Math 8
Calculus of functions of one and several variables

Last updated January 18, 2017

Announcements:
• Week #9 assignments are posted.

### Homework Descriptions

• 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. 3/8 + x-hour Wrap up 3/11 Final Exam

Dartmouth Math Department
Last updated January 18, 2017