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### Transcript of Calculus Basics

Calculus Basics

DIFFERENTIAL CALCULUS

What is calculus? Often the simplest questions are the hardest to answer. Calculus is a branch of

mathematics which uses operators to extract information from functions.

Branch of Mathematics Mathematical Elements Used

Arithmetic Numbers (0, 1, 101.7, , e, ..)

Algebra Variables (x, y, z, t, L, A, ..)

Analysis Functions (f(x), v(t), a(t), ..)

Calculus Operators (derivative, integral, Laplacian, ..)

There is a very large number of operators but we will focus on the derivative and integral. There is also a

very large number of functions, but the only ones typically needed are constant, polynomial, trigonometric,

exponential and logarithmic. These can also be combined by addition, subtraction, multiplication, division

and/or composition.

The derivative of a function is defined in terms of a limit.

(1.4)

Unless a new function is encountered, the limit definition of the derivative is never used. Instead, the limits

are determined once for each class of functions, and the resulting expressions, which are called shortcuts,

are used to calculate them.

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

The derivatives of the other four trigonometric functions (tangent, cotangent, secant and cosecant) can be

determined by using one or more of the chain rules (see below). Notice the derivative of ex is itself. This is

a very important property of this very special function. NOW YOU TRY IT

What is the derivative of ?

(A)

(B)

(C)

(D)

(E) 10x

When the basic types of functions are combined by addition, subtraction, multiplication, division and/or

composition, chain rulesare used to determine the derivatives of the combinations.

(2.0)

(2.1)

(2.2)

(2.3)

EXAMPLE: DETERMINING THE DIVISION CHAIN

RULE

If , then f(x)? Write f(x) as the product of two functions: f(x) = g(x)(h(x))

-1.

Apply the multiplication and composition chain rules: f(x) = g(x)(h(x))-1

- g(x)(h(x))-2

h(x).

Multiply the first term by h(x) and divide by h(x): f(x) = g(x)(h(x))-2

h(x) - g(x)(h(x))-2

h(x).

Factor and write as a fraction:

(2.4)

This is the Division Chain Rule.

EXAMPLE: USING THE CHAIN RULES

If f(x) = 5xe2x

, then f(x) =? The derivative of 5x is 5 and applying the composition chain rule the derivative of e

2x is

2e2x

.

Applying the multiplication chain rule: f(x) = (5)(e2x

) + (5x)(2e2x

).

Factoring yields: f(x) = 5e2x

(1 + 2x). This is the answer written in its simplest form.

NOW YOU TRY IT

If f(x) = x2sinx, then f(x) =?

(A) 2xsinx

(B) x2cosx

(C) xsinx(2 + x)

(D) xcosx(2 + x)

(E) x(2sinx + xcosx)

In order to take the derivative of a function that contains the product of two sub-functions, the product rule

can be applied as follows:

Where u and v are functions of the independent variable x.

EXAMPLE: USING THE PRODUCT RULE

The differential equation f(y) dx = f(x)dy (x and y) and the corresponding differential

variables (dx and dy) are not on their respective sides of the equation. Separation of

variables is a technique that uses simple cross multiplication in order to arrange the

differential equation into the desired form. Therefore:

f(y) dx = f(x)dy

can also be expressed as:

EXAMPLE: USING THE SEPARATION OF

VARIABLES TECHNIQUE

The differential equation that would be used in this case is:

Substitute the equation a=v2 into the above equation adx = vdv:

v2dx = vdv

Using the separation of variables, the equation becomes:

Integrate both sides of the equation (with integration limits):

Calculate the integral:

APPLICATIONS OF DIFFERENTIAL CALCULUS

Two applications of the derivative used in physics are determining the slope of the graph of a function at a

point and the minimum and/or maximum values of a function.

To determine the slope of the graph of a function at a point, calculate the derivative of the function and

substitute the value of the variable.

EXAMPLE: SLOPE OF A FUNCTION AT A POINT

If f(x) = 5x2 + 2x - 3, then what is the slope at x = 2?

Using the shortcut for polynomials combined with the addition chain rule yields f(x) = 10x + 2. The slope at x = 2 is then 10(2) + 2 = 22.

To determine the minimum and/or maximum values of a function, take its derivative, set it

equal to zero, solve for the variable by factoring and using the Zero-Product Rule, and

substitute the value or values of the variable into the function.

EXAMPLE: MAXIMUM STRESS

During normal operation, the main cable of the Golden Gate Bridge in San Francisco is

subjected to a tensile stress [MPa] described by the following function:

f(x) = 50x0.5

- 50x + 100

where f(x) is the tensile stress and x is the strain in the main cable.

Determine:

(i) The strain at which the maximum stress develops

(ii) The maximum stress the main cable is subjected to at this strain.

Solution:

(i) To calculate the strain at which the maximum stress develops, we will take the

derivative of f(x) with respect to x, and set it equal to 0:

f(x) = 50x0.5

- 50x + 100

f (x) = 25x-0.5

- 50

Let f (x) = 0:

25x-0.5

- 50 = 0

25x-0.5

= 50

x-0.5

= 2

x = (1/2)2 = 0.25 . . . strain at which maximum stress is achieved

(ii) To calculate the maximum stress, substitute x = 0.25 into f(x).

Maximum stress = f(0.25)

= 50(0.25)0.5

- 50(0.25) + 100

= 112.5 MPa

NOW YOU TRY IT

What's the maximum value of f(x) = 10x(2 - x)?

(A) 1

(B) 2

(C) 5

(D) 10

(E) 20

Evaluating the integral of a function or a set of data points is often impossible to solve in closed form.

Therefore, numerical integration is required. In this section, the popular and reasonably accurate

Trapezoidal Rule will be explained.

First and foremost, it must be stressed that an integral is simply an infinite sum as follows:

The Trapezoidal Rule is a discreet method that is used to evaluate an integral; the technique uses the

right hand side of the above equation with .

Considering a random individual area, it is assumed that a straight line segment can estimate the

function from fi to fi+1. Notice that there is an error, which represents the difference between the area

that is bounded by the line segment and the area that is bounded by f(x). The estimated shape is a

trapezoid. The formula for the area of a trapezoid is:

Estimate the area under the curve by combining the two formulas above:

Collect terms:

Therefore, in general, the Trapezoidal Rule can be expressed as:

It is important to note that a smaller will result in a larger number of areas, which will provide a

more accurate result. This is because a smaller line segment from fi to fi+1 will more closely represent

the exact value of f(x).

INTEGRAL CALCULUS

The indefinite integral of a function can also be defined in terms of a limit, but for our purposes a more

useful definition relates it to the derivative. The indefinite integral is the inverse of the derivative, and vice

versa. That is to say, the derivative of the indefinite integral of a function is equal to the function itself and

the indefinite integral of the derivative of a function is equal to the function itself, within a constant called

the constant of integration.

The symbol for the derivative is d, which is an abbreviation for difference, and the symbol for the

indefinite integral is , which looks much like an s, which is an abbreviation for sum. Using these symbols, the inverse relationships are:

(2.5)

(2.6)

This pair of equations is called the Fundamental Theorem of Calculus and provides an intuitive way of

calculating the indefinite integrals of many functions. Since the indefinite integral is the inverse of the

derivative, it's often called the antiderivative.

Unless a new function is encountered, the limit definition of the indefinite integral is never used. Instead,

the limits are determined once for each class of functions, and the resulting expressions, which are

called shortcuts, are used to calculate them.

(2.7)

(2.8)

(2.9)

(3.0)

(3.1)

You should verify these shortcuts are correct by applying the derivative to the indefinite integrals of each

function. If the shortcut is correct, then the derivative of the indefinite integral will be equal to the