Basic Review Of Calculus Page 2
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1
2If f (x) = 0 and f (x) > 0 at a point, then the point corresponds to a local
minimum.
If f (x) = 0 and f (x) < 0 at a point, then the point corresponds to a local
maximum.
5. You should understand limits; How to compute them and how they relate to calculus.
When computing limits it may be useful to try the following techniques:
(a) If the function is continuous at the point, just plug in the point.
(b) If it is not continuous at the point try the following:
Simplify by factoring or rationalizing.
0
Write in the form “
” or “ ” and apply L’Hospital’s Rule. That is, take the
0
derivative of the top and bottom separately and then try to take the limit.
A major application of limits in Calculus I comes from the definition of the derivative.
In particular, we defined the derivative of a function f (x) to be
f (x + h)
f (x)
f (x) = lim
.
h
h
0
6. A common problem for calculus students is remembering the properties of trigonomet-
ric and logarithmic functions. I review some of these key ideas here:
Given a right triangle and a non-right angle, θ, in the triangle, you should under-
stand what is meant by
opposite
adjacent
opposite
sin(θ)
sin(θ) =
cos(θ) =
tan(θ) =
=
hypotenuse
hypotenuse
adjacent
cos(θ)
1
1
1
csc(θ) =
sec(θ) =
cot(θ) =
sin(θ)
cos(θ)
tan(θ)
x
sin(x) cos(x)
You also should know all the following basic trigono-
0
0
1
metric values. Note that you can use these values
1
3
π/6
2
2
along with the rules above and the symmetry of lo-
2
2
π/4
cations on a circle to find several other values of the
2
2
3
1
π/3
trigonometric functions.
2
2
π/2
1
0
Recall the basic trig identities. The first identity should be very familiar, you can
2
2
get the others in the first row by dividing the first identity by cos
(x) and sin
(x),
respectively. The second row contains the so-called “double angle identities”. We
will make use of these during this course.
2
2
2
2
2
2
sin
(x) + cos
(x) = 1
tan
(x) + 1 = sec
(x)
1 + cot
(x) = csc
(x)
2
1
2
1
1
sin
(x) =
(1
cos(2x)) cos
(x) =
(1 + cos(2x)) sin(x) cos(x) =
sin(2x)
2
2
2
Finally remember the basic ln(x) and power rules:
b
ln(1) = 0
ln(e) = 1
ln(a
) = b ln(a) ln(ab) = ln(a) + ln(b)
a
b
a
b
a+b
x
a b
1
a
a
ab
x
x
= x
= x
= x
(x
)
= x
b
a
x
x
There is no way to simplify
a
2
+ b
2
in general. In particular,
a
2
+ b
2
= a + b.
2
2
2
2
Notice that
3
+ 4
=
25 = 5, this is not the same at
3
+
4
= 7. DO NOT
MAKE THE MISTAKE OF SIMPLIFYING
a
2
+ b
2
.
2
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