Estimate delta f using linear approximation

F(x) = x^ (1/3)e^(x-1), a=1, delta x = 0.4
linear approximation for delta f =
Error is =
Percentage error is =
please explain tburoughly

To estimate delta f using linear approximation, we can use the formula:

delta f = f'(a) * delta x

First, let's find the derivative of the function f(x) = x^(1/3) * e^(x-1).

To do this, we can apply the product rule and chain rule:

f'(x) = (1/3) * x^(-2/3) * e^(x-1) + x^(1/3) * e^(x-1) * (d/dx)(x-1)

Simplifying the second term using the chain rule, we get:

f'(x) = (1/3) * x^(-2/3) * e^(x-1) + x^(1/3) * e^(x-1)

Now, to estimate delta f, we need to evaluate f'(a) at a=1:

f'(1) = (1/3) * 1^(-2/3) * e^(1-1) + 1^(1/3) * e^(1-1)
= (1/3) * 1 * 1 + 1 * 1
= 1/3 + 1
= 4/3

Now, we can plug this value into the linear approximation formula:

delta f = f'(a) * delta x
= (4/3) * 0.4
= 4/3 * 2/5
= 8/15

Therefore, the linear approximation for delta f is 8/15.

Now, let's calculate the error of this approximation. The error is given by the difference between the actual change delta f and the linear approximation delta f. In this case, it is:

Error = actual delta f - linear approximation delta f

To find the actual delta f, we can evaluate f(x) at x=a+delta x and subtract f(a):

actual delta f = f(a + delta x) - f(a)
= f(1 + 0.4) - f(1)

Plugging the values into the original function, we get:

actual delta f = (1.4)^(1/3) * e^(1.4-1) - 1^(1/3) * e^(1-1)

Calculating this expression, we find:

actual delta f ≈ 0.719

Now, we can find the error:

Error = actual delta f - linear approximation delta f
= 0.719 - 8/15

To simplify the calculation, we can convert 8/15 to a decimal:

8/15 ≈ 0.5333

Subtracting this from 0.719, we get:

Error ≈ 0.719 - 0.5333
≈ 0.1857

Therefore, the error of the linear approximation is approximately 0.1857.

To find the percentage error, we can divide the error by the actual delta f, and then multiply by 100:

Percentage error = (Error / actual delta f) * 100
= (0.1857 / 0.719) * 100

Calculating this, we find:

Percentage error ≈ 25.8%

Therefore, the percentage error of the linear approximation is approximately 25.8%.

To estimate delta f using linear approximation, we can first find the derivative of the given function, F(x), and then use the formula for linear approximation:

Linear Approximation Formula: delta f ≈ f'(a) * delta x

Step 1: Finding the derivative of F(x):
To find the derivative of F(x) = x^(1/3) * e^(x-1), we can use the product rule and the chain rule.

Let's denote f(x) = x^(1/3) and g(x) = e^(x-1).

Derivative of f(x):
f'(x) = (1/3) * x^(-2/3)

Derivative of g(x):
g'(x) = e^(x-1)

Now, using the product rule:

F'(x) = f'(x) * g(x) + f(x) * g'(x)
= (1/3) * x^(-2/3) * e^(x-1) + x^(1/3) * e^(x-1)

Step 2: Applying the linear approximation formula:
To estimate delta f, we need to evaluate F'(a) at x = a and then multiply it by delta x.

Given a = 1, we need to find F'(1).

F'(1) = (1/3) * (1)^(-2/3) * e^(1-1) + (1)^(1/3) * e^(1-1)
= (1/3) * e^0 + 1
= (1/3) + 1
= 4/3

Step 3: Calculating delta f:
delta f = f'(a) * delta x
= (4/3) * 0.4
= 4/7

Therefore, the linear approximation for delta f is 4/7.

Step 4: Calculating the error:
The error associated with the linear approximation can be found by subtracting the actual value of delta f from the estimated value using linear approximation.

Error = Actual delta f - Linear approximation for delta f

Since we don't have the actual value of delta f, we can't calculate the exact error. However, we can approximate the error using the concept of percentage error.

Step 5: Calculating the percentage error:
Percentage error = (Error / Actual delta f) * 100

Since we don't know the actual delta f, we can't calculate the percentage error either.

To summarize:
- The linear approximation for delta f is 4/7.
- The error and percentage error cannot be calculated without knowing the actual value of delta f.