APPM 1350
Summer 2014
Study Guide for Exam 4
1. u-Substitution:
(a) The Substitution Rule: If u = g(x) is a differentiable function whose range is an interval I and if f
is continuous on I, then
Z
Z
0
f (g(x)) g (x) dx = f (u) du.
(b) You should be familiar with how to compute integrals using u-substitution for definite and indefinite
integrals.
(c) Symmetric functions: Suppose f is continuous on [−a, a].
Z a
Z a
f (x) dx.
f (x) dx = 2
• If f is even, then
0
Z −a
a
f (x) dx = 0.
• If f is odd, then
−a
2. Inverse functions:
(a) A function f is one-to-one if it never takes on the same value twice. That is,
f (x1 ) 6= f (x2 )
when x1 6= x2 .
• A function is one-to-one if it passes the horizontal line test (i.e. no horizontal line intersects
the graph of y = f (x) more than once).
(b) Definition of an inverse function: If f is one-to-one, then its inverse function is defined by
y = f −1 (x)
⇐⇒
x = f (y)
OR
y = f (x)
⇐⇒
x = f −1 (y).
• Domain of f = Range of f −1
• Range of f = Domain of f −1
(c) Cancellation equations:
f −1 (f (x)) = x
f f −1 (x) = x
for every x in domain (f )
for every x in domain f −1
(d) How to find the inverse of a one-to-one function f :
i. Write y = f (x)
ii. Solve the equation for x in terms of y (if possible).
iii. Interchange x and y. The resulting equation is y = f −1 (x).
(e) The graph of f −1 is found by reflecting the graph of f about the line y = x.
(f) Theorem: If f is one-to-one and continuous on an interval, then its inverse f −1 is also continous.
(g) Theorem: Let f be a one-to-one, differentiable function with inverse function f −1 . If f (b) = a and
f 0 (b) 6= 0, then the inverse function is differentiable at a and
0
f −1 (a) =
1
1
.
f 0 (b)
3. Natural logs and exponentials
(a) The natural log function:
Z
x
1
dt, where x > 0.
1 t
• f (x) = ln x is an increasing, continuous function.
• By definition, ln x =
x
= ln x − ln y
(2) ln
y
• Laws of logs: (1) ln (xy) = ln x + ln y
(3) ln (xr ) = r ln x
• lim ln x = ∞ and lim ln x = −∞.
x→∞
x→0+
d
1
•
(ln x) = .
dx
x
d
g 0 (x)
•
[ln (g(x))] =
dx
g(x)
Z
d
1
1
•
(ln |x|) =
=⇒
dx = ln |x| + C
dx
x
x
(b) The natural exponential function:
• The number e is defined by ln e = 1.
• The natural exponential is the inverse function of ln x:
y = ex
•
•
•
•
•
⇐⇒
x = ln y.
Cancellation equations: eln x = x for x > 0 and ln (ex ) = x for all real x.
f (x) = ex is an increasing, continuous function.
Domain of ex = R = Range of ln x.
Range of ex = (0, ∞) = Domain of ln x.
lim ex = ∞ and lim ex = 0. (Note that the latter implies lim e−x = 0.
x→∞
x→∞
x→−∞
• Laws of exponents: (1) ex+y = ex ey
(2) ex−y
ex
= y
e
(3) (ex )r = erx
d x
(e ) = ex
dx
d g(x) = g 0 (x)eg(x)
•
e
Zdx
•
•
ex dx = ex + C.
(c) Be able to perform logarithmic differentiation.
(d) Know the graphs of ln x and ex .
(e) Theorem: The only solutions of the differential equation
dy
= ky,
dt
where k is a constant, are the exponential functions y(t) = y(0)ekt .
• Understand exponential growth and decay problems (i.e. population growth, radioactive decay,
half-life, Newton’s Law of Heating and Cooling).
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4. General logs and exponentials
(a) General exponentials with base a:
• Assume a > 0. Then by definition ax = ex ln a .
d x
• Using the definition, you should know how to derive:
(a ) = ax ln a
dx
Z
ax
•
ax dx =
+ C (a 6= 1).
ln a
ax
• Laws of exponents: (1) ax+y = ax ay (2) ax−y = y (3) (ax )y = exy
a
• If a > 1, then lim ax = ∞ and lim = 0.
x→∞
(4) (ab)x = ax bx
x→−∞
x
• If 0 < a < 1, then lim a = 0 and lim = ∞.
x→∞
x→−∞
(b) General logs with base a:
• loga x is the inverse function of ax . That is, y = loga x
x = ay .
ln x
• Change of base formula: For a > 0 and a 6= 1, we have loga x =
.
ln a
d
1
• Using the change of base formula, know how to derive
(loga x) =
.
dx
x ln a
⇐⇒
5. Inverse Trig Functions:
(a) Trig functions are only one-to-one on a restricted domain, so their inverses are defined there.
(b) Inverse Sine:
y = arcsin x ⇐⇒ x = sin y, − π2 ≤ y ≤ π2
Domain of sin x = − π2 , π2 = Range of arcsin x.
Range of sin x = [−1, 1] = Domain of arcsin x.
Cancellation equations: arcsin (sin x) = x, x in − π2 , π2 and sin (arcsin x) = x, x in [−1, 1].
d
1
•
(arcsin x) = √
when −1 < x < 1. (You should be able to derive this.)
dx
1 − x2
Z
1
√
•
dx = arcsin x + C
1 − x2
(c) Inverse Cosine:
•
•
•
•
•
•
•
•
y = arccos x ⇐⇒ x = cos y, 0 ≤ y ≤ π
Domain of cos x = [0, π] = Range of arccos x.
Range of cos x = [−1, 1] = Domain of arccos x.
Cancellation equations: arccos (cos x) = x, x in [0, π] and cos (arccos x) = x, x in [−1, 1].
d
1
(arccos x) = − √
•
when −1 < x < 1.
dx
1 − x2
(d) Inverse Tangent:
y = arctan x ⇐⇒ x = tan y, − π2 < y < π2
Domain of tan x = − π2 , π2 = Range of arctan x.
Range of tan x = R = Domain of arctan x.
Cancellation equations: arctan (tan x) = x, x in − π2 , π2 and tan (arctan x) = x, x in R.
π
π
• lim arctan x = and lim arctan x = − .
x→∞
x→−∞
2
2
•
•
•
•
3
d
1
(arctan x) =
for all x.
1 + x2
Zdx
1
dx = arctan x + C
•
1 + x2
•
6. Hyperbolic Functions
(a) Definition: sinh x =
ex − e−x
ex + e−x
and cosh x =
.
2
2
sinh x
1
• The others are defined similarly to the trig functions: tanh x =
, cschx =
,
cosh x
sinh x
1
1
sechx =
, and coth x =
.
cosh x
tanh x
(b) Symmetry: sinh x (like sin x) is odd, while cosh x (like cos x) is even.
(c) cosh2 x − sinh2 x = 1.
d
d
d
(d) Derivatives:
(sinh x) = cosh x,
(cosh x) = sinh x,
(tanh x) = sech2 x
dx
dx
dx
(e) Inverses: Note that sinh x and tanh x are one-to-one on R, while cosh x is one-to-one on the
restricted domain [0, ∞).
y = sinh−1 x
⇐⇒
x = sinh y,
y = cosh−1 x
⇐⇒
x = cosh y,
⇐⇒
x = tanh y.
−1
y = tanh
(f) Inverse derivatives:
x
y>0
d
1
d
1
d
1
sinh−1 x = √
,
cosh−1 x = √
,
tanh−1 x =
2
2
dx
dx
dx
1
−
x2
1+x
x −1
7. L’Hôpital’s Rule:
Suppose f and g are differentiable and g 0 (x) 6= 0 near a (except possibly at a). Suppose that we have
an indeterminate form of type 00 or ±∞
±∞ . Then
f (x)
f 0 (x)
= lim 0
,
x→a g (x)
x→a g(x)
lim
if the limit on the right-hand side exists (or is ∞ or −∞).
(a) Know how to apply L’Hôpital’s Rule to indeterminate products of the form 0 · ∞ (that is, write
f
g
or 1/f
, in order to obtain the form 00 or ±∞
f g = 1/g
±∞ ).
(b) Also know how to deal with indeterminate differences (∞ − ∞) and indeterminate powers (00 , ∞0 ,
or 1∞ ).
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