#### To determine

**To write: **Expression for the linearization of f at a.

#### Answer

Lx=fa+f'(a)(x-a)

#### Explanation

Close to the point of tangency a curve is nearby to its tangent line. In other words, the graph of a differentiable function looks like its tangent line near the point of tangency. From this we can find the approximate values of functions.

We can calculate a value f(a) of a function, but it might be hard to find the nearby values of f. So we consider the tangent which is a linear function. It is easy to calculate it’s values.

In other words, use tangent line at (a,fa) as an approximation to the curve y=f(x) when x is near a. An equation of tangent line is

y=fa+f'(a)(x-a)

and the approximation

f(x)≈fa+f'(a)(x-a)

called **linear approximation **or **tangent approximation** of f at a.The linear function whose graph is tangent line at x=a, that is,

Lx=fa+f'(a)(x-a)

is called linearization of f at a.

**Conclusion:**

Expression for the linearization of f at a is Lx=fa+f'(a)(x-a)

#### To determine

**To write: ** Expression for differential dy.

#### Answer

dy=f'xdx

#### Explanation

Consider y=f(x). Here f is a differentiable function. Hence dx becomes an independent variable and it can be given the value of any real number.

The differential dy is then defined as:

dy=f'x dx

Here,dy is a dependent variable. It will depend on the values of x and dx. **Conclusion:**

Expression for differential dy is dy=f'x dx

#### To determine

**To write: ** Draw a picture showing the geometrical meaning of ∆y and dy.

#### Answer

dy ≈∆y

#### Explanation

**Given:**dx=∆x

The, geometric meaning of differentials is shown above.

Let P(x,f(x)) and Q(x+∆x,fx+∆x) be points on the graph of f and let dx=∆x. The corresponding change in y is

∆y=fx+∆x-f(x)

The derivative f’(x) is the slope of the tangent line PR. Thus the directed distance from S to R is f'x=dy.

The value of the tangent line which rises or falls is given by dy, when x changes by an amount dx, whereas ∆y represents the amount that the curve y=f(x) rises or falls when x changes by an amount dx=∆x.

**Conclusion:**

If dx=∆x then dy ≈∆y.