Calculus Basics

download Calculus Basics

of 15

  • date post

    08-Nov-2015
  • Category

    Documents

  • view

    10
  • download

    1

Embed Size (px)

description

calculus

Transcript of Calculus Basics

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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)

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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:

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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:

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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 .

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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).

  • Copyright 2011 John Wiley & Sons, Inc. All rights reserved.

    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