CALCULUS I (In a Nutshell)

CALCULUS I (in a nutshell)

We’ve spent this semester learning two main concepts:

1. DERIVATION

2. INTEGRATION

DERIVATION:

The derivative of a function f(x) with respect to the variable x is a FUNCTION denoted

df f 0(x) or . dx

The value of the derivative at a given point x = a tells us how the function f(x) is growing at the point a. More precisely, f 0(a) is the slope of the tangent line to the graph of f at the point (a, f(a)).

For this reason, the sign of the derivative is connected to the fact that the function f is increasing or decreasing. In particular:

• f is increasing when f 0 is positive;

• f is decreasing when f 0 is negative.

The points where the derivative is 0 are special, and are called “critical points”. Here the tangent line to the graph of the function is horizontal. A critical point P can be:

• a MAXIMUM, if f’ is positive before P and negative after P ;

• a MINIMUM, if f’ is negative before P and positive after P ;

• an INFLECTION POINT with horizontal inﬂectional tangent if f 0 mantains the same sign before and after P .

Careful: Points of maximum and minimum can also occur elsewhere: besides where f 0 = 0 you should always check also the following points, to see if they are maxima or minima:

• points where the derivative is not deﬁned!!

• endpoints of the domain of the function. (if included!)

1 The derivative of the derivative of the function f is called the SECOND derivative of f and denoted by f 00. The second derivative gives us information about how the graph of the function “bends”:

• f is CONCAVE UP if f 00 is positive;

• f is CONCAVE DOWN if f 00 is negative.

Studying concavity can also help us decide whether a critical point is a maximum or a minimum. Let P be a critical point:

• if f 00(P ) is positive, then P is a MINIMUM;

• if f 00(P ) is negative, then P is a MAXIMUM;

• if f 00(P ) = 0, then I do not know!!

Also, concavity is related to whether the graph of your function is above or below the tangent line at a given point:

• If f is concave up at a point P , then f is above the tangent line to the graph at the point (P, f(P )).

• If f is concave down at a point P , then f is below the tangent line to the graph at the point (P, f(P )).

Points where f 00 goes from positive to negative are called INFLECTION POINTS. In particular, if P is an inﬂection point, then f 00(P ) = 0.

(But f 00(P ) = 0 does NOT allow us to conclude that P is an inﬂection point !!)

Remember, the second derivative is the first derivative of the first derivative, so, the second derivative going from positive to negative corresponds to the first derivative having a maximum, and vice-versa, if f 00 goes from negative to positive, then f 0 has a minimum. Therefore, INFLECTION POINTS for f correspond exactly to MAXIMA and MINIMA for f 0.

Computing derivatives is not too bad. There are a few functions for which we had to compute derivatives “by hand”, using the deﬁnition of derivative as the limit of a rate of change. Then we developed a bunch of rules that help us taking the derivatives of more complex functions, starting from the easy ones we computed by hand. In particular, let me remind you:

2 • the derivative of a sum of functions IS the sum of the derivatives of each summand;

• the derivative of a constant times a function IS that same constant times the derivative of the function;

• the derivative of a product of two functions IS NOT the product of the derivatives of the factors; there is a rule (PRODUCT RULE) that allows us to take the derivative of a product of functions;

• the derivative of a quotient of two functions IS NOT the quotient of the derivatives of the factors; there is a rule (QUOTIENT RULE) that allows us to take the derivative of a quotient of functions;

• there is a rule that allows us to deal with composition of functions. This rule is called the CHAIN RULE.

The CHAIN RULE is undoubtedly the coolest and most important of all of the above. It is EXTREMELY IMPOR- TANT to know how to use it well!!!!! There are two equivalent ways of stating the chain rule:

1. (f(g(x)))0 = f 0(g(x))g0(x)

2. df df dg = dx dg dx .

The second form of the chain rule is the one that is most suited to problems of related rates and implicit diﬀerentiation, so I reccommend that you are very warmly acquainted with it.

INTEGRATION:

We have talked about two diﬀerent types of integrals:

1. DEFINITE INTEGRALS.

2. INDEFINITE INTEGRALS (a.k.a ANTIDERIVATIVES).

A DEFINITE INTEGRAL is a NUMBER and it represents an area. Well, more than an area, to be honest. It is an “area with sign”. By Z b f(x)dx a we mean the area between the graph of the function f(x) and the x-axis. This area acquires a sign according to:

• whether f(x) is above (+) or under (-) the x-axis.

• wheter a is before b (+) or b is before a (-).

3 The fact that the integral is an “intelligent” area with sign, allows it to have these beautiful properties:

1. you can break up an interval into pieces. The integral over the whole interval is the sum of the integral over each piece. Yes, even if chopping up your interval consists in subtracting an interval from another!! The formula is:

Z b Z c Z c f(x)dx + f(x)dx = f(x)dx. a b a

2. you can break up the integral of a sum of functions into the sum of the integrals. In a formula:

Z b Z b Z b (f(x) + g(x))dx = f(x)dx + g(x)dx. a a a

3. if you multiply a function by a constant number (5, 17, 23 or even the mystical c or N!!), such a number can harmlessly come out of the integral sign:

Z b Z b Nf(x)dx = N f(x)dx a a

Huge Caution: the beautiful rules pretty much end here. Don’t be creative in making up new ones. Statements such as the integral of a product is the product of the integrals, or a function can come out of the integral sign under certain circumstances are dreadfully wrong!!! Symmetry is cool. We like symmetry. The following observations often come in handy!

• Even functions are symmetric about the y-axis. Therefore the integral of an even function on an interval of the form [−a, a] is twice the integral of the same function on the interval [0, a].

• Odd functions are symmetric with respect to the origin. The integral of an odd function on an interval of the form [−a, a] is always 0.

An INDEFINITE INTEGRAL, on the other hand, is NOT a number, but a FUNCTION...actually not quite even that...it is a family of functions. The indeﬁnite integral of the function f(x) consists of all possible functions having derivative equal to f(x). Such functions are called ANTIDERIVATIVES of f(x).

If we take any antiderivative of f(x) and shift it vertically, you obtain another antiderivative of f(x) (the slope of the tangent line at each point does not change, since also the tangent lines are simply shifted vertically!!!). As a matter of fact, by doing so you ﬁnd ALL antiderivatives of f(x). Hence, if in any possible way you can get your hands on one antiderivative F (x) of f(x), then you can safely conclude:

Z f(x)dx = F (x) + C, where C is just any ol’ constant.

4 Finding antiderivatives is quite a tricky business...basically so far the only tool we have up our sleeve is...guessing. However, if you do have a guess for an antiderivative of f(x), then checking whether the guess is right is no problem. Just take the derivative of the antiderivative and see if you get f(x) back. For example: is Z xn+1 xndx = + C? n + 1 YES, because d xn+1 + C = xn!! dx n + 1 Finally, what brings the two types of integration together is the

!!! FUNDAMENTAL THEOREM OF CALCULUS !!!

If F(x) is an antiderivative of f(x), then Z b f(x)dx = F (b) − F (a). a Now, remember, a deﬁnite integral is a number, not a function, therefore you must always remember to plug in the appropriate values into the antiderivative. The right thing to do is, ﬁrst plug in F the value that appears in the upper side of the integral sign, then subtract F evaluated at the bottom extreme of integration.

If it’s easy, (and a lot of times it is!!) check whether the sign of the integral you obtain makes sense!

A useful way to interpret the FUNDAMENTAL THEOREM OF CALCULUS is that it tells you that, between point a and point b, the antiderivative F (x) grows exactly by the area between the graph of f(x) and the x-axis between a and b.

All this said, GOOD LUCK GUYS! Do your best, I hope you enjoyed this semester. I sure enjoyed you!

Renzo