Introduction:
AP Physics C is a calculus-based physics course, but the fact is that you can
do pretty well, grade-wise, in AP Physics C- particularly the mechanics part -
without any knowledge of calculus. Of course it won't hurt you to know
what is going on with all those strange mathematical symbols and concepts, and
there will definitely be problems on he AP Test that require a knowledge of
calculus. Fortunately, you can get by really well -even on the AP Test -
with very rudimentary calculus skills that can be learned in a few minutes.
You need to know something about:
- Derivatives - a derivative is a rate of change, or graphically, the
slope of the tangent line to a graph. Although physics is "chock full"
of application of the derivative, you need to be able to calculate only very
simple derivatives in this course.
- Definite Integrals - a definite integral represents an area, and
there a myriad of applications of integrals in physics. In this course,
you need to be able to evaluate only very elementary integrals.
Derivatives:
A derivative is a rate of change, which, geometrically, is the slope of a
graph. In physics, velocity is the rate of change of position, so
mathematically belocity is the derivative of position. Acceleration is the
rate of change of velocity, so acceleration is the derivative of velocity.
Net force is the rate of change of momentum, so the derivative of an object's
momentum tells you the net force on the object. These are only a few
applications of the derivative in physics.
Derivative Rules:
Finding the derivative of a function ("differentiating" in calculus language)
is a rule-based operation. In other words, you need to recognize what
derivative rule applies, and then apply it. In order to recognize what
derivative rule applies, you need to know some derivative rules. The tables
below list the derivative rules that you will need to use in this course, and
shows some examples of their use. These rules are stated using "t" as a
variable (the derivative is "with respect to", t, in calculus language), since
most of the functions that we will use are functions of time. If you are
taking a derivative whose variable is "s," simply substitute "x" for "t" in the
derivative rule
- Rules for Particular Functions:
|
Rule in
English |
Rule in Math |
Example |
|
The derivative of a
constant is zero |
 |
If x(t) = 5, then v(t)
= 0
If v(t) = -3, then a(t)
= 0 |
|
The derivative of t is
one |
 |
If x(t) = t, then v(t)
= 1 |
|
The derivative of t to a
power is the power time t to the “one less” power. |
 |
If x(t) = t2,
then v(t) = 2t1 =2t. (n = 2)
If v(t) = t4,
then a(t) = 4t3. (n = 4)
If x(t) = t-3,
then v(t) = -3t-4. (n = -3)
|
|
The derivative of the
sine of t is the cosine of t |
 |
If x(t) = sin t then,
v(t) = cos t |
|
The derivative of the
cosine of t is negative of the sine of t |
 |
If v(t) = cos t, then a(t) = -sin t. |
|
Rule in
English |
Rule in Math.
Notation |
Example |
|
The derivative of a
constant times a function equals the constant times the derivative of the
function. |
 |
If x(t) = 3t2,
then v(t) = 3(2t1) = 6t (c =3 and u =t2)
If v(t) = 4sin t, then
a(t) = 4 cos t.
(c = 4, u = sin t) |
|
The derivative of the
sum (or difference) of two functions is the sum (or difference) of their
derivatives. |
 |
If x(t) = t + sin t,
then v(t) = 1+ cos t, (u = t, w = sin t)
If v(t) = t2-
4t, then a(t) = 2t1 – 4(1) = 2t – 4. (u = t2, w =
4t) |
|
The derivative of a
composite (one function inside another function) function equals the
derivative of the “outside” function leaving the “inside” function alone,
time the derivative of the “inside” function. (The Chain Rule) |
If
 |
If x(t) = (t+2)2,
then v(t) = 2 (t+2)1 (1 +0) = 2(t+2). (u = t2 w = t
+2)
If v(t) = sin (2t3),
then a(t) = cos(2t3)(2)(3t2) = 6t2 cos(2t3)
(u = cos t, w = 2t3 |
Definite Integrals:
A definite integral represent an area, and evaluating a
definite integral (“integrating” in calculus language) is the inverse of finding
the a derivative – like subtraction is the inverse of addition. In physics the
area under a velocity vs. time graph represents displacement, so the definite
integral of velocity gives displacement. The area under an acceleration vs.
time graph equals change in velocity, so the definite integral of acceleration
tells you the change in velocity. The area under a force vs. position graph
equals work done by the force, so the definite integral of force (with respect
to position) tells you the work done by the force. There are many, many
applications of the definite integral in physics.
Rules for Definite Integrals:
Antiderivative Rules for Particular Functions. If you are
a calculus student, you will notice that we are ignoring an important
mathematical point in the following rules.
|
Rule in
English |
Rule in Math.
Notation |
Example |
|
The Antiderivative of a
constant is the constant times t. |
 |
 |
|
The Antiderivative of t
to a power is t to the power-plus-one, divided by the power plus one.
|
 This is the inverse of the derivative formula.
 |
 |
|
The Antiderivative of
the sine of t is the negative of cosine t |
 this is the inverse of the derivative formula
 |
|
|
The Antiderivative of
the cosine of t is the sine of t. |
 This is the inverse of the derivative formula
 |
 |
Antiderivative Rules for Combinations of Functions:
In the rules below, u and w represent functions of time, t.
|
Rule in
English |
Rule in Math.
Notation |
Example |
|
The antiderivative of a
constant times a function equals the constant times the antiderivative of
the function. |
 |
|
|
The antiderivative of
the sum (or difference) of two functions equals the sum (or difference) of
their antiderivatives. |
 |
  |
|
Rule in
English |
Rule in Math.
Notation |
Example |
|
The antiderivative of a
constant times a function equals the constant times the antiderivative of
the function. |
|
|
|
The antiderivative of
the sum (or difference) of two functions equals the sum (or difference) of
their antiderivatives. |
|
|
The Fundamental Theorem of Calculus - tells you how to
evaluate definite integrals based on what you know about antiderivative
To Be Completed
