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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

3.1 Defining the Derivative

  • The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h.h.
  • The derivative of a function f(x)f(x) at a value aa is found using either of the definitions for the slope of the tangent line.
  • Velocity is the rate of change of position. As such, the velocity v(t)v(t) at time tt is the derivative of the position s(t)s(t) at time t.t. Average velocity is given by
    vave=s(t)s(a)ta.vave=s(t)s(a)ta.

    Instantaneous velocity is given by
    v(a)=s(a)=limtas(t)s(a)ta.v(a)=s(a)=limtas(t)s(a)ta.
  • We may estimate a derivative by using a table of values.

3.2 The Derivative as a Function

  • The derivative of a function f(x)f(x) is the function whose value at xx is f(x).f(x).
  • The graph of a derivative of a function f(x)f(x) is related to the graph of f(x).f(x). Where f(x)f(x) has a tangent line with positive slope, f(x)>0.f(x)>0. Where f(x)f(x) has a tangent line with negative slope, f(x)<0.f(x)<0. Where f(x)f(x) has a horizontal tangent line, f(x)=0.f(x)=0.
  • If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
  • Higher-order derivatives are derivatives of derivatives, from the second derivative to the nthnth derivative.

3.3 Differentiation Rules

  • The derivative of a constant function is zero.
  • The derivative of a power function is a function in which the power on xx becomes the coefficient of the term and the power on xx in the derivative decreases by 1.
  • The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative.
  • The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
  • The derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g.
  • The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
  • The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
  • We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.

3.4 Derivatives as Rates of Change

  • Using f(a+h)f(a)+f(a)h,f(a+h)f(a)+f(a)h, it is possible to estimate f(a+h)f(a+h) given f(a)f(a) and f(a).f(a).
  • The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.
  • The population growth rate and the present population can be used to predict the size of a future population.
  • Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.

3.5 Derivatives of Trigonometric Functions

  • We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. The results are
    ddxsinx=cosxddxcosx=sinx.ddxsinx=cosxddxcosx=sinx.
  • With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

3.6 The Chain Rule

  • The chain rule allows us to differentiate compositions of two or more functions. It states that for h(x)=f(g(x)),h(x)=f(g(x)),
    h(x)=f(g(x))g(x).h(x)=f(g(x))g(x).

    In Leibniz’s notation this rule takes the form
    dydx=dydu·dudx.dydx=dydu·dudx.
  • We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.
  • The chain rule combines with the power rule to form a new rule:
    Ifh(x)=(g(x))n,thenh(x)=n(g(x))n1g(x).Ifh(x)=(g(x))n,thenh(x)=n(g(x))n1g(x).
  • When applied to the composition of three functions, the chain rule can be expressed as follows: If h(x)=f(g(k(x))),h(x)=f(g(k(x))), then h(x)=f(g(k(x))g(k(x))k(x).h(x)=f(g(k(x))g(k(x))k(x).

3.7 Derivatives of Inverse Functions

  • The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
  • We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

3.8 Implicit Differentiation

  • We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
  • By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.

3.9 Derivatives of Exponential and Logarithmic Functions

  • On the basis of the assumption that the exponential function y=bx,b>0y=bx,b>0 is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
  • We can use a formula to find the derivative of y=lnx,y=lnx, and the relationship logbx=lnxlnblogbx=lnxlnb allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
  • Logarithmic differentiation allows us to differentiate functions of the form y=g(x)f(x)y=g(x)f(x) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
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