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    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

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    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)

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    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:

    adx=vdv

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

    v2dx = vdv

    Using the separation of variables, the equation becomes:

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    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:

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    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

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    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 .

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    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).

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    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 original

    function. Also, notice the indefinite integral ofex is itself. This is another very important property of this

    very special function. DEFINITE INTEGRAL

    In many engineering applications, a physical quantity, f, can be expressed as a function of another

    independent variable, x. Examples include:

    Velocity, v, as a function of time, t

    o For instance, v(t) = t2+2t (velocity of a car on a highway)

    Force, F, as a function of displacement, s

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    o For instance, F(s) = 3s (in the case of the stretching of a spring)

    Temperature, T, as a function of time, t

    o For instance, T(t) = 100(1 - e-t) (temperature of a glass of ice water being microwaved)

    Electricity usage, P, in a house as a function of the number of people, n, in a house

    o For instance, P(n) = 2n + 4

    The definite integral is a technique used in aggregating the product of f(x) and x between a lower and an upper limit. For instance, when velocity, v(t), of a car is integrated over the time interval between 1 second to 3 seconds, we obtain the displacement of the car between 1 second and 3

    second.

    Properties of the Definite Integral

    1. When the limits of an integral are reversed, a negative sign needs to

    be assigned.

    2. Integrating between similar limits results in 0.

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    3. Multiple definite integrals can be decoupled and

    aggregated separately.

    4. If n is a constant, it can be taken out of the operation.

    5. A definite integral can be broken down into steps and

    aggregated, as long as the lower and upper limits are completely covered in the operation.

    Some Common Definite Integrals

    EXAMPLE 1:

    f (x) = 1

    EXAMPLE 2:

    f (x) = x

    EXAMPLE 3:

    f(x) = nxn, where n is a constant

    EXAMPLE 4:

    f(x) = ex

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    EXAMPLE 5:

    f(x) = enx

    EXAMPLE 6:

    EXAMPLE 7:

    f(x) = sin x

    EXAMPLE 8:

    f(x) = cos x

    EXAMPLE 9:

    f(x) = tan x

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    NOW YOU TRY IT

    What is the indefinite integral of ?

    (A)

    (B)

    (C)

    (D)

    (E) 10x

    (F) 10x + C

    APPLICATIONS OF INTEGRAL CALCULUS

    Two applications of the integral used in physics are determining area on the graph of a function and the

    average value of a function.

    The area on the graph of a function is defined as the area enclosed by the graph of the function, the

    horizontal axis, and two vertical lines. (See Figure 0.6.)

    Figure 0.6

    If the area is above the horizontal axis, then it's positive and if it's below, then it's negative. If the two

    vertical lines are located in the same place, then the area is zero.

    One way to determine this area is to calculate the definite integral of the function. This integral is the

    same as the indefinite integral except it results in a specific value rather than a generalized function. For

    this kind of integral, the constant of integration is omitted, the two values of the variable defining the

    vertical lines are substituted and the results are subtracted.

    The two values defining the vertical lines are called limits of integration. The larger value is called

    the upper limit and the smaller is called the lower limit. The result using the lower limit is subtracted

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    from the one using the upper limit. The subset of the domain on the horizontal axis between the two

    vertical lines is called the interval.

    EXAMPLE: THE AREA UNDER A CUBIC

    If f(x) = x3, then what is the area enclosed by the graph of the function, the horizontal axis,

    and vertical lines at x = 1 and x = 2? (See Figure 0.7.)

    Figure 0.7

    Using the shortcut for polynomials, the indefinite integral of the function (omitting the

    constant of integration) is . Substituting and .

    The area is the difference: .

    NOW YOU TRY IT

    What is the area enclosed by the graph of , the horizontal axis, and vertical

    lines at x = 1 and x = 2?

    (A) 1

    (B)

    (C) 5

    (D)

    (E) 10

    To determine the average value of a function over some interval, calculate the definite integral over the

    interval and divide by the length of the interval.

    The average value of the function can be thought of as the average height of the graph, measured with

    respect to the horizontal axis. This method finds the average height by calculating the enclosed area and

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    dividing by the length of the interval. Since area is length multiplied by height, dividing the area by the

    length of the interval gives the average height.

    EXAMPLE: THE AVERAGE VALUE OF A CUBIC

    Determine the average value of f(x) = x3 over the interval 1 < x < 2.

    Using the shortcut for polynomials, the indefinite integral of the function (omitting the

    constant of integration) is . Substituting and .

    The area is the difference: . The length of the interval is 2 - 1 = 1. The

    average value of the function over the interval is then .

    NOW YOU TRY IT

    What is the average value of over 1 < x < 2?

    (A) 1

    (B)

    (C) 5

    (D)

    (E) 10