MATH 53 SPRING 2018 MIDTERM 1 LIST OF TOPICS

NIKHIL SRIVASTAVA

The first midterm will be based on the first ten lectures of the class, except for partial differential
equations. Here is a list1 of things you should know / know how to do; if you are comfortable with
everything on this list, you should be fine.
The emphasis will be on computational problems, conceptual questions, and occasionally visu-
alization, and not on formal proofs. However, you should know all of the definitions and informal
proofs that were covered in lecture.
• Know the distance formula for Euclidean geometry in two and three dimensions,
• For each of the vector operations (length, addition, multiplication by scalars, dot product,
determinant, cross product), compute it and understand its geometric meaning.
• Algebraically manipulate expressions involving vector operations, avoiding undefined oper-
ations (e.g. the sum of a vector and a scalar).
• Know how to find the projection of one vector along another vector.
• Write the equation for a line in parametrized form, given a point on the line and a tangent
vector to the line.
• Write the equation for the line or line segment between two points.
• Write the equation of a plane in the form ax + by + cz + d = 0 given a point on the plane
and a normal vector, and vice versa.
• Find the equation of the plane through three points.
• Understand what a parametrized curve is, as opposed to just a curve.
• Understand how to write parametrized curves in space as vector-valued functions.
• Write down a formula for a parametrized curve from a description.
• Calculate the length of a parametrized curve.
• Compute the tangent line to a parametrized curve at a point.
• Compute the length of a parametrized curve in space.
• Compute the velocity and speed of a parameterized curve at a point.
• Know how to reparameterize a curve in terms of arc length.
• Know the rules of calculus for vector-valued functions, namely three versions of the product
rule and an analogue of the fundamental theorem of calculus.
• Sketch the graph of a function of two variables. Highly accurate drawings are not required.
• Sketch the level sets of a function of two or three variables. Understand how some properties
of a function are reflected in the appearance of its level sets.
• Understand the different ways that a given surface or curve might be presented (i.e., as the
set of points satisfying some equation, as the graph of a function, or as a parameterized
curve or surface), and how to switch between them.
• Plot traces (obtained by fixing one variable when a surface is given by equations) and projec-
tions (obtained by ignoring one variable when a curve is given as a parameterized equation)
as well as other visualization heuristics such as plotting tangent directions, discussed in
lecture.
• Derive the parametric equation of the trajectory of a particle given a verbal description, by
decomposing the motion into simpler motions using vectors.
1
This list is a modified version of a similar list produced by Prof. Hutchings for a previous version of this course.
1
• Know the basic limit properties and the definition of a continuous function.
• Use limit properties and continuity to compute limits when they exist.
• Prove in some cases that the limit of a function as (x, y) ? (a, b) does not exist by
considering the limits along different curves approaching (a, b).
• Know the definition of partial derivatives.
• Compute partial derivatives.
• Know Clairauts theorem.
• Compute partial derivatives of functions that are defined implicitly in basic examples.
• Determine the tangent plane to the graph of a function of two variables at a point.
• Determine the tangent plane to the level surface of a function of three variables at a point.
• Understand that a function is differentiable at a point when the graph is well approximated
by the tangent plane.
• Use linear approximation to approximate the value of functions of two and three variables.
• Perform calculations using the different versions of the chain rule in multivariable calculus,
and know where to evaluate the various functions that appear in the formulas.
• Calculate partial derivatives of functions defined implicitly in the general case, using the
total differential and the chain rule.
• Know the definition of directional derivatives, total differential, and gradient, and the rela-
tionship between them.
• Know the geometric interpretation of the gradient, namely that it is perpendicular to the
level sets and points in the direction in which the directional derivative is largest.
• Know the vector version of multivariable chain rule.
• Use the gradient to find the tangent line/plane to a level set of a function of two or three
variables.
• Know the definitions of local and global minima and maxima, and critical points.
• Know the second derivative test for functions of two variables.
• Know the statement of the Extreme Value Theorem and understand why the hypotheses of
closed, bounded, and continuous are necessary.
• Find global minima and maxima of a function on a domain by looking for critical points
(and points where the partial derivatives are not both defined) and minima and maxima on
the boundary.
• Understand why the method of Lagrange multipliers works.
• Solve constrained optimization problems using Lagrange multipliers.
• Know the significance of the Lagrange multiplier.