Download Inverse Trigonometric Functions and their Derivatives

Inverse Trigonometric Functions and their Derivatives
1. Warm Up. Think of two different numbers θ such that sin θ = 21 .
2. Review of arcsin.
(a) On the axes below, sketch y = sin x, and state the domain and range of the function sin x.
3π
2
π
Domain of sin x:
π
2
Range of sin x:
− 3π
2
−π
π
2
− π2
π
3π
2
− π2
Domain of arcsin x:
−π
Range of arcsin x:
− 3π
2
(b) Dorothy says, “arcsin x is the inverse function of sin x.” Why isn’t this right?
(c) What is the actual definition of arcsin x?
(d) On the axes above, sketch y = arcsin x. What are the domain and range?
1
(e) What is arcsin
?
2
B The function arcsin x is also written as sin−1 x.
It’s important to remember that sin−1 x refers
to arcsin x, not to (sin x)−1 ; (sin x)−1 is the same as sin1 x , which is the same as csc x. (Yes, this
notation is inconsistent: sin2 x is the same thing as (sin x)2 . This is one of those quirks that
you just have to remember!)
1
(f) What is cos sin−1
3
5
?
3. Similarly, we define:
• arccos x =
• arctan x =
4. (a) Sketch y = arctan x, and label any horizontal and vertical asymptotes on your graph. Then,
sketch a rough graph of the derivative of arctan x.
(b) Use implicit differentiation to find the derivative of arctan x. Does this match the graph you
sketched in (a)?
2
p
5. Now that you know the derivative of arctan x, find the derivative of h(x) =
not need implicit differentiation!)
arctan(4x)
. (You should
e5
Derivatives of the inverse trigonometric functions. (You should memorize these.)
d
1
(arcsin x) = √
dx
1 − x2
d
1
(arccos x) = − √
dx
1 − x2
d
(arctan x) =
dx
(You’ll do arccos x on your homework.)
6. (a) We looked at the limit lim
x→0
(b) Can you evaluate lim
x→1 π
4
sin x
a while ago; what was it?
x
sin(πx)
?
− arctan x
7. (a) What do we mean when we say “ 00 ” is an indeterminate form? Come up with a couple of examples
that show that “ 00 ” is an indeterminate form.
(b) Why is the indeterminate form
0
0
important in calculus?
(c) If f (x) = xx , what is f (0)?
What is lim f (x)?
x→0
(d) What other indeterminate forms can you think of?
8. Let’s look some more at the limit lim
x→1 π
4
sin(πx)
.
− arctan x
3
(a) If you were to graph sin(πx) and π4 − arctan x and zoom in around x = 1, what would the graphs
resemble? Can you approximate the graphs by something simpler near x = 1?
(b) Using your approximations, what do you think lim
x→1 π
4
sin(πx)
is?
− arctan x
f (x)
0
∞
f (x)
f 0 (x)
is a “ ” or “ ” type of limit, then lim
= lim 0
, if the
x→a g(x)
x→a g(x)
x→a g (x)
0
∞
latter limit exists or is ±∞. (This rule also works for one-sided limits.)
L’Hôpital’s Rule. If lim
+ Remember that L’Hôpital’s Rule only applies to limits of the form
(c) Use L’Hôpital’s Rule to evaluate lim
x→1 π
4
0
0
and
∞
∞!
sin(πx)
.
− arctan x
9. Evaluate the following limits. Warning: L’Hôpital’s Rule does not apply to both of them!
(a)
(b)
lim xe−2x
x→−∞
lim x2 ex
x→−∞
4
10. Evaluate lim
x→0+
√
x ln x. What form is this limit?
The strategy for dealing with a limit of this form is to rewrite the product f (x)g(x) as a quotient;
f (x)
g(x)
f (x)g(x) = 1/g(x)
or f (x)g(x) = 1/f
(x) . (Both equations are true; often, one will be easier to work
with than the other.)
11. Sketch arcsin x and
d
dx (arcsin x),
and then use the idea of #4(b) to prove
5
d
dx (arcsin x)
=
√ 1
.
1−x2
12. Let f (x) = arcsin(4x2 ).
(a) What is the domain of f (x)?
(b) Is f (x) an even function, an odd function, or neither?
(c) On what intervals is f increasing? On what intervals is it decreasing?
(d) Does f have an absolute maximum and absolute minimum on its domain? If so, find the absolute
maximum and minimum values, and say where they occur.
(e) In (a), you should have found that the domain of f was a closed interval [a, b]. What are lim+ f 0 (x)
x→a
and lim f 0 (x)? (Note that this is asking about f 0 , not about f .) What does this tell you about
x→b−
the graph of f ?
(f) Use the above information to sketch a rough graph of f . (Your sketch need not accurately reflect
the concavity of the graph.)
6