:

So

Example

Use the tangent plane to estimate where .

solution
Consider for our approximation.

Finding the partial derivatives:

actual:

ERROR:

One major application of the linear approximation is error propagation using differentials.

With the Gradient Vector

(See Gradient Vector)

We can develop a short hand to write the tangent plane at the point , with .

(see Dot Product)

The components of the gradient vector give us the slope of tangent plane in the x, y directions respectively.

Equation

The linear approximation of the point knowing is: