Math 100 ~ Single Variable Calculus I

Finding the area between two functions

• Example: 6.1 - 1.
• Example: 6.1 - 3 done with respect to x and with respect to y.
• A = ∫_a^b f(x)-g(x) dx = ∫_c^d h(y)-l(y) dy
  where f and h are the upper functions respectively.
• Remember to consider whether the integral expression will be easier to evaluate in terms of x or in terms of y.

Sigma notation

• 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.

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.

Definite integrals by substitution

• Just remember to change the limits of integration from limits on the original variable (x) to limits on the substitution variable (u).

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)

Optimization examples

• This problem is difficult because of the algebraic manipulations and because there are so many symbols, including many letters (R and H) that aren't variables.
• The challenge in this problem is in formulation. In class we used trig functions rather than the pythagorean theorem and similar triangles.
• Like the previous problem the challenge in this one is formulation. Also like the previous problem it relies on a pair of related triangles. This time we used similar triangles and the pythagorean theorem rather than trig.

Optimization

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.

4.1 - 4.3 Function 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.

Taking up the midterm

Midterm! (sitting 2)

Midterm! (sitting 1)

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).

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.
8. Examples done in class: 3.8 ~ 12, 16, 25.

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.

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"

Derivatives of trigonometric functions

• Start with the derivative of sin x using the definition of the derivative. Do the same for cos x. Now use earlier theorems to extend to tan x (i.e. apply quotient rule to sin x/cos x), cot x, sec x, csc x.

3.3-3.4 The Derivative

• Theorems: The derivative of ...
  • Sum and differences of functions
  • xn
  • Products of functions
  • Quotients of functions

3.1-3.3 The Derivative

• The idea
• Sketching derivatives
• Derivative as a function of its own
• Geometric interpretation
• Derivation from m_sec
• Notations
• Theorems: Derivative of ...
  • constant function
  • identity function
  • cf(x)
  • x²

2.6 Continuity and Limits of Trigonometric Functions

• Show lim (sin x)/x = 1
  • Squeezing theorem
  • Geometric derivation
• Show lim (1 - cos x)/x = 0 (Use result above)
• Examples

2.6 Continuity and Limits of Trigonometric Functions

• Show sin x is continuous for all x (requires alternative continuity test).
• Continuity of remaining trig fns found in terms of sin x.
• 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

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
• Removable discontinuities.
• Examples

2.3 Limits at infinity

• Limits of polynomials at infinity: Look at highest power term.
• 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.
• Algebra trick: Factoring powers of x out of radicals.

2.2 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.

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.
• Axioms and theorems

Review

• The six basic trig functions, their graphs and relationships, and their use with right-angled triangles.

Problems 11-13 from Quiz 0:

Review

Problems 1-10 from Quiz 0:

• Solving inequalities.
• Manipulating expressions involving absolute value.
• Graphs of straight lines.
• Completing the square.
• Graphing by the method of successive transformations (easier than it sounds!).

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.