About Exam 3

Bring a calculator (a graphing one is fine, but not necessary). You may prepare a page of notes ahead of time to be used during the exam as long as you make it yourself:

  • both sides of a standard (8.5×11) sheet of paper,

Some starting suggestions for your notes:

  • formulas for derivatives and antiderivatives.
  • the statement for the second derivative test (max/min/neither?) for functions of two variables.

You may re-submit up to 2 problems on the exam that you'd like to redo outside of class.

Review

Exam Content

The exam will cover primarily the topics since exam 2:

  • Maxima, minima, inflection points [4.1-3],
    Partial derivatives and finding max/mins in multivariable functions [9.3-6],
  • The fundamental theorems [5.5,FT],
    Antiderivatives and techniques of integration [7.1-7.5].
  • In particular, you need to be able to:
    1. Find critical points, local max/min points, and inflection points of functions of one variable...
      • from a graph of $f(x)$ or $f'(x)$ (Review Exer 8)
      • algebraically.
  • Estimate partial derivatives from a table or a contour diagram. Estimate locations and values of local/global max/mins from contour diagram or table. (Exer 7 and Exer 8.)

    I may ask you to evaluate the signs of $f_{xx}$ or $f_{yy}$ (but not $f_{xy}$) from a contour diagram such as this variation of Problem 26, from section 9.2:

      A patient with a blood pressure of 8 mm Hg went into shock. Now they've been in shock for four hours.
    • What is the patient's cardiac output? (What is f?)
    • As more time passes, is their cardiac output going down or up? (What is the sign of $f_x$?)
    • Is $f_{xx}$ +, -, or $\approx 0$ for this patient at 4 hours after going into shock? Practically speaking, what does that mean?
    show / hide
    If the patient's initial pressure was 8 mm, and it's been four minutes, the patient's cardiac output is 8 l/min. On the bottom contour plot, this is the point circled in green.

    If you plot cardiac output along the (green) line for 8 mm of pressure (top plot above), you can see the slope, $f_x$ is negative.

    The slope, $f_x$, is gradual and negative to the left of 4 minutes, and steeply negative to the right of 4 minutes. So, $f_{xx}=(f_x)'$, which is the rate of change of the slope is also negative. (This also means that the graph is concave down).

    This means that the patient's cardiac output is going down, and accelerating: Going down faster and faster! So you'd better do something quickly to help them!

    I might ask you to calculate $f_x$, $f_y$, $f_{xx}$, $f_{yy}$, or $f_{xy}$ given an expression for $f(x,y)$.

  • Compute 1st and 2nd order partial derivatives algebraically. Find critical points and use 2nd deriv test to determine local max/min/neither.
  • Give values of antiderivatives of f(x) from graph of f(x) and an initial value.
  • Evaluate indefinite integrals:
    • ones that do not require substitution
    • ones that do require substitution
    and definite integrals
    • ones that do not require substitution
    • ones that do require substitution