estimate change in f using the linear approximation and compute both error and the % error
f(x)= (3+x)^1/2
a=1
change in x=0.5
I tried to answer but Jiskha will not allow me to post my reply.
In general calculate f(1) = 2
f(1.5) = 2.1 something
that is exact to the accuracy of your calculator
then calculate f(x+dx) = f(x)+ dx (df/dx)
where df/dx = .5/(3+x)^.5
which is (.5 / 2).5 = .125
so f(1.5) = 2.125 linear approximation
error = 2.125 - exact
percent = 100 * error/2.1whatever)
Help someone, I can no longer reply to questions as me.
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To estimate the change in f using linear approximation, we can use the equation for linear approximation:
f(x + Δx) ≈ f(a) + f'(a) * Δx
where f'(a) represents the derivative of f with respect to x evaluated at x = a.
First, let's find the derivative of f(x). The function f(x) = (3 + x)^(1/2) can be rewritten as:
f(x) = (3 + x)^(0.5)
To find the derivative, we can use the power rule for differentiation:
f'(x) = 0.5 * (3 + x)^(-0.5)
Now we need to evaluate f(a) and f'(a). In this case, a = 1, so we have:
f(a) = f(1) = (3 + 1)^(0.5) = 2^(0.5) = √2
f'(a) = f'(1) = 0.5 * (3 + 1)^(-0.5) = 0.5 * 4^(-0.5) = 0.5 * 0.5 = 0.25
Now let's calculate the change in f, Δf, using the linear approximation formula:
Δf ≈ f(a) + f'(a) * Δx
Δf ≈ √2 + 0.25 * 0.5
Δf ≈ √2 + 0.125
Next, let's compute the error in our approximation. The error is given by the difference between the actual change in f and the estimate we calculated using linear approximation:
error = Δf_actual - Δf
To find the actual change in f, we can calculate f(a + Δx) - f(a):
f(a + Δx) = f(1 + 0.5) = f(1.5) = (3 + 1.5)^(0.5) = 4.5^(0.5) = √4.5
Δf_actual = f(a + Δx) - f(a) = √4.5 - √2
Now we can calculate the error:
error = Δf_actual - Δf = (√4.5 - √2) - (√2 + 0.125)
Finally, to compute the percent error, we can use the formula:
percent error = (error / Δf_actual) * 100
percent error = (error / (√4.5 - √2)) * 100