Use linear approximation, i.e. the tangent line, to approximate 8.4^(1/3) as follows:
Let f(x)= x^(1/3) . The equation of the tangent line to f(x) at x=8 can be written in the form y=mx+c where m=1/12 and c=4/3:
Using this, find our approximation for 8.4^(1/3).
To use the tangent line approximation method, you need to follow these steps:
Step 1: Define the function
Let f(x) = x^(1/3)
Step 2: Find the equation of the tangent line
To find the equation of the tangent line at x = 8, we need to find the slope (m) and the y-intercept (c).
The slope (m) is given as 1/12.
To find the y-intercept (c), substitute the point (8, f(8)) = (8, 2) into the equation y = mx + c. Solve for c.
2 = (1/12)(8) + c
2 = 2/3 + c
c = 8/3 - 2/3
c = 6/3
c = 2
So, the equation of the tangent line is y = (1/12)x + 2.
Step 3: Approximate 8.4^(1/3)
Now that we have the equation of the tangent line, we can use it to approximate 8.4^(1/3).
Substitute x = 8.4 into the equation y = (1/12)x + 2:
y = (1/12)(8.4) + 2
y = 0.7 + 2
y = 2.7
Therefore, the linear approximation of 8.4^(1/3) is approximately 2.7.
To use linear approximation to approximate 8.4^(1/3), we can use the equation of the tangent line to approximate the value.
First, let's find the equation of the tangent line to f(x) at x = 8.
The equation of a tangent line can be written in the form y = mx + c, where m represents the slope and c represents the y-intercept.
Given that m = 1/12 and c = 4/3, we can write the equation of the tangent line as:
y = (1/12)x + 4/3.
Now, we can use this equation to approximate the value of 8.4^(1/3).
Substituting x = 8.4 into the equation, we get:
y = (1/12)(8.4) + 4/3
= 7/10 + 4/3
= (21 + 40)/30
= 61/30.
Therefore, our approximation for 8.4^(1/3) is 61/30.