CALCULUS III - REVIEW FOR TEST I

The University of Akron
Department of Mathematics
Calculus III
Spring 2026, Golovaty

REVIEW PROBLEMS FOR TEST 1

Problem 1. Calculate the given quantity if ${\bf {a}}=<1,1,-2>\,$ , ${\bf {b}}=3{\bf {i}}-2{\bf {j}}+{\bf {k}}\,$ , and ${\bf {c}}=<0,1,-5>\,$ :

$\begin{displaymath}\begin{array}{ll} \mbox{\bf (a) }\bf {a\cdot({b}\times{c})}\,... ...{The vector projection of $\bf b$\ onto $\bf a$}\,. \end{array}\end{displaymath}$

Problem 2. Given the points $A(1,0,1)\,,\ B(2,3,0)\,,\ C(-1,1,4)\,,\ \mbox{and }D(0,3,2)\,,$ find the volume of the parallelepiped with adjacent edges $AB\,,\ AC\,,\ \mbox{and }AD\,.$

Problem 3. Find both parametric and symmetric equations of the line that satisfies the given conditions.

$\begin{displaymath}\begin{array}{l} \mbox{\bf (a) }\mbox{Passing through $(1,2,4... ...Passing through $(-6,-1,0)$\ and $(2,-3,5)$}\,. \\ \end{array}\end{displaymath}$

Problem 4. Find an equation of the plane that satisfies the given conditions

$\begin{displaymath}\begin{array}{l} \mbox{\bf (a) }\mbox{Passing through $(-4,1,... ...rough $(-1,2,0)$, $(2,0,1)$, and $(-5,3,1)$}\,. \\ \end{array}\end{displaymath}$

Problem 5. Determine whether the lines given by the symmetric equations

$\begin{displaymath}\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\end{displaymath}$

and

$\begin{displaymath}\frac{x+1}{6}=\frac{y-3}{-1}=\frac{z+5}{2}\end{displaymath}$

are parallel, skew, or intersecting. If the lines are intersecting, find the point of intersection and the angle between the lines.

Problem 6. Identify and sketch the graph of each surface

(a) x=7, (b) y=z, (c) y=x²+z².
(d) x²-y²+z²=0, (e) x²+z²=1.

Problem 7. A surface consists of all points such that the distance from to the plane is twice the distance from to the point . Find an equation for this surface and identify it.

Problem 8. Sketch the curve with vector function

$\begin{displaymath}{\bf r}(t)=2{\bf i}+\sin{t}\,{\bf j}+\cos{t}\,{\bf k}\,.\end{displaymath}$

Problem 9. The helix ${\bf r_1}(t)=<\cos{t},\sin{t},t>$ intersects the curve ${\bf {r_2}}(t)=<1+t,t^2,t^3>$ at the point . Find the angle of intersection of these curves as well as a curvature of each curve at the point of their intersection. Also, find the unit normal vector to r₁(t) when t=0.

Problem 10. A particle starts at the origin with initial velocity and its acceleration is ${\bf a}(t)=<t,1,t^2>$ . Find its position function.

Problem 11. For the function $f(x,y)=\frac{xy\,\ln{(x+2)}}{x^2+y^2}$

Sketch the domain of .
Find $\lim_{\,(x,y)\rightarrow{(0,0)}}{f(x,y)}$ if it exists.

Problem 12. Sketch several level curves (surfaces) of the function

$f(x,y)=\ln{(1-x^2-y^2)}$

Problem 13. Suppose that $f(x,y)=e^{-x}\sin{y}$ . Show that the function satisfies the differential equation $f_x=f_{yy}$ .

Problem 14. Find a given partial derivative.

$\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x,y)=\sqrt{x+y}+y$ .
$\frac{\partial^2f}{\partial x\partial z}$ for $f(x,y,z)=x^{\ln{yz}}\,.$
$\frac{\partial u}{\partial z}$ for $z\sin{(x+u)}+u\cos{(y-z)}=1\,.$

Dmitry Golovaty 2026-02-12