Use a linear approximation of f(x)=^3√x at x=8 to approximate ^3√7.
Express your answer as an exact fraction.
what are you using this symbol ^ for?
its cubed
do you meant the f(x)=cubrtx ?
then f'= 1/ 3(cubrtx)^2
f(7)=f(8)-f'(1)=cubrt8-1/3(cubrt1)2
= 2-1/3=1 2/3=5/3
To use the linear approximation of f(x) = ∛x at x = 8 to approximate ∛7, we can follow these steps:
Step 1: Find the derivative of f(x) = ∛x.
The derivative of ∛x can be found by using the power rule for differentiation. The power rule is a general rule for finding the derivative of a function of the form f(x) = x^n, where n is a constant:
d/dx (x^n) = n*x^(n-1)
Applying the power rule to f(x) = ∛x, we get:
f'(x) = (1/3) * x^(-2/3)
Step 2: Find the equation of the tangent line at x = 8.
To find the equation of the tangent line, we need a point on the line and the slope of the line.
a) Point on the line: We know that the point (8, f(8)) lies on the tangent line because we want to approximate ∛7 using x = 8. So, the point is (8, ∛8).
b) Slope of the line: The slope of the tangent line is equal to the value of the derivative at x = 8. So, the slope is f'(8).
So, we have the point (8, ∛8) and the slope f'(8) to find the equation of the tangent line.
Step 3: Write the equation of the tangent line.
Using the point-slope form of a linear equation, the equation of the tangent line can be written as:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - ∛8 = f'(8) * (x - 8)
Step 4: Use the equation of the tangent line to approximate ∛7.
We want to approximate ∛7, so we substitute x = 7 into the equation of the tangent line and solve for y.
y - ∛8 = f'(8) * (7 - 8)
y - ∛8 = f'(8) * (-1)
y = ∛8 - f'(8)
Now, we substitute the known values into the equation:
y = ∛8 - f'(8)
y = ∛8 - [(1/3) * 8^(-2/3)]
Simplifying further:
y = ∛8 - (1/3) * (1/∛8^2)
y = ∛8 - (1/3) * (1/64)
y = ∛8 - 1/192
So, ∛7 ≈ ∛8 - 1/192, expressed as an exact fraction.