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Final exam and solutions
Final exam information:
Date: Wednesday 12/12, 3:30 - 5:30
Location: Ell Hall 411
There will be a review session on Monday 12/10, 9:15-10:20, in Ryder Hall 245 (usual classroom).
The exam is cumulative, and covers material roughly equally from the whole semester.
The exam will have 11 problems. It is pretty long, so please do some timed practicing.
You can use a calculator and a one-sided sheet of notes (handwritten or typed).
Here is the final exam from Fall 2017 (solutions), and Spring 2017 (solutions).
Here is the final exam from Fall 2016 (solutions), and Fall 2015 (solutions).
Here is the (hard) practice exam I handed out in class, and here are the solutions.
Important topics:
Vector geometry
Use the cross product and dot product.
Find the equation of a plane given three points or two vectors on the plane.
Find the equation of a line through two points.
Related to the gradient
Given a function f(x,y) of two variables, find the tangent plane to its graph at a given point (x,y,f(x,y)).
Given a function f(x,y), find the tangent line to a level curve at a given point (x,y).
Given a function f(x,y,z), find the tangent plane to a level surface at a given point (x,y,z).
Given a function f(x,y) or f(x,y,z), find the direction of steepest ascent at a given point.
Given a function f(x,y) or f(x,y,z), find the directional derivative at a given point, in a given direction.
Given a function f(x,y) or f(x,y,z), write down the linear approximation at a given point.
Find critical points of a function f(x,y), and classify each critical point as a maximum, minimum, saddle point, or unknown using the second derivative test.
Use Lagrange multipliers (one constraint only) to find critical points of a function subject to a constraint.
Find the absolute minimum and maximum of a function on a region with boundary, like a disc.
Multiple integrals
Set up double integrals over various regions in the plane, including in polar coordinates, remembering dA = r dr dθ. (That is, find the volume above a region and below the graph of a function.) Convert between rectangular and polar coordinates when convenient.
Switch orders of integration.
Set up triple integrals over various regions in 3-space, including in cylindrical and spherical coordinates, remembering dV = r dz dr dθ = ρ^2 sin(ϕ) dρ dϕ dθ. Convert between these coordinates when convenient.
Given an integral, visualize the region of integration.
Parametrize surfaces, and find the surface area.
Vector Calculus
2 dimensions:
Given a vector field, decide if it is conservative. If so, find a potential function.
Set up line integrals of vector fields over various curves.
Use a potential function to compute line integrals, using the fundamental theorem of line integrals.
Use Green's Theorem to convert line integrals to double integrals and vice versa, when the curve encloses an area.
3 dimensions:
Given a vector field, decide if it is conservative. If so, find a potential function.
Calculate the curl and divergence of vector fields.
Set up line integrals of vector fields over various curves, and surface integrals of vector fields (i.e. flux integrals) over various surfaces.
Be able to use the three methods for computing line integrals:
Fundamental theorem of line integrals, if F is conservative.
Directly, using a parametrization r(t).
Stokes' theorem, if C is a closed loop (the boundary of a surface).
Be able to use the three methods for computing surface/flux integrals:
Stokes' Theorem, if F is the curl of some other vector field.
Directly, using a parametrization r(u,v).
Divergence theorem, if S encloses a volume.
Class time and location
Class time and location
Monday, Wednesday, Thursday, 9:15-10:20AM, Ryder Hall 245
Recitation sessions (Brian Hepler):
Recitation sessions (Brian Hepler):
Tuesday 1:35-3:15PM, West Village G 104
Office hours:
Office hours:
Mondays 10:35-12:15, Wednesdays 1:30-2:50 (Lake Hall 463)
First Midterm Exam: Wednesday, 10/3
Material covered (the emphasis will be on the later topics):
Vectors, dot product, cross product
Equations of lines and planes
Parametric curves
Graphing functions of two variables, level curves
Partial derivatives, equation of the tangent plane, linear approximation
Gradient vector, directional derivative
Implicit differentiation
This material will all be covered by 9/27.
Here is a practice midterm.
Doing homework #4 will also help you prepare for the midterm.
Midterm 2: Wednesday, 11/7
Practice exam. Solutions are here.
Here is a long file of extra problems on all material covered. It is roughly in the order covered. (Some problems will not make sense or will not be possible with what we've covered, but most of them will.) Solutions to odd-numbered problems are here.
Material covered:
Local extrema, critical points, the second derivative test
Lagrange multipliers
Double integrals, Cartesian and polar coordinates
Triple integrals, Cartesian, cylindrical, and spherical coordinates
Parametrized surfaces and surface area.
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