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Calculus Volume 1

Key Concepts

Calculus Volume 1Key Concepts
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  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

1.1 Review of Functions

  • A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
  • If no domain is stated for a function y=f(x),y=f(x), the domain is considered to be the set of all real numbers xx for which the function is defined.
  • When sketching the graph of a function f,f, each vertical line may intersect the graph, at most, once.
  • A function may have any number of zeros, but it has, at most, one y-intercept.
  • To define the composition gf,gf, the range of ff must be contained in the domain of g.g.
  • Even functions are symmetric about the yy-axis whereas odd functions are symmetric about the origin.

1.2 Basic Classes of Functions

  • The power function f(x)=xnf(x)=xn is an even function if nn is even and n0,n0, and it is an odd function if nn is odd.
  • The root function f(x)=x1/nf(x)=x1/n has the domain [0,)[0,) if nn is even and the domain (−∞,)(−∞,) if nn is odd. If nn is odd, then f(x)=x1/nf(x)=x1/n is an odd function.
  • The domain of the rational function f(x)=p(x)/q(x),f(x)=p(x)/q(x), where p(x)p(x) and q(x)q(x) are polynomial functions, is the set of xx such that q(x)0.q(x)0.
  • Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
  • A polynomial function ff with degree n1n1 satisfies f(x)±f(x)± as x±.x±. The sign of the output as xx depends on the sign of the leading coefficient only and on whether nn is even or odd.
  • Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the xx- and yy-axes are examples of transformations of functions.

1.3 Trigonometric Functions

  • Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180°180° has a radian measure of ππ rad.
  • For acute angles θ,θ, the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is θ.θ.
  • For a general angle θ,θ, let (x,y)(x,y) be a point on a circle of radius rr corresponding to this angle θ.θ. The trigonometric functions can be written as ratios involving x,y,x,y, and r.r.
  • The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period 2π.2π. The tangent and cotangent functions have period π.π.

1.4 Inverse Functions

  • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
  • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
  • For a function ff and its inverse f−1,f(f−1(x))=xf−1,f(f−1(x))=x for all xx in the domain of f−1f−1 and f−1(f(x))=xf−1(f(x))=x for all xx in the domain of f.f.
  • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
  • The graph of a function ff and its inverse f−1f−1 are symmetric about the line y=x.y=x.

1.5 Exponential and Logarithmic Functions

  • The exponential function y=bxy=bx is increasing if b>1b>1 and decreasing if 0<b<1.0<b<1. Its domain is (,)(,) and its range is (0,).(0,).
  • The logarithmic function y=logb(x)y=logb(x) is the inverse of y=bx.y=bx. Its domain is (0,)(0,) and its range is (,).(,).
  • The natural exponential function is y=exy=ex and the natural logarithmic function is y=lnx=logex.y=lnx=logex.
  • Given an exponential function or logarithmic function in base a,a, we can make a change of base to convert this function to any base b>0,b1.b>0,b1. We typically convert to base e.e.
  • The hyperbolic functions involve combinations of the exponential functions exex and ex.ex. As a result, the inverse hyperbolic functions involve the natural logarithm.
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