Use differential, (i.e. linear approximation), to approximate cube root of 64.1 as follows:
Let f(x) = cube root of x. The linear approximation to f(x) at x = 64 can be written in the form y = mx+b. Compute m and b and find cube root of 64.2 via linear approximation
dy = 1/3 x^-2/3 dx
At x=4, dy = 1/48 dx
So, use dx=0.2 to find dy, and add that to y(64)=4
Note also that since the slope is 1/48,
y-4 = 1/48 (x-64)
To use linear approximation to approximate the cube root of 64.1, we need to find the equation of a line that approximates the function f(x) = cube root of x near x = 64.
Step 1: Find the derivative of f(x)
The derivative of f(x) = x^(1/3) can be calculated using the power rule:
f'(x) = (1/3)x^(-2/3)
Step 2: Evaluate the derivative at x = 64
Substituting x = 64 into the derivative equation:
f'(64) = (1/3)(64)^(-2/3) = 4/(3 * (64)^(2/3))
Step 3: Form the equation of the tangent line
The equation of the tangent line can be written as: y = f(64) + f'(64)(x - 64)
Step 4: Calculate f(64) and substitute the values into the equation
f(64) = 64^(1/3) = 4
Substituting the values in:
y = 4 + (4/(3 * (64)^(2/3)))(x - 64)
Step 5: Approximate the cube root of 64.2
To approximate the cube root of 64.2 using the linear approximation, substitute x = 64.2 into the equation:
y ≈ 4 + (4/(3 * (64)^(2/3)))(64.2 - 64)
Simplifying the equation gives the approximation of the cube root of 64.2 using linear approximation.