Applications of Derivatives, Approximations, Slope of Tangent and Normal

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Approximations

Let be a function of and be a small change in and let be the corresponding change in . Then,

,where as

is a very-very small quantity that can be neglected, therefore

We have , approximately

Some Important Terms:

Absolute Error:

The error in is called the absolute error in.

Relative Error:

If is an error in , then is called relative error in .

Percentage Error:

If is an error in , then is called percentage error in .

Example:

Using differentials find the approximate value of

Solution:

Take

Let and , then

When

Now

Hence (Approximate)

Slope of Tangent and Normal:

Slope of Tangent and Normal

Slope of Tangent and Normal

Slope of Tangent and Normal

Let be a continuous curve and let be a point on it then the slope at is given by

at

And (1) is equal to

We know that a normal to a curve is a line perpendicular to the tangent at the point of contact

We know that (From Fig.)

Slope of normal at or at

Example:

Show that the tangents to the curve at the points are parallel.

Solution:

The equation of the curve is

Differentiating (1) w.r.t. , we get

The tangents to the curve at are parallel as the slopes at are equal

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