The slope of the secant line
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Consider the function f(x)= sqrt(7x+6)
We will take steps to find the tangent line to the graph of f at the general point (x,f(x)), and use it to find a tangent line with a specific property.
(a) For any point (x,f(x)) on the graph of f, let (x+h,f(x+h)) be another point on the graph of f, where h [cannot]= 0 . The slope of the (secant) line joining the two points (x,f(x)) and (x+h,f(x+h)) can be simplified to the form A/(sqrt(7x+6))+(sqrt(7(x+h)+6)), where A is a constant. Find A.
Answer: A=_____________
b) By considering the slope of the secant line as h approaches 0, find the slope of the tangent line to the graph of f at the point (x,f(x)).
Answer: The slope of the tangent line to the graph of f at the point (x,f(x)) is _____
[answers should be expressed in x]
c) At which point on the graph of f is the tangent line parallel to the line y=7/8x?
Answer: The tangent line to the graph of f at the point (______,________) is parallel to the line y=7/8x.
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This solution thoroughly demonstrates a tangent line at the point.
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