Use linear approximation to approximate √64.1 as follows.
Let f(x)=√x. The equation of the tangent line to f(x) at x=64 can be written in the form y=mx+b. Compute m and b.
Using this find the approximation for √64.1.
since for small ∆x,
∆y/∆x ≈ dy/dx,
f(64+∆x) ≈ f(64) + f'(64)*∆x = 8 + 1/16 * 0.1 = 8.00625
check: √64.1 ≈ 8.00624756
the estimate seems reasonable, since √x is concave down, meaning the tangent line lies above the curve, making the estimate a bit too big.
but, doing the math, we have the slope of 1/16 at (64,8) making the line
y = 1/16 (x-64) + 8
which is the same as the estimate above
Alright, let's give it a go!
Step 1: Finding the slope (m)
To find the slope (m) of the tangent line to f(x), we first need to find the derivative of f(x). Since f(x) = √x, its derivative is given by f'(x) = 1/(2√x).
Now, let's plug in x = 64 into f'(x) to obtain the slope at x = 64:
f'(64) = 1/(2√64) = 1/128 = 0.0078125.
Therefore, the slope m is approximately equal to 0.0078125.
Step 2: Finding the y-intercept (b)
To find the y-intercept (b) of the tangent line, we need to know a point on the line. Fortunately, we have x = 64 which lies on the line.
To find y, we plug in x = 64 into f(x):
f(64) = √64 = 8.
Therefore, the point (64, 8) is on the tangent line.
Using the point-slope form of a line, we can find b:
y - y1 = m(x - x1)
y - 8 = 0.0078125(x - 64)
y - 8 = 0.0078125x - 0.5
y = 0.0078125x - 0.5 + 8
y = 0.0078125x + 7.5.
Hence, the y-intercept b is approximately 7.5.
Step 3: Approximating √64.1
Now that we have the equation of the tangent line, y = 0.0078125x + 7.5, we can use this to approximate √64.1.
Plugging in x = 64.1 into the equation, we have:
y = 0.0078125(64.1) + 7.5
y = 0.5 + 7.5
y ≈ 8.
Therefore, the linear approximation of √64.1 is approximately equal to 8.
Remember, this is just a rough approximation using linear approximation. The actual value of √64.1 is more precise. Anyway, hope this helps tickle your mathematical funny bone!
To find the equation of the tangent line to f(x)=√x at x=64, we need to compute the slope and y-intercept.
1. Slope:
The slope of the tangent line can be found using the derivative of f(x)=√x.
f'(x) = (1/2) * x^(-1/2)
Evaluate the derivative at x=64:
f'(64) = (1/2) * (64)^(-1/2) = (1/2) * (1/8) = 1/16
Hence, the slope (m) of the tangent line is 1/16.
2. Y-Intercept:
To find the y-intercept (b), substitute the coordinates (x=64, y=f(64)) into the equation of the line.
Using f(x)=√x, f(64)=√64=8.
Therefore, the point (64, 8) lies on the tangent line.
Now, we can substitute the slope (m=1/16) and the point (64, 8) into the equation of a line (y=mx+b) to find b:
8 = (1/16)(64) + b
8 = 4 + b
b = 8 - 4
b = 4
Hence, the y-intercept (b) is 4.
3. Equation of the Tangent Line:
The equation of the tangent line to f(x)=√x at x=64 can be written as:
y = (1/16)x + 4
Now, we can use this equation to approximate √64.1.
4. Approximation for √64.1:
Substitute x=64.1 into the equation of the tangent line:
y = (1/16)(64.1) + 4
y = 4.00625 + 4
y = 8.00625
Hence, using linear approximation, √64.1 is approximated to be 8.00625.
To find the equation of the tangent line to f(x) at x=64, we need to first find the slope of the tangent line. The slope of the tangent line represents the rate of change of the function at that specific point.
To find the slope, we can compute the derivative of f(x) with respect to x. The derivative of f(x) = √x is given by:
f'(x) = (1/2) * (x^(-1/2)) = (1/2√x).
Now, substitute x=64 into the derivative formula to find the slope at x=64:
f'(64) = (1/2√64) = 1/16.
This means the slope of the tangent line at x=64 is 1/16.
To find the y-intercept (b) of the tangent line, we need to use the point (64, f(64)).
Since f(x) = √x, we can find f(64) by plugging in x=64:
f(64) = √64 = 8.
Now, we have the point (64, 8) which lies on the tangent line.
Using the point-slope form of a line (y-y1 = m(x-x1)), we can write the equation of the tangent line as:
y - 8 = (1/16)(x - 64).
Simplifying this equation gives:
y = (1/16)x + 5.
Now, to approximate √64.1 using linear approximation, we substitute x=64.1 into the equation of the tangent line:
y = (1/16)(64.1) + 5.
Calculating this gives:
y ≈ 4.00625 + 5 ≈ 9.00625.
Therefore, using linear approximation, √64.1 is approximately 9.00625.