use the limit process to find the slope of the graph of 'square root' x+8 at (8,4)
a. 4
b. 1/4
c. the slope is undefined at this point
d. 1/8
e. infinity
do we have y = √(x+8) ?
I will assume that, if (8,4) lies on it
slope
= lim ( √(16+h) - 4)/h , as h --->0
multiply top and bottom by (√(16+h) + 4)
= lim ( √(16+h) - 4)/h * (√(16+h) + 4)/(√(16+h) + 4)
= lim (16 + h - 16)/(h(√(16+h) + 4))
= lim h/(h(√(16+h) + 4))
= lim 1/(√(16+h) + 4) , as h --->0
= 1/(4+4)
= 1/8
or in general:
slope =
Lim ( √(x+h + 8) - √(x+8) )/h , as h ---> 0
multiply top and bottom by ( √(x+h + 8) + √(x+8) )
= Lim ( √(x+h + 8) - √(x+8) )/h * ( √(x+h + 8) + √(x+8) ) / ( √(x+h + 8) + √(x+8) )
= lim (x+h + 8 - (x+8) / (h( √(x+h + 8) + √(x+8) ))
= lim h/(h( √(x+h + 8) + √(x+8) ))
= lim 1/( √(x+h + 8) + √(x+8) ) , as h --->0
= 1/(√(x+8) + √(x+8) )
= 1/ (2√(x+8))
at (8,4) we get
1/(2√16) = 1/8
To find the slope of the graph of the square root function at a specific point, we can use the limit process. The slope of a function at a particular point is given by the derivative of the function at that point.
In this case, we are asked to find the slope of the square root function at the point (8,4). To do this, we need to find the derivative of the square root function and evaluate it at x = 8.
The square root function is given by f(x) = √x.
To find the derivative of f(x), we can use the power rule for derivatives:
1. Take the exponent, which is 1/2 in this case, and multiply it by the coefficient in front of the variable, which is 1.
2. Subtract 1 from the exponent to get -1/2.
So, the derivative of f(x) with respect to x is f'(x) = (1/2) * x^(-1/2).
Let's calculate the derivative at x = 8:
f'(8) = (1/2) * 8^(-1/2) = (1/2) * (1/√8)
Now, simplify the value:
f'(8) = (1/2) * (1/(√2 * √4)) = (1/2) * (1/(√2 * 2)) = 1 / (4√2)
To determine the slope at the point (8,4), we find the value of f'(8) and verify if it matches any of the answer choices:
f'(8) = 1 / (4√2)
Simplifying by rationalizing the denominator:
f'(8) = 1 / (4√2) * (√2/√2) = √2 / (4 * 2) = √2 / 8
Since the slope is √2 / 8, which cannot be reduced further, none of the answer choices match the slope.
Therefore, the correct answer is not listed among the given options.