Sturr, Fall 2014
Math 1C Notes on Exam #1
Test Date: Thursday, September 25
Format: You will have the full class time for the exam. The test will be given in two parts.
Part I: No notes or calculator. This will cover hand computing of partial derivatives (possibly with the chain
rule) and implicit differentiation.
Part II: This will cover all other topics. A calculator will be needed and you may use one 8½ x 11 inch sheet
of prepared notes, written on one side only. Notes may be typed or handwritten. They must, however, be
your own creation. Do not use photocopies of textbooks or copies of another person’s notes.
Topics: The exam will cover chapter 14. Major topics are listed below for your convenience. Note that this
list may not be complete. Any mathematics discussed in class or found in the homework sets can appear on
the exam.
1.
Introduction to Multivariable Functions (section 14.1) Be able to:
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2.
Limits and Continuity (section 14.2) Be able to:
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3.
Apply the chain rule in its various forms to find derivatives
Use partial derivatives to perform implicit differentiation
Solve application problems involving rate of change and the chain rule
Directional Derivatives, Tangents and Normals (sections 14.4 and 14.6). Be able to:
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7.
Find the equation of a tangent plane to f(x,y) at a given point
Find the linear approximation of f(x,y) or f(x,y,z) near a point
Chain Rule and Implicit Differentiation (section 14.5) Be able to:
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6.
Calculate partial derivatives (first, second or higher) by hand.
Estimate partial derivative values from a table
Analyze a contour plot to estimate the sign of first and second partials
Solve and interpret application problems that use partial derivatives as rates of change
Linear Approximations and Differentials (section 14.4) Be able to:
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5.
Recognize common continuous multivariable functions and evaluate limits for these functions
Use different paths to show that a limit does not exist at a given point for certain functions
Partial Derivatives (section 14.3). Be able to:
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4.
Find (and possibly sketch) the domain of a given multivariable function
Match a given multivariable function with a sketch of a graph
Use tables to interpret the meaning of application problems that use multivariable functions
Sketch and/or interpret level curves and contour plots for a given two-variable function
Sketch and/or interpret level surfaces for a given three-variable function
Find gradient vectors and directional derivatives for functions of two or three variables
Find the equation of a tangent plane or normal line to a surface defined as F(x,y,z) = k
Solve and interpret application problems involving directional derivatives and gradient vectors
(e.g. find direction of steepest ascent, tangent line on a curve of intersection and other
problems similar to hw in section 14.6 and quizzes)
Maximum and Minimum Values (section 14.7-8). Be able to:
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Find local maximums, local minimums and saddle points by finding critical points and applying
the 2nd Derivatives Test
Find absolute maximum and minimum values for a function over a closed bounded set
Solve application problems that require a maximum or minimum value (similar to 14. 7 hw)
Use LaGrange multipliers to solve optimization problems with a constraint equation.
Sample Exam: There is a sample exam from a previous semester (with key) posted on my website in the
handout folder for Math 1C.