#### To determine

**To find:**

The linearization of function at given point ‘a’ and approximate value for 1.033

#### Answer

Lx=1+x and approximate value is 1.01

#### Explanation

**1. Concept:**

Use of Linear Approximations and Differentials, Derivative of function

**2. Formula:**

(i) Linearization Formula:

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

(ii) Power rule:

ddx xn=n xn-1

(iii) Chain Rule:

dydx= dydu dudx

(iv) Absolute Value:

If ‘a’ is any positive integer, then | x | < a is equivalent to -a < x < a

**3. Given:**

fx= 1+3x3

4. **Calculation:**

fx= 1+3x3

fx=(1+3x)13

Let, u= 1+3x

Differentiating u with respect to x

dudx=3

So function becomes,

fx=u13

Now, differentiating with respect to x.

ddxfx=ddxu13

By using power rule and Chain Rule

f'(x) =13u13-1dudx

Simplifying and substituting the value of u and dudx,

f'(x)=131+3x-23 (3)

f'(x)=1+3x-23

So, value of fx and f'(x) at a=0 is fa and f'(a)

fa= 1+3a3

f0= 1+3(0)3

f0= 1

f'(a)=1+3(a)-23

f'(0)=1+3(0)-23

f'(0)=1

Now, linearization at a=0 is,

By Linearization Formula

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

Lx=f0+f'0(x-0)

Lx=1+1 (x-0)

Lx=1+x

Now, 1.03 can be split as 1 + 3 (0.01)

Therefore,1.033= 1+3(0.01)3. So, x = 0.01

L(0.01)=1+0.01=1.01

**Conclusion:**

Linearization Function is Lx=1+x

Approximate value for 1.033 = 1.01

#### To determine

The values of x for which the linear approximation is accurate to within 0.1

#### Answer

The linear approximation is accurate within 0.1 when -0.2<x<0.4

#### Explanation

**1. Concept:**

By using linear approximation

2. **Calculation:**

solve the inequality

fx-Lx<0.1

∴1+3x3-1+x<0.1

By Absolute Value

∴ -0.1<1+3x3-1+x<0.1

Now subtract 1+3x3 throughout

∴ -1+3x3-0.1< -1+x<0.1-1+3x3

Multiply whole equation by ‘-‘

∴ --1+3x3-0.1>1+x>-0.1-1+3x3

∴ 1+3x3+0.1>1+x>-0.1+1+3x3

Rewriting the above expression

-0.1+1+3x3<y=1+x<1+3x3+0.1

By plotting the graph of above expression

The line y=1+x intersects y=1+3x3+0.1 at points (-0.2, 0.7) and (0.4, 1.4). Thus we can say that linear approximation is accurate within 0.1 when -0.2<x<0.4.

**Conclusion:**

Therefore, the linear approximation is accurate within 0.1 when -0.2<x<0.4