Study guide and
147 practice problems
on:
Multivariable calculus
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Study Guide
Vectors
(55 problems)
Dot product
(41 problems)
Determinants
(23 problems)
Cross product
(17 problems)
Matrices and linear equations
(20 problems)
Lines, Planes, and Curves
(13 problems)
Functions of several variables
(36 problems)
Max/min problems
(8 problems)
Polar coordinates
(4 problems)
Double integrals
(3 problems)
Vector fields
(10 problems)
Line integrals
(8 problems)
Related topics
Vectors
(55 problems)
Dot product
(41 problems)
Cross product
(17 problems)
Determinants
(23 problems)
Matrices and linear equations
(20 problems)
Lines and planes
(10 problems)
Parameterized curves
(5 problems)
Functions of several variables
(36 problems)
Surfaces in 3d
(10 problems)
Max/min problems
(8 problems)
Polar coordinates
(4 problems)
Double integrals
(3 problems)
Vector fields
(10 problems)
Line integrals
(8 problems)
Lines, Planes, and Curves
(13 problems)
Practice problems
Let $y = \frac{1}{2} x^2+\frac{1}{2}$. For what value of $c$ is $\mathbf{i} + c \mathbf{j}$ a tangent vector to $y(x)$ at $x=1$?
Solution
Find the length of the 2d vector $2 \ \bfi + 3 \ \bfj$ and the 3d vector $\langle2, 3, 4 \rangle$.
Solution
Sketch all 3d vectors whose angle with respect to the vector $\bfi$ is
$\pi/6$
$\pi/2$
$5\pi/6$
Solution
Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
Solution
Estimate the angle between the following pairs of vectors:
Solution
Sketch all the unit vectors in 2d that have an angle of $\pi/4$ with respect to the vector $\bfi$.
Solution
The vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are shown below. Using only vector addition, express one of the vectors in terms of the others.
Solution
The following parallelogram has one corner at the origin. The two neighboring corners are given by vectors $\mathbf{a}$ and $\mathbf{b}$. Express the fourth corner as a vector.
Solution
A river flows with speed $10$ m/s in the northeast direction. A particular boat can propel itself at speed $20$ m/s relative to the water. In which direction should the boat point in order to travel due west.?
Solution
Consider a pyramid with square base formed by the points $(1,1,0), (1,-1, 0), (-1, 1, 0), (-1, -1, 0),$ and $(0,0,1)$. What is the length of each edge connecting the base to the apex?
Solution
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