Suppose g (x) is increasing and concave up everywhere and
g (A) = 9, g '(A) = 17, h = 0.01
estimate g(A-h)
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To estimate g(A-h), we can use the tangent line approximation method.
Step 1: Find the slope of the tangent line at point A.
Since g'(A) = 17, the slope of the tangent line at point A is 17.
Step 2: Determine the change in x.
Since h = 0.01, the change in x is -0.01 (A minus h).
Step 3: Use the slope and change in x to calculate the change in y.
The change in y is given by the formula: change in y = slope * change in x.
So, change in y = 17 * (-0.01) = -0.17.
Step 4: Estimate g(A-h).
To estimate g(A-h), we need to subtract the change in y from g(A).
g(A-h) = g(A) - change in y
g(A-h) = 9 - (-0.17)
g(A-h) = 9 + 0.17
g(A-h) = 9.17
Therefore, g(A-h) is estimated to be 9.17.
To estimate g(A-h), we can use the concept of linear approximation.
Linear approximation can be expressed as:
f(x) ≈ f(a) + f'(a)(x - a)
In this case, we want to approximate g(A-h), so our values will be:
f(x) ≈ f(a) + f'(a)(x - a)
f(x) ≈ g(A) + g'(A)(x - A)
Given that g(A) = 9 and g'(A) = 17, we have:
f(x) ≈ 9 + 17(x - A)
Now, let's substitute the value of h = 0.01 into the approximation:
f(x) ≈ 9 + 17(0.01 - A)
To get a more accurate estimate, we need to know the value of A. If you provide that, we can calculate g(A-h) using the given information.