Question

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)?

Answer #1

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