L12-linearApprox.mws

__Calculus I__

**Lesson 12: Linear Approximation**

Assume that f(x) is differentiable.

Recall that :
y = f (x +
x) - f (x)

x = dx =( x +
x ) - x

dy = f ' (x) dx

For sufficiently small
x,

f ( x +
x) - f (x) is approximately equal to f ' (x) dx

and hence for sufficiently small
x

dy approximates
y

**Example**
**1**

Let
, x = 2, and
x = 0.4.

Calculate,
y, dy, plot f(x) and the tangent line when x = 2.

`> `
**restart: **

`> `
**f:= x -> x^5;**

`> `
**D(f);**

`> `
**f( 2 + 0.4) - f(2);**

`> `
**f(2.4);**

`> `
**df:= x -> 5*x^4;**

`> `
**df(2);**

`> `
**80 * .4;**

Hence,
y =
and dy = f '(2) (0.4) = 32;

note that
x = 0.4 is large.

`> `
**tl:= x -> 32 + 80*(x - 2);**

`> `
**tl(2.4);**

`> `
**with(plots): **

`Warning, the name changecoords has been redefined`

`> `
**A:= plot({f(x),tl(x)}, x= 1.5..2.7, color=[blue,brown]):**

`> `
**B:= plot([t,32,t = 2..2.4], color = magenta):**

`> `
**C:= plot([2.4,t,t= 32 ..79.62624], color = magenta):**

`> `
**F:= plot([t, 64, t= 2.4..2.6], color = black):**

`> `
**G:= plot([2.6,t,t=64..79.62624], color = black):**

`> `
**H:= plot([t,79.62624, t= 2.4..2.6], color = black): **

`> `
**K:= textplot([2.65,74,'dy'],color = red):**

`> `
**L:= textplot([2.5,50,'deltay'] , color = red):**

`> `
**M:= textplot([2.2,20,'deltax'],color = red):**

`> `
**display({A,B,C,F,G,H,K,L,M}, axes = boxed);**

The tangent line is the linear approximation to a function. We see here, that

for larger
x , the linear approxiamtion is NOT a good one.

**Example**
**2**

Let
.

a) Find the linear approximation to f(x) when x = 1.

b) Plot f(x) and the approximation on the same axes.

c) Use the linear approximation to estimate, sqrt(3.98).

Compare this estimate to the actual value.

d) Calculate
y and dy, for x = 1 and
x = 2.98.

Plot f(x), the linear approximation and show dy and
y .

`> `
**f:= x -> sqrt(x + 3);**

`> `
**D(f);**

`> `
**df:= x -> (0.5) * ( 1 / (sqrt(x + 3)));**

`> `
**f(1);**

`> `
**df(1);**

Thus the linear approximation (i.e., the tangent line) passes through the point (1,2)

and has slope .25 .

`> `
**L:= x -> 2 + 0.25 * ( x - 1);**

a) Thus, linear approximation is: L(x) = 1.75 + 0.25 x .

`> `
**plot({f(x),L(x)}, x = -5..5, color=[brown,blue]);**

The linear approximation to f (x) is NOT good for x not close to 1.

`> `
**plot({f(x),L(x)}, x = 0.5..1.5, color=[brown,blue]);**

For x values close to 1, the linear approximation is better.

`> `
**L(3.98);**

`> `
**f(3.98);**

c) The linear approximation function gives 2.7450 as an estimate for sqrt(3.98),

the actual value is approximately,

d) We now do part (d).

`> `
**f(3.98) - f(1);**

Thus,
y =
(for x = 1 and
x = 2.98).

`> `
**df(1) * 2.98;**

Thus, dy =
(for x = 1 and
x= 2.98).

`> `
**L(3.98);**

`> `
**f(1);**

`> `
**L(1);**

`> `
**f(3.98);**

`> `
**with(plots): **

`Warning, the name changecoords has been redefined`

`> `
**A:= plot({f(x),L(x)}, x = 0..6, color=[blue,brown]):**

`> `
**B:= plot([t,2, t = 1..3.98], color = magenta):**

`> `
**C:= plot([3.98,t,t = 2..2.7450], color = magenta):**

`> `
**F:= plot([t,2.641968963, t = 3.98..4.2],color = black):**

`> `
**G:= plot([4.2,t,t = 2.641968963..2.7450], color = black):**

`> `
**H:= plot([t,2.7450, t = 3.98..4.2],color = black):**

`> `
**K:= textplot([4.5,2.7,'dy'], align=RIGHT, color = red):**

`> `
**L:= textplot([4.2,2.3,'deltay'], align=RIGHT, color = red):**

`> `
**M:= textplot([2.3,1.9,'deltax'], color = red):**

`> `
**display( {A,B,C,F,G,H,K,L,M}, axes = boxed);**