Estimate delta f using the Linear Approximation and use a calculator to compute the error.
f(x)=sqrt(3+x). a=6. delta x=0.1.
delta f=
With these calculations, we have determined that the square root of ___________ is approximately _______
The error in Linear Approximation is:_____
To estimate delta f using the Linear Approximation, we can use the formula:
delta f = f'(a) * delta x
First, let's find f'(x), the derivative of f(x):
f'(x) = 1 / (2 * sqrt(3 + x))
Next, we will substitute the value of a = 6 into f'(x):
f'(6) = 1 / (2 * sqrt(3 + 6)) = 1 / (2 * sqrt(9)) = 1/(2 * 3) = 1/6
Now, we will calculate delta f using the formula:
delta f = f'(a) * delta x = (1/6) * 0.1 = 1/60
Therefore, delta f is approximately 1/60.
To compute the error, we can compare the actual value of delta f with the estimated value obtained using the calculator.
Using a calculator, we can find the actual value of delta f by evaluating f(x) at a + delta x:
f(6.1) = sqrt(3 + 6.1) = sqrt(9.1) ≈ 3.0166 (rounded to 4 decimal places)
The error in the Linear Approximation is given by:
error = actual delta f - estimated delta f = 3.0166 - 1/60 = 3.0166 - 0.0167 ≈ 2.9998 (rounded to 4 decimal places)
Therefore, the error in the Linear Approximation is approximately 2.9998.
To estimate delta f using the Linear Approximation method, we can use the formula:
delta f ≈ f'(a) * delta x
Where f'(a) is the derivative of f(x) evaluated at x = a, and delta x is the given change in x.
First, let's find the derivative of f(x) = sqrt(3+x). Using the power rule of differentiation, we have:
f'(x) = (1/2)*(3+x)^(-1/2)
Now, substitute x = 6 to find f'(a):
f'(6) = (1/2)*(3+6)^(-1/2) = (1/2)*(9)^(-1/2) = 1/6
Next, we can calculate delta f:
delta f = f'(a) * delta x = (1/6) * 0.1 = 1/60
So, delta f is approximately 1/60.
To compute the error in the Linear Approximation, we can use the formula:
error = |f(a+delta x) - f(a)|
First, let's calculate f(a+delta x):
f(a+delta x) = sqrt(3+(6+0.1)) = sqrt(3+6.1) = sqrt(9.1) = 3.01662 (rounded to 5 decimal places)
Now, calculate f(a):
f(a) = sqrt(3+6) = sqrt(9) = 3
Substitute the values in the error formula:
error = |3.01662 - 3| = 0.01662 (rounded to 5 decimal places)
Therefore, the error in the Linear Approximation is approximately 0.01662.