Vectors

Problem Zero: Read the section and make your own summary of the material. 

Read the section and make your own summary of the material. 

Read the section and make your own summary of the material.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k=1$.

(b) True or False: $\sum _{k=0}^n \frac{1}{k+1}+\sum _{k=1}^n k^2$ is equal to $\sum _{k=0}^n \frac{k^3+k^2+1}{k+1}$ .

(c) True or False: $\sum _{k=1}^n \frac{1}{k+1}+\sum _{k=0}^n k^2$ is equal to $\sum _{k=1}^n \frac{k^3+k^2+1}{k+1}$  .

(d) True or False:    $\left(\sum _{k=1}^n \frac{1}{k+1}\right)\left(\sum _{k=1}^n k^2\right)$  is equal to $\sum _{k=1}^n \frac{k^2}{k+1}$ .

(e) True or False: $\sum _{k=0}^m \sqrt{k}+\sum _{k=m}^n \sqrt{k}$ is equal to$\sum _{k=0}^n \sqrt{k}$.

(f) True or False: $\sum _{k=0}^n a_k=-a_0-a_n+\sum _{k=1}^{n-1} a_k$.

(g) True or False:     ${\left(\sum _{k=1}^{10} a_k\right)}^2=\sum _{k=1}^{10} a_k^2$ .

(h) True or False: $\sum _{k=1}^n {\left(e^x\right)}^2=\frac{e^x\left(e^x+1\right)\left(2e^x+1\right)}{6}$ . 

Second--derivative graphs: The three graphs shown are graphs of a function $f$ and its first and second derivatives $f^{\text{'}}$ and $f^{\text{'}\text{'}}$, in no particular order. Identify which graph is which.

True/False: Determine whether each of the statements that

follow is true or false. If a statement is true, explain why.

If a statement is false, provide a counterexample.

Use the Maclaurin series for $\sin \left(x\right), \cos \left(x\right),e^x$ to prove that

$$\begin{gathered} a)\frac{d\left(\sin x\right)}{dx}=\cos x \\ b)\frac{d\left(\cos x\right)}{dx}=-\sin x \\ c)\frac{d\left(e^x\right)}{dx}=e^x \end{gathered}$$

True / False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$

(b) True or False: $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$

(c) True or False: $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\underset{\mathrm{z}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{z}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{z}-\mathrm{x}}$

(d) True or False: If  $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3\text{ then }\mathrm{f}(\mathrm{x}+\mathrm{h})=(\mathrm{x}+\mathrm{h}{)}^3$

(d) True or False: If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3$ then $\mathrm{f}\text{'}\left(\mathrm{x}\right)=\underset{\mathrm{h}\to 0}{\lim }\frac{\mathrm{f}\left({\mathrm{x}}^3+\mathrm{h}\right)-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$

(e) True or False: A function f is differentiable at x = c if and only if ${{\mathrm{f}}^{\text{'}}}_{+}\left(\mathrm{c}\right) \text{and }{{\mathrm{f}}^{\text{'}}}_{-}\left(\mathrm{c}\right)$ both exist.

(f) True or False: If f is continuous at x = c, then f is differentiable at x = c. 

(g) True or False: If f is not continuous at x = c, then f is not differentiable at x = c 

(h) True or False: If f is not continuous at x = c, then f is not differentiable at x = c 

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $f\left(x\right)\to 0^{+},$ then $\frac{1}{f\left(x\right)}\to \infty .$

(b) True or False: If $f\left(x\right)\to {\infty }^{+},$ then $\frac{1}{f\left(x\right)}\to 0^{+}.$

(c) True or False: If a limit initially has an indeterminate form, then it can never be solved. 

(d) True or False: A limit “does not exist” if there is no real number that it approaches.

(e) True or False: As limit forms, ${\infty }^2\to \infty .$

(f) True or False: As limit forms, $2^{\infty }\to \infty .$

(g) True or False: As limit forms, $\infty -\infty \to 0$

(h) True or False: The limit of a function $f$ as $x\to c$ is always equal value $f\left(c\right),$ provided that $f\left(c\right)$exists.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: $f\text{'}\left(x\right)=\frac{f(x+h)-f\left(x\right)}{h}$

(b) True or False: $f\text{'}\left(x\right)=\underset{x\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$

(c) True or False: $f\text{'}\left(x\right)=\underset{z\to 0}{\lim }\frac{f\left(z\right)-f\left(x\right)}{z-x}$

(d) True or False: If $f\left(x\right)=x^3, then f(x+h)=x^3+h$

(e) True or False: If $f\left(x\right)=x^3, then f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x^3+h)-f\left(x\right)}{h}$

(f) True or False: A function $f$ is differentiable at $x=c$ if and only if $f\text{'}\_\left(c\right) and f{\text{'}}_{+}\left(c\right)$ both exists.

(g) True or False: If $f$ is continuous at $x=c$, then $f$is differentiable at $x=c$.

(h) True or False: If $f$ is not continuous at $x=c, then f$ is not differentiable at $x=c$.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: To find the derivative of \(sin x\) we have to use the definition of the derivative.

(b) True or False: To find the derivative of \(tan x\) we have to use the definition of the derivative.

(c) True or False: The derivative of \(\frac{x^{4}}{sinx}\) is \(\frac{4x^{3}}{cosx}\)

(d) True or False: If a function is algebraic, then so is its derivative. 

(e) True or False: If a function is transcendental, then so is its derivative.

(f) True or False: If \(f\) is a trigonometric function, then \(f'\) is also a trigonometric function.

(g) True or False: If \(f\) is an inverse trigonometric function, then \(f'\) is also an inverse trigonometric function. 

(h) True or False: If \(f\) is a hyperbolic function, then \(f'\) is also a hyperbolic function.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The del operator, $$\bigtriangledown$$, converts vectors into scalars.

(b) True or False: The del operator, $$\bigtriangledown$$, measures the rotation of a vector field.

(c) True or False: The divergence of a vector field is a scalar.

(d) True or False: The curl of a vector field is a vector.

(e) True or False: The curl of a gradient vector field is $$0$$.

(f) True or False: Both the Fundamental Theorem of Calculus (Theorem 4.24) and Green’s Theorem relate the integral of a function on a (mathematically well-behaved) region to a quantity measured on the boundary of that region.

(g) True or False: The curl of a vector field measures how much the field is compressing or expanding.

(h) True or False: The conclusion of Green’s Theorem does not depend on the direction of parametrization of the boundary curve in question.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{\pi }}\right)=0$

(b) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{z}}\right)={\mathrm{e}}^{\mathrm{z}}$

(c) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{lnx}$

(d) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\left|\mathrm{x}\right|}$

(e) True or False: If f is an exponential function, then f ' is a constant multiple of f . 

(f) True or False: If f ' is a constant multiple of f , then f is an exponential function. 

(g) True or False: Logarithmic differentiation is required in order to differentiate complicated products and quotients. 

(h) True or False: Logarithmic differentiation is required in order to differentiate expressions that have a variable in both the base and the exponent. 

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function that is decreasing on $(-\infty ,0)$, increasing on $(0,\infty )$, and undefined at $x=0$.

(b) A function that is decreasing on $(-\infty ,0]$ and increasing on $[0,\infty )$.

(c) A function that is always positive and always decreasing, on all of $R$.

More second-derivative graphs: The three graphs shown are graphs of a function $f$ and its first and second derivatives $f^{\text{'}}$ and width="18" style="max-width: none; vertical-align: -5px;" $f^{\text{'}\text{'}}$, in no particular order. Identify which graph is which.

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading:

(a) The graph of a function with $f\left(4\right)=2$ that has a removable discontinuity at $x=4.$

(b) The graph of a function that is continuous on its domain but not continuous at $x=0.$

(c) The graph of a function that is continuous on $(0,2] and (2,3)$ but not on $(0,3).$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading:

(a) A function $f$ whose area accumulation function is negative on $[0,5].$

(b) A function $f$ whose area accumulation function is decreasing on $[0,5].$

(c) Three antiderivatives of $e^{{\sin }^{2}x}.$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A simple closed surface that is smooth.

(b) A simple closed surface that is not smooth, but is piecewise smooth.

(c) A simple closed surface that is neither smooth nor piecewise smooth.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 1)$$ that trace the unit circle exactly once clockwise, starting at the point $$(1, 0)$$.

(b) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 2π)$$ that trace the circle centered at $$(2, −3)$$ with radius 5 exactly once counterclockwise, starting at the point $$(7, −3)$$.

(c) Parametric equations $$x = f(t), y = g(t)$$ whose graph is not the graph of a function $$y = f(x)$$.

A kind of derivative for a function of three variables: Explain why the derivative of the function xe^{-4z} \sin{y} is

\begin{align}

    e^{-4z}\sin{y}

\end{align}

 if x is the variable and y and z are constants, and the derivative is xe−4z cos y if y is the variable and x and z are constants, and the derivative is −4xe−4z sin y if z is the variable and x and y are constants. What is the derivative if x, y, and z are all constants? 

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) A sum that would not be suitable for expressing in sigma notation. 

(b) Two different sigma notation expressions of the same sum. 

(c) A sum from k = 1 to k = n that is smaller for n = 10 than it is for n = 5 

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible. 

the formal$\delta -M, N-\in ,$, and N–M definitions of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=\infty ,\underset{x\to \infty }{\lim }f\left(x\right)=L,$ and$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$, respectively 

Fill in the blanks to complete each of the following theorem statements: 

For $\epsilon >0, f\left(x\right)\in (L-\epsilon ,L+\epsilon )$if and only if $\underline{ }<\epsilon .$

Use the definition of the derivative to calculate the derivative of $f\left(x\right)=\sqrt{x}$ at $c=4$. At some point you will need to multiply numerator and denominator by the conjugate of $\sqrt{4+h}-2$, which is $\sqrt{4+h}+2$.

CC

Find the equation of the sphere center at \((2,-3,4)\) and radius \(6\).

The distance between two points in the plane: What is the formula for computing the distance between points \(\left ( x_{1},y_{1} \right ) and \left ( x_{2},y_{2} \right )\)?

The sides of a 2 × 3 × 4 rectangular solid are parallel to the coordinate planes. The coordinates of four of its vertices are (1, −2, 3), (−1, −2, −1), (−1, 1, 3), and (1, −2, 3). What are the coordinates of the other four vertices? 

The distance between a point and a line in the plane: Describe a method for computing the distance between the point \(\left ( x_{0},y_{0} \right )\) and the line y = mx + b.

Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)$. You may have to guess and check to find such a function

$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\sqrt{1+{\mathrm{x}}^2}$

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible:

what it means for a function $f$ to be differentiable, left differentiable, and right differentiable at a point $x=c$

What is the definition of a sphere? 

Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)$. You may have to guess and check to find such a function.

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2\sqrt{1+{\mathrm{x}}^3}$

What is the definition of a cylinder? What is the directrix?

What is a ruling?

what it means, in terms of limits, for a function to have a removable discontinuity, a jump discontinuity, or an infinite discontinuity at x = c 

Why do we use the terminology "separable" to describe a differential equation that can be written in the form

Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)$. You may have to guess and check to find such a function.

$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\left(2-5\mathrm{x}\right)}^3}$

Consider the equation  $$x^2+y^2 =4 $$

(a) What does this equation represent in a two-

dimensional system?

b) What does this equation represent in a three-

dimensional system?

Consider the sequence of sums  $\frac{1}{3},\frac{1}{3}+\frac{1}{9},\frac{1}{3}+\frac{1}{9}+\frac{1}{27},\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81},\dots$

(a) What happens to the terms of this sequence of sums as k gets larger and larger? 

(b) Find a sufficiently large value of k which will guarantee that every term past the kth term of this sequence of sums is in the interval (0.49999, 0.5). 

Consider the equation $$z= y^2$$.

(a) What does this equation represent in the yz-plane?

(b) What does this equation represent in a three-

dimensional system?

If the initial point of the vector \(\langle2, 3, −5\rangle\) is the point \(\left(-3, 2, 4\right)\), what is the terminal point of the vector? 

Suppose f and g are functions such that $\underset{x\to 3}{\lim }f\left(x\right)=5,\underset{x\to 4}{\lim }f\left(x\right)=2$ and $\underset{x\to 3}{\lim }g\left(x\right)=4$

Given this information, calcuate the limits that follow, if possible. If it is not possible with the given information, explain why. 

$\underset{x\to 3}{\lim }-2f\left(x\right)$

Find the terminal point of a vector of magnitude \(5\) that is parallel to the vector \(\langle1,2,3\rangle\) and whose initial point is \(\left(0, 3, −2\right)\). 

Consider the function f shown in the graph next at the right. Use the graph to make a rough estimate of the average value of f on [−4, 4], and illustrate this average value as a height on the graph.

A function f that satisfies the hypotheses of Rolle’s Theorem on [−2, 2] and for which there are exactly three values c ∈ (−2, 2) that satisfy the conclusion of the theorem .

If an object is moving along the graph of a vector function

$$r(t)$$ defined on an interval $$[a, b]$$, how can the distance the

object travels from $$t = a$$ to $$t = b$$ be calculated?

dsfgh

$$\int 56$$

Find the mass of a 30-centimeter rod with square cross sections of side length 2 centimeters, given that the density of the rod x centimeters from the left end is ρ(x) =$10·5+0·01527x^2$ grams per cubic centimeter.

Perform the following steps for the power series in $x-x_0$ in $\sum _{k=0}^{\infty }{(-1)}^k\frac{k!}{\left(2k\right)!}{(x-7)}^{2k}$

Find the tangential and normal components of acceleration for the position functions in Exercises 18–22

\(r\left ( t \right )=\left<t,t^{2} \right>\)