Math 100 ~ Single Variable Calculus I

Exam preparation

Work

• Work = Force × distance, when force is constant, but Work = ∫ F(x) dx when force is variable. Example: 6.7-1b).
• Problems involving springs. Hooke's law. Other (nonlinear) springs. Examples: 6.7-8.
• Problems involving pumping and lifting. Work = work done in lifting all layers of liquid required height.
  W = ∫ F × d
  W = ∫ Volume × Weight-density × d
  W = ∫ Area × thickness × Weight-density × d
  W = ∫ Area × dx × Weight-density × d
  Area and distance depend on the geometry of the problem.
  Example: 6.7-15.

Surface area of a solid of revolution

Arc length of a curve

• Derivation via Riemann sum.
• Parametric equations.
• Alternative formula for Arc length of parametric equations.

Volumes of solids using cylinders

V = 2π ∫ radius × height × thickness
  = 2π ∫ x f(x) dx

Finding volumes of solids using disks

• Derivation: From Riemann sum to integral.
• Example: Volume of a cone.
• Example: Volume of a sphere.
• Example: Volume of a square-base pyramid.

Finding the area between two functions

• A = ∫_a^b f(x)-g(x) dx = ∫_c^d h(y)-l(y) dy
  where f and h are the upper functions respectively.
• Example: Area between x² and x³ over [0,2]. Note well the importance of being sure which function is above the other.
• Suggestion: If you are unsure of what functions look like, look for their points of intersection, and then use a test point in each interval between them.
• Example: 6.1 - 3 done with respect to x and with respect to y.
• Example: 6.1-11.
• Remember to consider whether the integral expression will be easier to evaluate in terms of x or in terms of y.

The First Fundamental Theorem of Calculus

• Area as a sum of areas of rectangles: inscribed rectangles, circumscribed rectangles, rectangles of height given by an arbitrary value of x on each subinterval (via squeezing theorem).
• Riemann sum.
• The first fundamental theorem of calculus: where F is an antiderivative of f, i.e. F'(x) = f(x).
• Proving the first fundamental theorem: Appeal to Mean-Value theorem, which appeals to Rolle's theorem, which appeals to extreme value theorem.
• Example: The area beneath sin x over [0,π]

Towards the First Fundamental Theorem of Calculus

• Area as limit of sum of areas of inscribed rectangles
• Example: Area beneath y = 9 - x² via a) the limit formula, and b) the antiderivative. b) is much nicer so let's try to show it always works.

Sigma notation

But first integration by substitution review: 5.3 ~ 13, 7, 23, 27, 31.

• Sigma notation: index, lower limit, upper limit, (discrete) function.
• Just six theorems:
  1. Σ c a_k = c Σ a_k
  2. Σ a_k + b_k = Σ a_k + Σ b_k
  3. Σ a_k - b_k = Σ a_k - Σ b_k
  4. Σ k = n(n+1)/2
  5. Σ k² = n(n+1)(2n+1)/6
  6. Σ k³ = [n(n+1)/2]²
• Technique: changing the lower limit of a sum to be 1 so we can use the formulae above: add/subtract the same amount from both the upper and lower limit, and subtract/add this amount from each occurence of k in the summation expression.

Examples: 5.4 ~ 14, 16, 18, 24.

Integration!

• Derivation: Trying to find a formula for A(x), the area beneath a function f(x) and the x-axis from a given point a to a point x, reveals that A'(x) is f(x), i.e. the derivative of the area function is the function itself.
• This means the area function is the function whose derivative is f(x), or the antiderivative or integral of f(x).
• Note that there are an infinite number of antiderivatives of f(x), but they differ from each other other only in the value of a constant, i.e. they all have the pattern F(x)+C where F'(x)=f(x).
• We choose among the antiderivatives by using a boundary condition, typically that A(a) = 0 (because the area of line is 0).
.
• Integration by recognition. The simplest way to integrate a function is to recognize it as the result of a differentiation formula. E.g. if cos x is the result of differentiating a function then we know the function was sin x + C because (sin x)' = cos x.
• Integration by substitution. Some integrals are too complex to recognize. Our first strategy for dealing with them is to make a change of variable that produces a simpler integral to evaluate.
• The trick is to see one part of the expression as the derivative of another part (preferably an 'inside function' in the chain rule sense) and to set the new variable u equal to the inside function.
• Then substitute every occurence of the original variable of integration an equivalent expression using u.
• If the resulting integral expression is easier to evaluate than the original, proceed, otherwise look for another u substitution.
• Integrate the new expression in terms of u, hopefully by recognition.
• Back-substitute to replace all occurences of u with expressions in terms of x.

Newton's Method

• An iterative method for finding the zeroes (x-intercepts) of a function. Start with an initial guess x₁ and iteratively improve it using the formula: x_n+1 = x_n - f(x_n)/f '(s_n)
• Possible failures: (i) divergence, (ii) getting stuck on a local minimum.

Optimization

Example problems:

• 4.5-6 Note the similar triangles!
• 4.5-14 Formulation only.
• 4.5-42 Mucking about in details.
• 4.5-26 No numbers, so be careful to keep track of which letters are constants and which are variables.

Optimization

• Extreme value theorem: If f is continuous on [a,b] then f has a maximum and a minimum on [a,b]. (Note that this is 'just' an existence theorem, i.e. it guarantees us that something exists, but does not tell us how to find it.

• Solution procedure:

  1. Summarize given information in diagrams and equations.
  2. Identify quantity to be optimized
  3. Find an equation for the quantity to be optimized.
  4. Identify the constraint.
  5. Express the constraint as an equation (or inequality).
  6. If the equation contains more than one independent variable, look for a relationship relating the independent variables, and use it to eliminate all but one by substitution.
  7. Differentiate the equation for the dependent variable.
  8. The optimum value will occur at either,
     1. a stationary point (so set the derivative to zero and solve to find them),
     2. another critical point, e.g. where the derivative does not exist, or
     3. an endpoint of the interval of the independent variable.

• Blackboard pictures: 1, 2.

Taking up the midterm & 4.1-4.3 Curve Sketching

To sketch a function:

1. Find the y-intercept (by setting x = 0).
2. Find the x-intercept (by setting y = 0).
3. Check the function for symmetries,
   1. about y by seeing if f(x) = f(-x)
   2. about x by seeing if g(y) = g(-y)
   3. about origin by substituting -x for x and -y for y.
4. Check for vertical asymptotes. If found take limit from left and right at point where asymptote occurs.
5. Look to see if there are any ranges of values for x for which f(x) will not exist, e.g. √x DNE for negative values of x
6. Find horizontal asymptotes by taking lim as x→+∞ and as x→-∞
7. Find stationary points by finding f '(x), setting it to zero and solving.
8. Look for other critical points, i.e. points where f '(x) DNE. Test to see if they are cusps.
9. If it would be helpful, use the first derivative to identify the intervals on which the function is increasing and decreasing.
10. Find points of inflection by finding f ''(x), setting it to zero and solving.
11. If it would be helpful, use the second derivative to check the concavity of the function at stationary points, and over intervals.

Remember to keep track of the information and check its consistency using a set of axes.

Midterm!

3.9 Differentials

• Definition: dy = f '(x) dx
• Graphical interpretation (see figure 3.9.7 in the text).
• Local linear approximation: f(x + Δx) ≈ f(x) + f '(x) Δx
• Error propagation. An error Δx will result in an absolute error Δy in y which can be approximated by dy. The relative errors are x/Δx and y/Δy respectively and the percentage errors are 100(x/Δx) and 100(y/Δy).

Implicit differentiation

• Method for finding dy/dx even when we can't solve for y explicitly in terms of x (i.e. we aren't given, and can't find, y=f(x)).
• Use it to extend the power rule to cover rational exponents.

Related rates by example

1. Draw a picture (when possible), and summarize given information.
2. Identify known rate.
3. Identify unknown rate.
4. Find an equation relating the variable whose rate of change is known to the one whose rate of change is not known.
5. (Eliminate other variables if necessary, as it was in the conical water tank problem).
6. Differentiate this equation (usually with respect to time) to get an equation relating the known and unknown rates of change.
7. Solve for the unknown rate of change.

Examples done in class: 3.8 ~ Example 2, Problems 17, 25.

3.6 The Chain Rule

• dy/dx = dy/du du/dx
• [f(g(x))]'=f '(g(x))g'(x)
• "the derivative of the outside times the derivative of the inside"

3.4-3.5 Product and Quotient Rules; Derivatives of Trig Functions

• Derivation of the product rule. Remember 3 moves,
  1. Start with the definition.
  2. Add and subtract something from the top of the fraction.
  3. Factor and group terms to reveal f'(x) and g'(x).
• Derivation of the quotient rule. Same method as above.
• Derivation of (sin x)'. Write definition, use addition formula for sin(x+h).
• Derivation of (cos x)'. Same as for sin x.
• Derivation of (tan x)'. Rewrite as (sin x/cos x)' and use quotient rule and known derivatives for sin x and cos x.
• Derivation of (sec x)', (csc x)' and (cot x)'. Same process as for (tan x)'.

3.1-3.3 The Derivative

• Theorems: Derivative of ...
  • sqrt(x)
  • constant function
  • identity function
  • cf(x)
  • x²
  • Sum and differences of functions
  • xⁿ

3.1-3.3 The Derivative

• The idea
• Sketching derivatives
• Derivative as a function of its own
• Geometric interpretation
• Derivation from m_sec
• Definition of the derivative: lim f(x+h)-f(x)/h
• Derivative of f(x) = x² + 1

2.6 Continuity and Limits of Trigonometric Functions

• sin x is continuous for all x (which means cos x is too, as is tan x except where cos x = 0)
• Limits of trig expressions that are indeterminate can often be found by rewriting them in terms of two basic limits both as x -> 0:
  • lim (sin x)/x = 1
  • lim (1-cos x)/x = 0
• Show lim (sin x)/x = 1
  • Squeezing theorem
  • Geometric derivation
• Show lim (1 - cos x)/x = 0 (Use result above)
• Examples

Intermediate Value Theorem

• The Intermediate Value Theorem or How to make sense of a theorem by drawing a picture.
• Application to finding the roots of functions.

2.5 Continuity of Functions

• The idea: Continuous functions are smooth.
• A definition of continuity must eliminate functions with:
  • asymptotes
  • holes
  • gaps (regions where f is undefined)
  • vertical jumps
  • displaced values
• We can do this if we require that at the point of interest,
  • f exists
  • f's two-sided limit exists
  • the value of f and its limit are equal
• Continuity of piecewise functions.

2.2 & 2.3 Limits of algebaics functions

• Polynomials => Just substitute the limit location for x.
• Forms:
  • c/d => limit = c/d
  • 0/c => limit = 0
  • c/0 => limit magnitude is infinite, check signs of top and bottom for sign of infinity
  • 0/0 => Top and bottom share a common factor. Factor them and cancel it, then evaluate limit of new expression.
• Algebraic technique: Long division of polynomials.
• Limits of polynomials at infinity: See if highest power term is odd or even; check sign of its coefficient.
• Limits of rational expressions at infinity:
  • Form infinity/infinity => Divide top and bottom by highest power of x.
  • Entire expression inside radical => move limit inside radical.
  • Radical top or bottom => Divide by highest power of x, but prorate power inside root.
  • Difference of radicals => Consider treating as a fraction and multiplying top and bottom by complex conjugate of numerator. Graphical interpretation.

2.2 Limit axioms and theorems

• Axioms about three functions.
• Five basic theorems.
• Extension to lim kx
• Extension to lim kf(x)
• Extension to lim xⁿ
• Extension to lim axⁿ (monomial)
• Extension to lim p(x) = p(c) (polynomial)

2.1 Introduction to limits

• The idea.
• Notation.
• Finding limits by looking, i.e. sketch the function.
• 'Finding' limits by calculation.

Review: Whirlwind tour of Trigonometry

• Graphs of sec, csc and tan.
• sin, cos and tan defined on triangles.
• The unit circle.
• 30-60-90 and 45-45-90 triangles.
• Pythagorean and complement identities.
• Problems 11-13 from the review handout.
• To simplify trig expressions: express all quantities in terms of sin and cos, cancel where possible and express final result as compactly as possible.
• in f(x) = a sin bx, a determines the amplitude of the function and the frequency, e.g. if b = 2 the frequency is doubled.
• Problem 14: Remember that fractional powers correspond to roots, and negative powers to reciprocals.
• Problem 15: Remember that logs are just another way of talking about exponentiation, i.e. log_ba = k just means that b^k=a.

Review

• Problems 8-10 from the review sheet illustrating
  • conic sections,
  • completing the square, and
  • graphing by the method of successive transformations (see also pp 32-34 in the text).
• The three basic trig functions, their graphs and relationships.

Review

Problems 1-7 from the review sheet:

• Solving inequalities.
• Manipulating expressions involving absolute value.

Course introduction

• Formal course description.
• The three key ideas this semester:
  • Limits: The value a function approaches (whether or not it reaches it).
  • The derivative: the slope of the tangent line to a function, or the rate of change of a physical quantity.
  • The integral: The area beneath a function, or the sum of a quantity.