SOLUTION: A1. Find an example of a function f such that:
-the line y = 2 is a horizontal asymptote of the curve y = f(x), and
-the cures intersects the line y = 2 at an in nite number o
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Question 1038078: A1. Find an example of a function f such that:
-the line y = 2 is a horizontal asymptote of the curve y = f(x), and
-the cures intersects the line y = 2 at an in nite number of points.
Note: For full marks, prove that lim x->1 f(x) = 2, find all intersection points, and sketch a graph of the curve and the line y = 2.
A2. Let f(x) = x3 - bx for some real number b.
a. Find f(x) using the limit def inition of the derivative.
b. If the curve y = f(x) has a horizontal tangent line at a point with x = 2 fi nd the
y-coordinate of this point. (Your answer should be a real number, not an expression
involving b.)
Been struggling with these 2 questions...Thankyou
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Here is the second one.
y=x^3-bx
f(x+h)-f(x)/h
numerator=(x+h)^3-x^3=x^3+3x^2h+3xh^2+h^3-b(x+h)-x^3-bx
This is 3x^2h+3xh^2+h^3-bx-bh-bx=3x^2h+3xh^2+h^3-bh
Divide that by h
=3x^2+3xh+h^2-b
as h goes to 0, f'(x^3) goes to 3x^2-b, which is the derivative.
f(x) when x=2 is 8-2b
The derivative must equal zero, so 3x^2-b=0 for x=2. Therefore b=12
the y-coordinate of the point when x=2 is 8-24 or -16. The point is (2,-16)
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