Given points #(4, 70), (6, 69), (8, 72), (10, 81)# on the graph of a function #f(x)#, how do you find an approximate value for #f'(x)# ?
Bonus
Given points:
Write down the sequence of differences between consecutive terms:
Write down the sequence of differences between those differences:
Write down the sequence of differences between those differences:
graph{1/24 x^3  1/4 x^2  7/6 x + 76 [1, 12, 67, 83]}
Then:
and:
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To find an approximate value for ( f'(x) ), the derivative of the function ( f(x) ), using the given points, you can use finite difference approximations. One commonly used method is the forward difference formula:
[ f'(x) \approx \frac{f(x + h)  f(x)}{h} ]
where ( h ) is the interval between the points. Since the points are evenly spaced with a difference of 2 between each xvalue, we can use ( h = 2 ).
Using the forward difference formula with the given points:

For the point (4, 70): [ f'(4) \approx \frac{f(4 + 2)  f(4)}{2} = \frac{f(6)  f(4)}{2} = \frac{69  70}{2} = \frac{1}{2} ]

For the point (6, 69): [ f'(6) \approx \frac{f(6 + 2)  f(6)}{2} = \frac{f(8)  f(6)}{2} = \frac{72  69}{2} = \frac{3}{2} ]

For the point (8, 72): [ f'(8) \approx \frac{f(8 + 2)  f(8)}{2} = \frac{f(10)  f(8)}{2} = \frac{81  72}{2} = \frac{9}{2} ]
Now, you can use the same process to find ( f'(10) ). This will give you approximate values for the derivative ( f'(x) ) at each of the given points.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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