Daily assignments: Just as in any other subject, developing proficiency in mathematics requires sustained, consistent effort. We will assign a few problems from the textbook every day for practice. These problems will mostly be
computational in nature and provide opportunities for you to review that material from that day's lecture.
Weekly assignments: The weekly problems will require more engagement with and test a deeper understanding of the material. You will be expected to write up complete solutions to these problems.
Homework Assignments
Week
Date
Sections
Topics
HW problems (Due Monday)
Daily Problems (Due MWF)
1
1/4
Math 3 Review
Solve these problems:
FTC
Give a careful statement of the Fundamental Theorem of Calculus.
Is every continuous function on $[0,1]$ differentiable? Why or why not?
Is every continuous function on $[0,1]$ integrable? Why or why not?
Discuss the statement that integration and differentiation are inverse operations.
Discuss whether integration or differentiation is a more difficult process/concept.
Compute the indefinite integral: $$\int (x^2-3x+4)\sin(4x)dx$$
Compute the indefinite integral: $$\int \sin(x)\cdot\cos(\cos(x))dx$$
Compute the fourth order Taylor polynomial, $T_4$, for $f(x)=x\sin(x)$ at $a=0$.
Using the software program here: link for this problem.
Plot some Taylor polynomials for $\sin(x)$ and $\cos(x)$.
Compare the results for even and odd choices of $n$.
Does these patterns change if you change the value of $x_0$?
When $x_0=0$ what value of $n$ would you use to match the function
$f(x)=\sin(x)+\cos(x)$ on the entire interval $[-5,5]$?
Solve these problems:
Section 4.3 #27: Evaluate the integral $$\int_0^1 (u+2)(u-3)du$$
Section 4.4 #39: Evaluate the integral $$\int_2^5 |x-3|dx$$
Section 4.5 #15: Evaluate the indefinite integral $$\int \cos(1+5t)dt$$
Section 7.1 #3: Evaluate the indefinite integral $$\int x\cos(5x)dx$$
x-hour
No x-hour this week
1/6
11.11
Taylor Polynomials
Solve these problems:
Section 11.11 #3,5,7 Compute the third order Taylor polynomials, $T_3$, for the following functions at the given centers $a$:
$f(x)=e^x$ at $a=1$
$f(x)=\cos(x)$ at $a=\frac{\pi}{2}$
$f(x)=\ln(x)$ at $a=1$
2
1/9
11.10 and 11.11
Taylor Error
Solve these problems: (Due Wednesday 1/18)
Use Taylor Polynomials to compute the integral below to at least 4 digits of accuracy:
$$\int_0^1 e^{x^2}dx$$
Estimate $e^{.1}$ and $e^{-.1}$ to 4 digits of accuracy.
Express $3.\overline{142857}$ as a ratio of integers.
Sequence Algebra:
If $\sum a_n$ and $\sum b_n$ are convergent series, can $\sum a_n+b_n$ diverge?
If $\sum a_n$ and $\sum b_n$ are divergent series, can $\sum a_n+b_n$ converge?
If $\sum a_n$ is a convergent series and $f(x)=cx+d$ for some real constants $c$ and $d$, for what values
of $c$ and $d$ does$ \sum f(a_n)$ converge?
If $\sum a_n$ is a convergent series with $1\geq a_n\geq 0$ does $\sum a_n\cdot a_{n+1}$ converge?
For what values of $p$ does $\sum n^p\cdot\ln(n)$ converge?
For what values of $p$ does $\sum (n\cdot(\ln(n))^p)^{-1}$ converge?
Does the series $\sum\frac{n!}{n^n}$ converge?
If $f(x)$ is a degree four polynomial and $g(x)$ is a degree 6 polynomial with all positive coefficients, does the
series $\sum \frac{f(n)}{g(n)}$ converge?
Solve these problems:
Section 11.11 #15: Approximate the function $f(x)=x^{\frac23}$ at the point $a=1$ with a third order Taylor polynomial.
What is the error associated to this approximation for $.8\leq x\leq 1.2$?
Section 11.11 #25: Use Taylor's inequality to determine the number of terms necessary to estimate $e^{.01}$ to within
$.00001$.
Section 11.11 # 27: Use Taylor's inequality to estimate the range of $x$ values where the approximation to $\sin(x)$
given by $\sin(x)\approx x-\frac{x^3}{6}$ has error less than $.01$.
1/11 + x-hour
11.2 and 11.4
Infinite Series and the comparison test
Solve these problems:
Section 11.2 #5: Calculate the first 8 terms of the sequence of partial sums of $$\sum_{n=1}^\infty \frac{1}{n^4+n^2}$$
to at least 4 decimal places. Does it look like this sum is converging?
Section 11.2 #19: Determine whether the series $10-2+.4-.08+\cdots$ converges. If it does, compute the sum.
Section 11.2 #35: Determine whether the series $\sum_{k=1}^\infty(\sin(100))^k$ converges. If it does, compute the sum.
Section 11.4 #7: Determine whether the series converges: $$\sum_{n=1}^\infty \frac{9^n}{3+10^n}$$
Section 11.4 #13: Determine whether the series converges: $$\sum_{n=1}^\infty \frac{1+\cos(n)}{e^n}$$
1/13
11.3
Integral Test
Solve these problems: (Due Wednesday 1/18)
Section 11.3 #3: Use the integral test to determine if the sum converges: $$\sum_{n=1}^\infty n^{-3}$$
Section 11.3 #7: Use the integral test to determine if the sum converges: $$\sum_{n=1}^\infty \frac{n}{n^2+1}$$
3
1/16
No class - MLK Day
1/18 + x-hour
11.5 and 11.6
Alternating Series; Root and Ratio tests
Solve these problems:
An alternating $p-$series is defined as:
$$\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{n^p} $$
For what values of $p$ do these series converge?
Estimate the following series to within 4 digits of accuracy:
$$\sum_{n=1}^\infty \dfrac{(-1)^n}{n^2} $$
Explain why we cannot apply the alternating series test to:
$$\sum_{n=1}^\infty \dfrac{\cos(n)}{n^2+3n+2} $$
Explain why a series with all negative terms cannot be conditionally convergent.
Determine whether the series below converge or diverge:
$$\sum_{n=1}^\infty \dfrac{n!}{50^n}$$
$$\sum_{n=1}^\infty \dfrac{n!}{n^{50}}$$
$$\sum_{n=1}^\infty (\arctan(n))^n$$
Determine whether the series is absolutely convergent, conditionally convergent, or divergent:
$$\sum_{n=2}^\infty \dfrac{(-1)^n}{\ln(n)\cdot n}$$
Explain why it is not possible to have a power series whose interval of convergence is $[0,\infty)$.
Find a power series whose interval of convergence is $[1,12)$.
Find a power series whose interval of convergence is $(-5,17)$
Find the radius and interval of convergence for:
$$\sum_{n=1}^\infty\dfrac{\sqrt{x}(x+4)^n}{12^n} $$
Solve these problems:
Section 11.5 #25: Show that the series is convergent and determine how many terms are necessary to estimate the sum with
$|\operatorname{error}|<.00005$: $$\sum_{=1}^\infty \dfrac{(-1)^{n-1}}{n^22^n}$$
Section 11.6 #3: Determine whether the series is conditionally or absolutely convergent:
$$\sum_{n=0}^\infty \dfrac{(-1)^n}{5n+1}$$
Section 11.6 #11: Determine whether the series converges:
$$ \sum_{k=1}^\infty \dfrac{1}{k!}$$
Section 11.6 #25: Determine whether the series converges:
$$\sum_{n=1}^\infty\left(\dfrac{n^2+1}{2n^2+1}\right)^n$$
1/20
11.8
Power Series
Solve these problems:
Section 11.8 #5: Find the radius and interval of convergence for:
$$ \sum_{n=1}^\infty\dfrac{x^n}{2n-1}$$
Section 11.8 #7: Find the radius and interval of convergence for:
$$ \sum_{n=1}^\infty\dfrac{x^n}{n!}$$
Section 11.8 #29: If $\sum_{n=0}^\infty c_n4^n$ is a convergent series can we conclude that the following series
are convergent?
$\sum_{n=0}^\infty c_n(-2)^n$
$\sum_{n=0}^\infty c_n(-4)^n$
4
1/23
11.9
Power Series
Solve these problems:
Sketch the surface in $\mathbb{R}^3$ described by $x-y=2$. Describe the plane figure formed by the intersection of this surface and the plane $z=0$.
Sketch the surface in $\mathbb{R}^3$ described by $y^2+z^2=16$. Describe the plane figure formed by the intersection of this surface and the plane $x=0$.
Find the equation of a sphere which has the line segment connecting the points $(5,4,3)$ and $(1,6,-9)$ as a diameter.
Find the distance from the point $(1,2,3)$ to each of the following planes:
The $xy$ plane
The $xz$ plane
The $yz$ plane
The $x$ axis
The $y$ axis
The $z$ axis
Solve these problems:
Section 11.9 #9: Find the power series and determine the interval of convergence for:
$$f(x)=\dfrac{(x-1)}{x+2}$$
Section 11.9 #15: Find the power series and determine the interval of convergence for:
$$\ln(5-x)$$
1/25 + x-hour
11.11 and Exam review
Infinite Series and the comparison test
Solve these problems:
Section 11.11 #37: Using Table 1 in the book compute a Maclaurin series for $x\cos(2x)$.
1/27
12.1
Introduction to $\mathbb{R}^n$
Solve these problems:
Section 12.1 #9: Find the lengths of the sides of the triangle $PQR$. Is it a right triangle or an isocsceles triangle?
$$P=(3,-2,-3)\qquad Q=(7,0,1)\qquad R=(1,2,1)$$
Section 12.1 #13: Find an equation of a sphere with center $(-3,2,5)$ and radius $4$. What is the intersection of this sphere and the $yz$--plane?
Section 12.1 #17: Find the center and radius of the sphere defined by:
$$x^2+y^2+z^2-2x-4y+8z=15$$
5
1/30
12.2
Vectors
Solve these problems:
Find a vector that has the same direction as $\langle 1,-22,3\rangle$ that has length 10.
For two arbitrary vectors $a$ and $b$, show that $c=b-\operatorname{proj}_ab$ is orthogonal to $a$.
Find the angle between a diagonal of the unit cube and one of its edges.
Show that for any two vectors $a$ and $b$, that $|a\cdot b|\leq|a|\cdot|b|$. When does equality hold?
If $(a+b)\cdot (a-b)=0$ what can be said about the lengths of $a$ and $b$?
Find two unit vectors that are parallel to the tangent line to the parabola $y=x^2-1$ at the point $(3,8)$.
Solve these problems:
Section 12.2 #9: Find a vector with representation given by the directed line segment $\vec{AB}$. Draw $\vec{AB}$ and and the
equivalent representation at the origin $A=(-2,1)$ and $B=(1,2)$.
Section 12.2 #19: Find $a+b$, $4a+2b$, $|a|$, and $|a-b|$ for $a=\langle -3,4\rangle$ and $b=\langle 9,-1\rangle$.
Section 12.2 #25: Find a unit vector that has the same direction as $a=\langle 8,1,-4\rangle$.
2/1 + x-hour
12.3 and 12.6
Dot products and surfaces
Solve these problems:
Section 12.3 #9: Find $a\cdot b$ if $|a|=7$, $|b|=4$ and the angle between $a$ and $b$ is $20^\circ$.
Section 12.3 #25: Decide whether the triangle with vertices $P=(1,-3,-2)$, $Q=(2,0,-4)$, $R=(6,-2,-5)$ is a right triangle.
Section 12.3 #27: Find a unit vector that is orthogonal to both $i+j$ and $i+k$.
2/3
12.3
Dot products and projections
Solve these problems:
Section 12.3 #29: Find the acute angle between the lines $2x-y=3$ and $3x+y=7$.
Section 12.3 #39: Find the scalar and vector projections of $b$ onto $a$ for $a=\langle -5,12\rangle$ and $b=\langle 4,6\rangle$
Section 12.6 #31: Reduce the equation to one of the standard forms, classify the surface, and sketch it: $y^2=x^2+\frac19z^2$.
6
2/6
12.4
Cross product
Solve these problems:
State whether each of these expressions is meaningful ($a$, $b$, and $c$ are all vectors). If not, explain why. If so, state whether the output is a scalar
or a vector.
$a\cdot(b\times c)$
$a\times (b\cdot c)$
$a\times (b\times c)$
$a\cdot (b\cdot c)$
$(a\cdot b)\times (c\cdot d)$
$(a\times b) \cdot (c\times d)$
Find an example of two vectors $a$ and $b$ where $a\times b\neq b\times a$. Find an example where $a\times b=b\times a$.
Use the scalar triple product to determine if the vectors $a=\langle1,2,3\rangle$, $b=\langle4,5,6\rangle$, and $c=\langle7,8,9\rangle$ are coplanar.
Read (do not solve, unless you want to) problems 23-40 in Section 12.5 of the textbook. Observe that there are many different ways to parameterize a plane.
Find the equation of the plane that passes through the points $(0,1,1), (1,0,1)$, and $(1,1,0)$.
Compute the distance from the point $(1,-2,3)$ to the plane $2x-5y+4z=\sqrt{3}$.
Find the domain of $r(t)=\langle \dfrac{t^2-1}{t-1},\ln(t)\sin(t) \rangle $ and compute the limit $\lim_{t\rightarrow 1} r(t)$.
Compute the derivative of $\langle1,t,t^2\rangle\cdot (\langle \sin(t),\cos(t),t\rangle\times \langle 3t,4t,5t\rangle)$
Compute the intergral from $t=0$ to $1$ of $r(t)=\langle \sin(t),\cos(t),t\rangle$.
Solve these problems:
Section 12.4 #3: Find the cross product $a\times b$ and verify that it is orthogonal to the original vectors for
$a=2j-4k$ and $b=-i+3j+k$.
Section 12.4 #19: Find two unit vectors orthogonal to both $\langle 3,2,1\rangle$ and $\langle -1,1,0\rangle$.
Section 12.4 #27: Find the area of the parallelogram with vertices $(-3,0),(-1,3),(5,2),(3,-1)$.
2/8 + x-hour
12.5
Lines and planes in standard forms
Solve these problems:
Section 12.5 #9: Find the parametric and symmetric equations for the line through the points $(-8,1,4)$ and $(3,-2,4)$.
Section 12.5 #21: Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. If they intersect find the point of intersection.
$$L_1:\dfrac{x-2}{1}=\dfrac{y-3}{-2}=\dfrac{z-1}{-3}$$
$$L_2:\dfrac{x-3}{1}=\dfrac{y+4}{3}=\dfrac{z-2}{-7}$$
Section 12.5 #35: Find the equation of the plane that passes through the point $(3,5,-1)$ and contains the line $x=4-t$
$y=2t-1$, and $z=-3t$.
2/10
13.1 and 13.2
Vector functions and calculus
Solve these problems:
Section 13.1 #3: Find the limit: $\lim_{t\rightarrow 0}\left( e^{-3t}i+\dfrac{t^2}{\sin^2(t)}j+\cos(2t)k\right)$.
Section 13.1 #33: Use a computer to graph the curve $r(t)=\langle \cos(t)\sin(2t),\sin(t)\sin(2t),\cos(2t)\rangle$.
Section 13.2 #39: Compute the intergral: $$\int \left(\sec^2(t)i+t(t^2+1)^3j+t^2\ln(t)k\right)dt$$
7
2/13
13.3
Arc Length and Curvature
Solve these problems:
Find and describe the domain of the following functions:
$f(x,y)=\ln(121-x^2-y^2)$
$g(x,y,z)= \frac{y+z-x}{1-3x+4y+z}$
$h(x,y,z)=\sqrt{10-x}+\sqrt{6-y}-\sqrt{2-z}$
Find the partial derivatives of:
$f(x,y)=(xy-y^2)^3$
$g(x,y,z)= \sqrt{xyz-z}$
$h(x,y,z)=\frac{x}{y}\cos(z)$
Use implicit differentiation to find $\frac{\partial z}{\partial x}$ and $\frac{\partial x}{\partial y}$ for $2x^2-3y^2+z^2-2z=0$.
Solve these problems:
Section 13.2 #17: Find the unit tangent vector at the point given by the parameter $t=2$ for $r(t)=\langle t^2-2t,1+3t,\frac13t^3+\frac12t^2\rangle$.
Section 13.3 #5: Find the length of the curve: $r(t)=i+t^2j+t^3k$ for $0\leq t\leq 1$.
Section 13.3 #17: Find the unit tangent vector, unit normal vector, and curvature for: $r(t)=\langle t, 3\cos(t), 3\sin(t)\rangle$.
Section 13.3 #21: Find the curvature of $r(t)=t^3j+t^2k$.
2/15 + x-hour
14.1 and 14.2
Limits and continuity of multivariate functions
Solve these problems:
Section 14.1 #11: Let $f(x,y,z)=\sqrt{x}+\sqrt{y}+\sqrt{z}+\ln(4-x^2-y^2-z^2)$. Evaluate $f(1,1,1)$ and determine the domain of $f$.
2/17
14.3
Partial Derivatives
Solve these problems:
Section 14.2 #11: Find the limit or show that it does not exist: $$\lim_{(x,y)\rightarrow(0,0)}\dfrac{y^2\sin^2(x)}{x^4+y^4}$$
Section 14.3 #31: Find the partial derivatives of $f(x,y,z)=x^3yz^2+2yz$.
Section 14.3 #51: Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=f(x)+g(y)$ and $z=f(x+y)$.
8
2/20
14.4
Tangent Planes
Solve these problems:
Does there exist a function $f(x,y)$ such that $f_x=3x^2+4y$ and $f_y= 3\cos(y)+2x$? Why or why not? What if the partial derivatives are
$f_x=3x^2$ and $f_y= 3\cos(y)$?
Explain the relationships between the tangent plane, linearization, and differential of a function $z=f(x,y)$. Which of these concepts seems to be the most useful?
Compute the linearization of $f(x,y)=-5e^{x^2-4y}+xy$ at the point $(2,1)$ and use it to approximate $f(2.1,1.1)$, $f(2.1,.9)$, $f(1.9,1.1)$, and $f(1.9,.9)$. Which of these approximations is most accurate?
The volume of ice cream in a one-scoop ice cream cone of radius $r$ and height $h$ is $\frac{\pi r^2h+2\pi r^3}3$. Use differentials
to determine the amount of additional ice cream that is needed if the radius and height of the cone both increase by
$.1$ from original values of $r=3$ and $h=5$.
Polar coordinates with $x=r\cos(\theta)$ and $y=r\sin(\theta)$ are often used to compute certain integrals. If
$f(x,y)=x^2+y^2-\frac{3}{xy}$ what are the partial derivatives of $f$ with respect to $r$ and $\theta$?
If $f$ is a differentiable function of variables $a,b,c,d,e,g,h$ and each of those variables is a
function of variables $x,y,z$ how many unique partial derivatives can $f$ have? How many summands appear in the
expression for $\frac{\partial f}{\partial z}$?
For the function $f(x,y)=x^2+y^2-2x-4y$ what are the points $(x_0,y_0)$ where the direction of fastest change is $\langle 1,1\rangle$?
Does every normal line to the unit sphere $x^2+y^2+z^2=1$ pass through the origin? Why or why not?
Solve these problems:
Section 14.3 #61: Verify Clairaut's Theorem by checking that $u_{xy}=u_{yx}$ for $u(x,y)=\cos(x^2y)$.
Section 14.4 #3: Find the equation of the tangent plane to$z=e^{x-y}$ at $(2,2,1)$.
Section 14.4 #21: Find a linear approximation of the function $f(x,y,z)=\sqrt{x^2+y^2+z^2}$ at $(3,2,6)$ and use it to approximate $f(3.02,1.97,5.99)$.
Section 14.4 #25: Find the differential of $z=e^{-2s}\cos(2\pi t)$.
2/22 + x-hour
14.5
Chain Rule
Solve these problems:
Section 14.5 #3: Use the chain rule to find $\frac{dz}{dt}$ for $z=\sin(x)\cos(y)$ and $x=\sqrt{t}$ $y=\frac1t$.
Section 14.5 #9: Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$ for $z=\ln(3x+2y)$ with $x=s\sin(t)$ and $y=t\cos(s)$.
Section 14.5 #27: Use the implicit differentiation method to find $\frac{dy}{dx}$ for $y\cos(x)=x^2+y^2$.
2/24
14.6
Directional Derivatives
Solve these problems:
Section 14.6 #9: Find the gradient of $f(x,y,z)=x^2yz-xyz^3$. Evaluate the gradient at $P=(2,-1,1)$. Find the rate of change of $f$ at$P$ in the direction of $\langle0,\frac45,-\frac35\rangle$.
Section 14.6 #21: Find the maximum rate of change of $f(x,y)=4y\sqrt{x}$ at $(4,1)$ and deterimne the direction in which it occurs.
9
2/27
14.6
Gradients
Solve these problems:
Find the equations for the tangent plane and normal line of $x+y+z=e^{xyz}$ at $(0,0,1)$.
If $f$ and $g$ are differentiable functions use the product rule to find an expression for the gradient of
$fg$.
Suppose that a differentiable function $f(x,y)$ has directional derivative of $4$ in the direction of $\langle 1,1\rangle$
at the point $(8,-13)$ and a directional derivative of $2$ in the direction of $\langle 3,4\rangle$ at the same point. What is the
gradient of $f$ at $(8,-13)$?
Find the absolute maximum and minimum of $f(x,y)=xy-x-y$ over the triangle with vertices: $(0,0)$, $(2,0)$ and $(2,2)$.
Find the closest point on the plane $z=1-x-y$ to the point $(0,2,-3)$.
Use Lagrange multipliers to finct the extreme values of $f(x,y)=3x+y$ if $x$ and $y$ are restricted to the circle
of radius 10.
Solve these problems:
Section 14.6 #43: Find the equations of the tangent plane and the normal line to the surface $xy^2z^3=8$ at the point (2,2,1).
Section 14.6 #23: Find the maximum rate of change of $\sin(xy)$ at $(1,0)$ and determine the direction in which it occurs.
3/1 + x-hour
14.7
Optimization
Solve these problems:
Section 14.7 #5: Find the local maximum and minimum values and saddle points of $x^2+xy+y^2+y$.
Section 14.7 #17: Find the local maximum and minimum values and saddle points of $xy+e^{-xy}$.
3/3
14.7
Lagrange Multipliers
Solve these problems:
Section 14.8 #3: Use Lagrange multipliers to find the extreme values of $x^2-y^2$ subject to $x^2+y^2=1$.
Section 14.8 #9: Use Lagrange multipliers to find the extreme values of $xy^2z$ subject to $x^2+y^2+z^2=4$.
10
3/6
14.8
Lagrange Multipliers
Solve these problems:
Write two problems that you think could appear on the final exam.