MAT 213 Brief Calculus Section 4.1 Approximating Change.
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Transcript of MAT 213 Brief Calculus Section 4.1 Approximating Change.
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