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MTH 124 Exam - 2

Concept List

From Section 3.4:
 

  1. Finding A.R.o.C of a function over the interval

    • [a,b] by using [f(b) - f(a)] / (b-a)

    • [a,a+h] by using [f(a+h) - f(a)] / h
       

  2. Know that the A.R.o.C over an interval [a,b] represent the slope of the secant line of the function f(x) over the interval [a,b] - Secant line of f(x) over [a,b] is the line that passes through the two points (a,f(a)) and (b,f(b))
     

  3. Units of A.R.o.C of a function f(x)  is = units of f / units of x

From Section 3.5:
 

  1. Instantaneous rate of change of a function f(x) at a point x = a is denoted by f '(a) and it is the:

    1. Slope of the tangent line to the graph of f(x) at x = a

    2. Equivalently the limit: lim_{h --> 0} [f(a+h) - f(a)] / h
       

  2. Approximating f '(a):

    1. Graphically: Draw the tangent line for the function f(x) at x = a and find (estimate) its slo

    2. Numerically: Approximating the limit: lim_{x --> a} [f(a+h) - f(a)]/h

      1. By approximating the limit for values of h closer to 0

      2. By using the Balance Difference Quotient : f '(a) ≈ [f(a+h) - f(a-h)] / [2h] for small values of h
        (h = 0.001 or h = 0.0001)
        Out of these two methods, I prefer using the Balance Difference Quotient.
         

  3. Tangent line to f(x) at x = a is the line that touches f(a) at x =a, but does not passes through (a,f(a)). Since f '(a) is the slope of the tangent line to the graph of f(x) at x = a, we have that the tangent line

    1. Has slope f '(a)

    2. Touches (a, f(a))
      So the       equation of the tangent line to f(x) at x = a is: y = f '(a)(x - a) + f(a)
       

From Section 3.6:
 

  1. Finding the derivative from the limit definition:
    All these limits are 0/0 form (see section 3.3).
    So the procedure is the same. Factor numerator and denominator. Cancel the h in the denominator. Plug h =0.

From Chapter 4:

  1. Chapter 4 gives us all the rules to differentiate functions. The main rule in chapter 4 is the Chain Rule. We almost always use Chain Rule in derivatives. Together with the Chain Rule, we have
    1. Sum/Difference rule [f(x) ± g(x)]' = f'(x) ± g'(x)
    2. Constant Multiples Rule [cf(x)]' = c'f(x)
    3. Product Rule [f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)
    4. Quotient Rule  [f(x) / g(x)]' = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
      I like to think of these results as "grammar"
  2. Then we have the "vocabulary" for derivatives:
    1. Power Rule: (x^n)' =nx^(n-1)
    2. (c)' = 0
    3. (mx +b)' = m
    4. |x|' = |x|/x
    5. (e^x)' = e^x
    6. [ln x]'  =1/x
    7. (a^x)' = (ln a)*(a^x)
    8. (log_a x)' = 1/ [x (ln a)]
  3. Combining the above results with the Chain Rule, one could also come up with these new rules:
    1. [e^(f(x))]' = f'(x) [e^(f(x))]
    2. [1/f(x)]' = -f'(x) / [f(x)]^2
    3. [ln f(x)]' = f'(x)/f(x)
  4. Other than the derivative rules, there are few other concepts (in chapter 4.2)
    1. Marginal cost and its meaning (C'(x))
    2. Marginal profit and its meaning (R'(x))
    3. Marginal revenue an its meaning (P'(x) = R'(x) - C'(x))
    4. Average cost = C(x) / x

From Section 5.1:
 

  1. Meaning of absolute/global minima/maxima.

  2. Meaning if relative/local minima/maxima.

  3. Critical point of a function f(x) are: x values where f '(x) does not exists or x values where f '(x) = 0

  4. Critical points where f '(x) does not exists are called singular critical points and critical points where f '(x0 = 0 are called stationary critical points.

  5. First derivative test for classifying stationary critical points.

  6. Understand that any kind of minima/maxima occurs on a function at

    1. Either the end point of it domain

    2. Or at a critical point

  7. How to find and classify all types of extrema
     

From Section 5.2:
 

  1. For a function f(x), x values where
    f ''(x) is positive  = f '(x) is increasing = Graph of f(x) is above its tangent lines
    In these places f(x) is said tot be concave up
  2. For a function f(x), x values where
    f ''(x) is negative = f '(x) is decreasing= Graph of f(x) is below its tangent lines
    In these places f(x) is said tot be concave down
  3. Inflection points of f(x) are the x values where:
    • graph of f(x) changes concavity (goes from being above its tangent lines to being below or being below its tangent lines to being above
    • f '(x) goes from being increasing to decreasing or being decreasing to increasing
    • f ''(x) goes from being positive to negative or negative to positive

Practice

A practice exam can be found under "Resources" in WebAssign. I will also post a list of practice questions that covers all the concepts listed above. Like in the exam 1 you are encouraged to do the list of questions multiple times to get used to your calculator, type of questions, etc. Labs are also a nice way to practice for the exam.

Practice Questions

Practice Questions

Solutions

Video Solutions

Labs

Lab 3

Lab 3 - Video

Lab 4

Lab 4 - Video

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