Outline/Study Guide for First-Year Calculus

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Study Guide: Precalculus :: What is a Derivative? :: Differentiation Techniques :: Applications of Derivatives :: What is an Integral? :: Integration Techniques :: Applications of Integrals

Please note: this study guide provides only as a brief review of important topics in first-year calculus and is intended primarily as a study aid for exams. You will not be able to learn calculus from this page alone! For more details, please refer to these topics' individual pages.

IMPORTANT PRECALCULUS TOPICS: (top)

  1. Lines
    1. slope (m) = ; also = tangent of angle of inclination of the line

    2. parallel lines~ slopes are equal:

    3. perpendicular lines~ slopes are negative reciprocals:

    4. distance formula: (from formation of right triangle)

  2. Functions & Graphs
    1. def. function~ a relation that assigns one unique range element for each domain element
      -> passes vertical line test

    2. def. one-to-one function~ a special kind of function where each range comes from a unique domain element
      -> the relation and its inverse are both functions
      -> passes horizontal and vertical line tests

    3. Even Functions
      &
      Odd Functions

      f(-x)=f(x)

      symmetric with respect to the y-axis

      f(-x)=-f(x)

      symmetric with respect to the origin


    4. Arithmetic with Functions (simultaneous)
      ex. Let (domain ) and (domain )
      Then . The new domain is ; the new restrictions are the combined restrictions of the individual functions.

      Note that division can create additional restrictions:
      with domain . The point x=0 has been removed from the domain because it would make the denominator of the new function 0.

    5. Composition of Functions (sequential)

      order matters:

      ex. Let and (note the restriction )
      To find f(g(x)), plug the equation for g(x) in for every x in the f(x) equation:

      . Note that the restriction from the "inside" function must be kept.


    6. Absolute Value
      1. Definitions
        1.    -> absolute value is always positive
        2. Also,
      2. Properties

        3. The inequality becomes an equality if x & y are either both positive or both negative
      3. Notation Using Inequalities
      4. Solving Inequalities w/Absolute Values
        Separate at places where components = 0 and solve separately.

        5. ex.
        First, find where each component is positive and negative.


        Then solve the three inequalities , , and .



    7. Function Inverses
      1. inverse functions undo each other:
      2. finding inverses algebraically:
        1. switch x's and y's, then...
        2. solve for y
      3. finding inverses graphically:
        1. flip graph over the line y=x
        2. this switches special points (ex. intercepts, max./mins.)

  3. Function Families
    1. Polynomials
      To graph polynomials...
      1. determine end behavior by degree of polynomial
        1. if odd, end behavior in opposite directions
        2. if even, end behavior in same direction
        3. which is which determined by sign of the leading coefficient
      2. find y-intercept by evaluating f(0)
      3. use factored form to find x-intercepts and determine behavior at roots
        1. if root has even multiplicity, bounces off axis like y=x2
        2. if root has odd multiplicity, passes through axis like y=x3

    2. Rationals
      To graph rationals...
      1. determine end behavior
        1. if degreenumerator > degreedenominator by 1 degree, approaches positive or negative infinity
        2. if degreedenominator > degreenumerator, approaches 0
        3. if degreenumerator = degreedenominator, approaches quotient of leading coefficients
        4. degreenumerator > degreedenominator, there will be an oblique asymptote
          1. find with synthetic division or long division

    3. Exponential Functions
      1. have the form
      2. definitions
        1. 0 exponent
        2. negative exponent
        3. fractional exponent
      3. laws
        3. power rule
      4. Compound Interest- Discrete
        1. d=deposit; r=interest rate; n=# compoundings/year; t=# years
      5. Compound Interest- Continuous

    4. Logarithms
      1. Logs can be rewritten as exponentials:
      2. Logarithms and exponentials are inverses:
      3. Properties
        1. Product Rule:
        2. Quotient Rule:
        3. Power Rule:
      4. Change of base: (where c is any desired new base)

  4. Parametric Equations
    1. parametric equations separate x & y into functions of a 3rd variable:
         x=f(t)  <-- horizontal position
         y=g(t) <-- vertical position
    2. to solve/graph:
      1. make a t|x|y chart
      2. eliminate the parameter, or
      3. use the extent~ range of possible values for x & y, forming a box into which the function must fit

WHAT IS A DERIVATIVE? (top)

  1. Instantaneous Rate of Change
    : To understand this, we will need greater knowledge of limits.

  2. Limits

    1. RHL must = LHL

    2. Properties
      1. for a constant k
      2. for a polynomial f(x)

        3. For the next properties, let .

      3. Sum Rule:
      4. Difference Rule:
      5. Product Rule:
      6. Quotient Rule:
      7. Power Rule:

    3. Limits of Trig Functions
      2. and
      3. ; ex.

  3. Limits and Continuity
    1. def. A function is continuous @ x=c if and only if:
      1. f(c) is defined
      2. exists (RHL = LHL)
      3. --> embodies other two restrictions

    2. Types of Discontinuity
      1. Point Discontinuity
      2. Jump Discontinuity
      3. Infinite Discontinuity (Asymptote)

  4. Definition of Derivative


     
     
     
     
    which is the definition of derivative.

  5. Tangent Lines & Normal Lines
    1. To find the tangent line if f(x) at x=a,
      1. Find the slope of the tangent line @ x=a using the definition of derivative (or differentiation rules)
      2. Evaluate f(a)
      3. Find equation of the tangent line using

    2. Normal Lines
      1. def. line perpendicular to the tangent line that passes through the point of tangency
      2. Find using y = mx + b as with tangent line, but with

  6. Derivatives Failing to Exist
       The derivative of f(x) does not exist when you have a...
    1. Corner
      ex.

      def. corner~ the left-hand derivative does not equal the right-hand derivative.

      Here, LHD ≠ RHD
      because but


    2. Cusp
      ex.

      def. cusp~ extreme case of corner where LHD ≠ RHD and LHD or RHD = .

      Here, LHD ≠ RHD
      because but


    3. Vertical Tangent
      ex.

      and , so


    4. Discontinuity
      point, jump, or infinite discontinuities all result in an undefined derivative because f(c)=DNE
  7. Numerical Derivatives w/the Calculator
    On a TI-83+, use nDeriv(f(x), x, c) to calculate the derivative of f(x) at x=c.
  8. Intermediate Value Theorems
    1. For Continuous Functions:

      If a function y=f(x) is continuous on [a, b], then it takes on every y-value between f(a) and f(b).

    2. For Derivatives:
      3. If a & b are any two points on an interval on which f is differentiable (and therefore also continuous), then f' takes on every value between f'(a) and f'(b).

TECHNIQUES OF DIFFERENTIATION: (top)

  1. Basic Rules
    1. Derivative of a Constant:

    2. Power Rule:

    3. Constant Multiple Rule:

    4. Sum/Difference Rule:

    5. Product Rule:

    6. Quotient Rule:

    7. Chain Rule:

  2. Derivatives of Trig Functions
    3. 1. 4.
    2. 5.
    3. 6.

  3. Chain Rule w/Parametrics
    1. 1st Derivative
      Have and , want .
      From , we can only get ; From , we can only get
      But since , we have that .
    2. 2nd Derivative

  4. Implicit Differentiation
    1. def. implicit form~ x's and y's are mixed together, rather than on separate sides of the equation

    2. Finding Derivatives of Equations in Implicit Form
      1. To find , take (the derivative with respect to x) of each side of the equation, then simplify to find .
      2. NOTE: By the chain rule,
      3. ex.

  5. Differentiating Inverse Trig. Functions
    1. Derivatives





    		2. Reasoning
    1. rewrite to remove the inverse
    2. find using implicit differentiation
    3. Use right triangles and the Pythagoren theorem to rewrite y's in the derivative in terms of x

    ex.
    First, rewrite to remove the inverse: .

    Next, differentiate implicitly:

    We must now replace "cos y" with some term involving x.
    Since , the triangle at left is formed. The bottom leg is found using the Pythagorean theorem. Using this triangle, we can see that .

    Substituting this into the equation for , we find that .

     

    3. Useful Tidbit

    These identities can be used to rewrite problems and make them easier to solve.

  6. Derivatives w/Exponents & Logarithms
    1. Derivatives of Exponents

    2. Derivatives of Logarithms

    3. Logarithmic Differentiation
      1. def. taking the natural logarithm of both sides of an equation and using properties of logarithms to simplify before and during differentiation.
      2. ex.



APPLICATIONS OF DERIVATIVES: (top)

  1. Maxima & Minima
    1. Existence
      1. If a function is continuous and has endpoints, there must be an absolute maximum and an absolute minimum.
      2. Extrema can occur at:
        1. horizontal tangents (y'=0)
        2. corners/cusps (y' DNE)
        3. endpoints
          NOTE: the term "critical points" refers to a Cartesian coordinate (x, y); "critical values" are values of x and given in the form x=c.

    2. 2 conditions must be met to have a local extremum in the interior:
      1. y'=0 or y' DNE
      2. slope changes from one side of critical value to the other
        2 ways to test:
        1. x|y' chart:
          + 0 – => maximum; – 0 + => minimum
        2. 2nd derivative test:
          y''<0 => max. b/c concave down; y''>0 => min. b/c concave up

  2. Geometric Significance of y'
    1. y' = change in y compared to change in x

    2. y' >0 means as x gets bigger, y gets bigger => POSITIVE SLOPE!

    3. y' <0 means as x gets bigger, y gets smaller = NEGATIVE SLOPE!
  3. Geometric Significance of y''
    1. y'' = change in y' (slope) as compared to change in x

    2. y'' > 0 -> as x gets bigger, y' gets bigger (more positive)
      => CONCAVE UP

    3. y'' <0 -> as x gets bigger y' gets smaller (more negative)
      => CONCAVE DOWN

    4. Points of Inflection
      1. def. a point on the graph where the curve changes in concavity (from concave up to concave down or vice-versa)
      2. 2 conditions:
        1. y''=0 or y'' DNE
          Note: when y''=0 or y'' DNE, the curve is neither concave up nor concave down at that point
        2. change in concavity: y''>0 -> y''<0 or vice-versa

  4. Mean Value Theorem

    If y=f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point (c, f(c)), a<c<b, such that ; that is, such that .

  5. Using y' and y'' to Graph Functions
    1. Precalculus
      1. end behavior
      2. x & y-intercepts, if easily found
        1. whether graph passes through or bounces off axis at roots
      3. asymptotes

    2. First Derivative, y'
      1. find critical values (y'=0 or y' DNE)
      2. make an x|y' chart to analyze
        1. find regions where function is increasing/decreasing
        2. find maxima and minima

    3. Second Derivative, y''
      1. find potential points of inflection
        1. make x|y'' chart to analyze
          1. find regions where funciton is concave up/concave down
          2. find points of inflection

  6. Max./Min. Word Problems
    Steps to solve:
    1. visualize problem & assign variables
    2. write equation to maximize/minimize
    3. list constraints
    4. rewrite equation in 2 variables
    5. differentiate to find critical values
    6. check to see if critical values are max's/mins.
    7. answer the questions (in words)

  7. Using Tangent Lines as Approximations of Curves
    1. Newton's Method for Approxmiating Roots (where f(x)=0)
      1. Use the Intermediate Value Theorem to choose an approxmiate for the root.
        1. use an x|y chart to find a change in sign (+ -> – OR – -> +)
        2. chose a vlaue in betwen the x-values that have y-values of opposite signs
      2. Find the y-value of this point to give an ordered pair that can be used as a point of tangency.
      3. Write the equation of the tangent line through that point.
        1. Find slope with 1st derivative
        2. Use y=mx+b to find the tangent line
      4. Since we don't know where the curve crosses the x-axis, find where the tangent line crosses the x-axis.
        -> result: approximate root
      5. Repeat steps b) through d) until the root is found to the desired level of precision

        NOTE: if f(c)=0 is found for a particular value of x=c, then c is the exact root of hthe function.

    2. Estimating Ugly Functions at Ugly Numbers
      1. Identify the ugly function and the ugly number
      2. find a nice number close to the ugly number and write the equation of hte tangent line at that nice number
        -> called "linearization of the function"
      3. substitute the ugly number into this nice, linear function to approximate the ugly function at the ugly number.


    3. Using Differentials to Estimate Change
      1. Definitions
        =derivative; dy = change in y on tangent line differentials
        dx = change in x on tangent line
      2. Differential Equation
        because
        dy is an approximate of
        i. if curve is concave up, dy is an underestimate
        ii. if curve is concave down, dy is an overestimate
        e = error, the amount by which dy is too big/small


  8. Finding Limits of the Indeterminate Form
    1. Limits of the Indeterminate Form:
      or ,
      as all of these forms can be written as 0/0.

    2. L'Hôpital's Rule
      1. using rate of change to evaluate limits
      2. MUST have limit of a fraction = 0/0 to use (can't just use instead of quotient rule), then take derivative of top and bottom
      3. ex.
      4. NOTE: may not always work with square roots

  9. Related Rate Problems
    1. For answering questions that ask "how fast is something changing";
      differentiate with respect to time

    2. Procedure:
      1. draw a picture and label; choose variables
      2. write an equation that relates the variables
      3. if more than 2 variables, rewrite in 2 variables, if possible
      4. implicitly differentiate both sides with respect to time (t)
      5. substitute known rate information and solve
      6. answer the question (in words)

WHAT IS AN INTEGRAL?: (top)

  1. Definitions
    1. indefinite integration~ method of retrieving function from wihch a given derivative came

    2. definite integration~ describing how instantaneous changes can accumulate over an interval
      1. answer is a number
      2. finite answer to an infinite summation

  2. Estimating Change w/Finite Sums
    1. Riemann Sums
      1. LRAM (left-endpoint rectangle approximation method)

        where xi=left x-value of the ith partition
        f(xi)= y-value at xi (=height of rectangle)
        ∆x = width of partition


      2. RRAM (right-endpoint rectangle approximation method)

        where xi=right x-value of the ith partition
        f(xi)= y-value at xi (=height of rectangle)
        ∆x = width of partition

      3. MRAM (midpoint rectangle approximation method)

        where xi=right x-value of the ith partition
        f(xi)= y-value at xi (=height of rectangle)
        ∆x = width of partition

    2. Trapezoidal Rule

    3. Simpson's rule--estimate area with a parabolic arch
      area under a parabolic arch is
      so with a coefficient pattern of 1..4..2..4..2.....4..2..1

  3. Definite Integrals
    1. To find the exact answer, take the limit of a Riemann sum as n -> .

      (Note: I=A if f(x)>0)
      Note: it doesn't matter which Riemann sum is used, as they are equivalent in the limit as n -> .

    2. Existence of Definite Integrals
      1. If a function f is continuous on [a, b], then its definite integral over [a, b] exists.
      2. If the function is not continuous, it may or may not be integrable.
        1. if there's just point discontinuity, the function is still integrable
        2. infinite discontinuities are sometimes integrable.

  4. The Fundamental Theorem of Calculus
    -> establishes the inverse relationship between integration and differentiation
    1. If f is continuous on [a, b], then the function has a derivative at every point , and .

    2. If f is continuous at every point on [a, b] and if F is the anti-derivative of f on [a, b], then .

INTEGRATION TECHNIQUES: (top)

  1. Basic Rules






  2. Integration by U-Substitution ("Chunking")
    1. Uses
      1. Undoes a chain rule problem
      2. A function and its derivative (or a linear multiple thereof) must be present for use

    2. Steps
      1. Chunk it!
      2. Integrate the Chunk
      3. Unchunk it!

  3. 17 Integrals
    1. 7. 13.
    2. 8. 14.
    3. 9. 15.
    4. 10. 16.
    5. 11. 17.
    6. 12.  

  4. Integrating Trig Functions
    1. Odd Powers of Sine/Cosine
      -> Manipulate to use u-substitution
      1. Separate one of the sines/cosines, leaving an even power
      2. Use the Pythagorean identity to rewrite the even powers as or
      3. Integrate with u-substitution

      ex.


    2. Even Powers of Sine/Cosine
      -> Use half-angle formulas
      The right-hand sides of each bottom equation are now easily integrable.

  5. Integration by Parts
    1. Formula:

    2. Usage
      1. when you have entirely unrelated functions, or
      2. when you have afunction you don't know how to integrate (which you must let be u) but do know how to differentiate

    3. Choosing u and dv
      1. u must be differentiable, and, ideally, should get simpler when differentiated
      2. everything that's not u must be dv
        1. will need to know how to integrate dv
      3. "LIPET" = log, inverse, polynomial, exponential, trig function

    4. Example

APPLICATIONS OF INTEGRATION: (top)

  1. Principle
    1. an integral is an infinite summation

    2. we can sum things besides areas

  2. Integral as Net Change
    1. area under a velocity curve = distance
      1. total D =
      2. net D = = displacement

  3. Area Between 2 Curves
    1. A=

    2. The same function must always be on top, or else one must partition

      ex. &

      PARTITION!




    3. Can do a dy problem when x=f(y) or to avoid dx-partitioning
      1. do right funciton minus left function
      2. again, must partition if the functions switch sides

  4. Finding Volumes of 3-D Figures
    1. Volume by Slicing
      Procedure:
      1. Draw a picture & sketch a sample cross-section
      2. Decide if the problem uses dx or dy
      3. Find limits of integration
      4. Find area of each cross-section
      5. Integrate
        or




    2. Volume by Shells
      Follow same procedure, but each cross-section is an open cylinder and


      where y1 is the line of rotation
      and y is an arbitrary y-value


      or


       



      where x1 is the line of rotation
      and x is an arbitrary x-value

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This page last updated 7 August, 2008 7:28 PM