The graph of
f(x)=−10x+e5sin(x)f(x)=−10x+e5sin(x)
is rotated counterclockwise about the origin through an acute
angle θθ. What is the largest value of θθ for which the rotated
graph is still the graph of a function? What about if the graph is
rotated clockwise?
To answer this question we need to find the maximal slope of
y=f(x)y=f(x), which is , and the minimal slope which
is .
Thus the maximal acute angle through which the graph can be rotated
counterclockwise is θ=θ= degrees.
Thus the maximal acute angle through which the graph can be rotated
clockwise is θ=θ= degrees. (Your answer should be negative to
indicate the clockwise direction.)
Note that a line y=mx+by=mx+b makes angle αα with the horizontal,
where tan(α)=mtan(α)=m.
Hints: Recall that a graph of a function is
characterized by the property that every vertical line intersects
the graph in at most one point. In view of this:
1. If ALL lines y=mx+by=mx+b of a fixed slope mm intersect a graph
of y=f(x)y=f(x) in at most one point, what can you say about
rotating the graph of y=f(x)y=f(x)?
2. If some line y=mx+by=mx+b intersects the graph of y=f(x)y=f(x)
in two or more points, what can you say about rotating the graph of
y=f(x)y=f(x)?
3. If some line y=mx+by=mx+b intersects the graph of y=f(x)y=f(x)
in two or more points, what does the Mean Value Theorem tell us
about f′(x)f′(x)?
Get Answers For Free
Most questions answered within 1 hours.