MAT 213Brief Calculus

Section 4.1

Approximating Change

• Recall that when we “zoomed in” on a differentiable graph, it became almost linear, no matter how much curve there was in the original graph

• Therefore a tangent line at x = a can be a good approximation for a function near a

• Let’s take a look at the function

and its tangent line at x = 1

2y x

Let’s zoom in

Let’s zoom in again

• Around x = 1 both graphs look almost identical

• Let’s find the tangent line at x = 0 and use it to approximate f(.5), f(.9), f(1.1), f(1.5), and f(2)

• We will then compare these to their actual function values

• Around x = 1 both graphs look almost identical

• Let’s find the tangent line at x = 0 and use it to approximate f(.5), f(.9), f(1.1), f(1.5), and f(2)

x f(x) f’(x)

0.5 0.25 0

0.9 0.81 0.8

1.1 1.21 1.2

1.5 2.25 2

2 4 3

The Tangent Line Approximation

Suppose f is differentiable at a. Then, for values of x near a, the tangent line approximation to f(x) is

f(x) ≈ f(a) + f’(a)(x - a)

The expression f(a) + f’(a)(x - a) is called the Local Linearization of f near x=a .

(We are thinking of a as fixed, so that both f(a) and f’(a) are constant)

The error, E(x) in the approximation is defined by:

E(x) = f(x) - f(a) ≈ f’(a)(x - a)

actual approximation

Now let’s use the same two graphs to talk about change

Δx

Δy

Δx

Δy

Now is the slope of our tangent line

y

x

Δx

Δy

f’ is ALSO the slope of our tangent line

Δx = h

f(x+h) – f(x)

Notice that f(x+h) – f(x) is close to Δy

Δx = h

f(x+h) – f(x)

Notice that f(x+h) – f(x) is close to Δy

Δy

Δx = h

f(x+h) – f(x)

So Δy ≈ f(x+h) – f(x)

Δy

Δx = h

f(x+h) – f(x)Δy

( ) ( )'( )

y f x h f xf x

x h

– Using our results we have

– Which can be rewritten to as

– Which approximates the change in the function values by multiplying the derivative by a small change in inputs, h

– Alternatively we can write

– Which says the output at x + h is approximately the output at f plus the approximate change in f

( ) ( )'( )

f x h f xf x

h

'( ) ( ) ( )f x h f x h f x

'( ) ( ) ( )f x h f x f x h

Marginal Analysis• Often a companies decision to continue to

produce goods is based on how much additional revenue they gain versus the additional cost

• The Marginal Cost is the change in total cost of adding one more unit

• Therefore it can be approximated by the instantaneous rate of change

• Marginal Cost = MC = C’(q)

• Marginal Revenue = MR = R’(q)

• Marginal Profit = MP = P’(q)

Example

• What is the marginal cost of q if fixed costs are $3000 and the variable cost is $225 per item?

• What is the marginal revenue if you charge $375 per item?

• What is your marginal profit?

EXAMPLES

a. Find the tangent line approximation for each of the following

b. Does the approximation give you an upper or lower-estimate?

Pg 239, #22

0near )(

-2near 42)(

1 near 31)(

3

2

xexk

xxxxg

xxxf

x