MATH 409 Lecture 21

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End Exam 2 content

Lecture Slides

Exam 2 Review

Topics

Derivatives

• Derivative of a function
• Differentiability Theorems
• derivative of inverse function
• Mean Value Theorem
  • Rolle's Theorem
  • Generalized Mean Value Theorem
• Taylor's Formula
• l'Hôpital's rule

Integrals

• Definitions
  • Darboux Sums
  • Riemann Sums
  • Riemann Integral
• Properties of Integrals
• Fundamental Theorem of Calculus (both parts)
• Integration by Parts
• Change of variable in an integral

Chapters 4.1–4.5, 5.1–5.3

Differentiability Theorems

• sum / difference rules
• product rule
• quotient rule
• chain rule
• differentiability implies continuity
• Rolle's Theorem: If a function $f$ is continuous on a closed interval $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then $f'(c)=0$ for some $c\in (a,b)$
• Mean Value Theorem: If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in (a,b)$ such than $f(b)-f(a)=f'(c)\,(b-a)$.
• $f$ is increasing on $[a,b]$ if and only if $f'\geq 0$ on $(a,b)$
• $f$ is decreasing on $[a,b]$ if and only if $f'\leq 0$ on $(a,b)$
• $f$ is constant on $[a,b]$ if and only if $f'=0$ on $(a,b)$

Properties of Integrals

• Linearity:
  • $\int _{a}^{b}\left(f(x)+g(x)\right)\,\mathrm {d} x=\int _{a}^{b}f(x)\,\mathrm {d} x+\int _{a}^{b}g(x)\,\mathrm {d} x$
  • $\int _{a}^{b}\left(\alpha \,f(x)\right)\,\mathrm {d} x=\alpha \,\int _{a}^{b}f(x)\,\mathrm {d} x$
• Subinterval property: $\int _{a}^{b}f(x)\,\mathrm {d} x=\int _{a}^{c}f(x)\,\mathrm {d} x+\int _{c}^{b}f(x)\,\mathrm {d} x$
• Comparison Theorem: If $f(x)\leq g(x)$ for all $x\in [a,b]$, then $\int _{a}^{b}f(x)\,\mathrm {d} x\leq \int _{a}^{b}g(x)\,\mathrm {d} x$

Fundamental Theorem of Calculus

• Part 1: $F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t$, $x\in [a,b]$ is continuously differentiable on $[a,b]$ and $F'(x)=f(x)$ for all $x\in [a,b]$
• Part 2: If a function $F$ is differentiable on $[a,b]$, and the derivative $F'$ is integrable on $[a,b]$, then $\int _{a}^{x}F'(t)\,\mathrm {d} t=F(x)-F(a)$ for all $x\in [a,b]$.

Sample Problems

Problem 1: Prove the Chain Rule

Theorem. If a function $f$...

Proved in class (except for one small part)

Problem 2: Find the Limits

$\lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}$

The function $f(x)=\left(1+x\right)^{\frac {1}{x}}$ is well-defined on $(-1,\infty )$ except at $0$. Since $f(x)>0$ for all $x>1$, a function $g(x)=\log {f(x)}$ is well-defined on $(-1,\infty )$ except at $0$ as well. For any $x>1$, we have $g(x)=\log {\left(1+x\right)^{\frac {1}{x}}}=x^{-1}\,\log {\left(1+x\right)}$. Hence $g={\frac {h_{1}}{h_{2}}}$, where $h_{1}(x)=\log {\left(1+x\right)}$ and $h_{2}(x)=x$ are continuously differentiable on $\left(-1,\infty \right)$. Since $h_{1}(0)=h_{2}(0)=0$, it follows that $\lim _{x\to 0}h_{1}(x)=\lim _{x\to 0}h_{2}(x)=0$. By l'Hôpital's rule, $h_{1}'(0)=1$ and $h_{2}'(0)=1$, so $\lim _{x\to 0}g(x)=\lim _{x\to 0}{\frac {h_{1}(x)}{h_{2}(x)}}={\frac {1}{1}}=1$

Since $f=\mathrm {e} ^{g(x)}$, a composition of $g$ with a continuous function, it follows that $\lim _{x\to 0}f(x)=\mathrm {e} ^{1}=\mathrm {e}$

$\lim _{x\to +\infty }\left(1+x\right)^{\frac {1}{x}}$ Similar to above, $\lim _{x\to +\infty }h_{1}(x)=\lim _{x\to +\infty }h_{2}(x)=+\infty$. At the same time, $h_{1}'(x)\to 0$ as $x\to +\infty$, while $h_{2}'$ is identically $1$. Using l'Hôpital's Rule, we obtain $\lim _{x\to +\infty }g(x)=\lim _{x\to +\infty }{\frac {h_{1}(x)}{h_{2}(x)}}={\frac {0}{1}}=0$

Since $f=\mathrm {e} ^{g(x)}$, a composition of $g$ with a continuous function, it follows that $\lim _{x\to 0}f(x)=\mathrm {e} ^{0}=1$

Problem 3: Limit of a sequence

Find the limit of a sequence $x_{n}={\frac {1^{k}+2^{k}+\dots +n^{k}}{n^{k+1}}}$ for $n\in \mathbb {N}$, where $k$ is a natural number.

The general element of the sequence can be represented as

$x_{n}={\frac {1^{k}+2^{k}+\dots +n^{k}}{n^{k}}}\cdot {\frac {1}{n}}=\left({\frac {1}{n}}\right)^{k}{\frac {1}{n}}+\left({\frac {2}{n}}\right)^{k}{\frac {1}{n}}+\dots +\left({\frac {n}{n}}\right)^{k}{\frac {1}{n}}$

which shows that $x_{n}$ is a Riemann sum of the function $f(x)=x^{k}$ on the interval $[0,1]$ that corresponds to the partition $P_{n}=\left\{0,{\frac {1}{n}},{\frac {2}{n}},\ldots ,{\frac {n-1}{n}},1\right\}$ and samples $t_{j}={\frac {j}{n}}$, $j=1,2,\ldots ,n$. The norm of the partition is $\left\|P_{n}\right\|={\frac {1}{n}}$. Since $\left\|P_{n}\right\|\to 0$ as $n\to \infty$ and the function $f$ is integrable on $[0,1]$, the Riemann sums $x_{n}$ converge to the integral:

$\lim _{n\to \infty }x_{n}=\int _{0}^{1}x^{k}\,\mathrm {d} x=\left.{\frac {x^{k+1}}{k+1}}\right|_{x=0}^{1}={\frac {1}{k+1}}$

Problem 4: Find/evaluate indefinite/definite integrals

Subproblem 1

$\int {\frac {x^{2}}{1-x}}\,\mathrm {d} x$

A standard way to evaluate this type of function is to split it into the sum of a polynomial and a simple fraction:

${\frac {x^{2}}{1-x}}={\frac {x^{2}-1+1}{1-x}}={\frac {x^{2}-1}{1-x}}+{\frac {1}{1-x}}=-x-1-{\frac {1}{x-1}}$

Since the domain of the function is $(-\infty ,1)\cup (1\infty )$, the indefinite integral has differetn representations on the intervals $(-\infty ,1)$ and $(1,\infty )$:

$\int {\frac {x^{2}}{1-x}}={\begin{cases}-{\frac {x^{2}}{2}}-x-\log {\left(1-x\right)}+C_{1}&x<1\\-{\frac {x^{2}}{2}}-x-\log {\left(x-1\right)}+C_{2}&x>1\end{cases}}$

Subproblem 2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\pi} \sin^2{(2x)} \,\mathrm{d}x}

To integrate this function, we use a trigonometric formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - \cos{2\alpha} = 2 \sin^2{\alpha}} and a new variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = 4x} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\pi \sin^2{(2x)} \,\mathrm{d}x = \int_{0}^{\pi} \frac{1-\cos{(4x)}}{2} \,\mathrm{d}x = \int_{0}^{\pi} \frac{1-\cos{(4x)}}{8} \,\mathrm{d}(4x) = \int_{0}^{4\pi} \frac{1-\cos{u}}{8} \,\mathrm{d}u = \left. \frac{u-\sin{u}}{8} \right|_{u=0}^{4\pi} = \frac{\pi}{2}}

Subproblem 3

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \log^3{x} \,\mathrm{d}x}

To find this indefinite integral, we integrate by parts:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int \log^3{x} \,\mathrm{d}x &= x \, \log^3{x} - \int x \,\mathrm{d}(\log^3{x}) \\ &= x \, \log^3{x} - \int x\left( \log^3{x} \right)' \,\mathrm{d}x \\ &= x \, \log^3{x} - \int 3 \log^2{x} \,\mathrm{d}x \\ &= x \, \log^3{x} - 3x \, \log^2{x} + \int x \,\mathrm{d}\left( 3 \log^2{x} \right) \\ &= x \, \log^3{x} - 3x \, \log^2{x} + \int 6 \log{x} \,\mathrm{d}x \\ &= x \, \log^3{x} - 3x \, \log^2{x} + 6x \, \log{x} - \int x \,\mathrm{d}\left( 6 \log{x} \right) \\ &= x \, \log^3{x} - 3x \, \log^2{x} + 6x \, \log{x} - \int 6 \,\mathrm{d}x \\ &= x \, \log^3{x} - 3x \, \log^2{x} + 6x \, \log{x} - 6x + C \end{align}}

Subproblem 4

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\frac{1}{2}} \frac{x}{\sqrt{1-x^2}} \,\mathrm{d}x}

To integrate this function, we introduce a new variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = 1 - x^2} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{0}^{\frac{1}{2}} \frac{x}{\sqrt{1-x^2}} \,\mathrm{d}x &= -\frac{1}{2} \, \int_{0}^{\frac{1}{2}} \frac{\left( 1-x^2 \right)'}{\sqrt{1-x^2}} \,\mathrm{d}x \\ &= -\frac{1}{2} \, \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1-x^2}} \,\mathrm{d}\left( 1-x^2 \right) \\ &= -\frac{1}{2} \, \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} \,\mathrm{d}u \\ &= \ldots \\ &= 1-\frac{\sqrt{3}}{2} \end{align}}

Subproblem 5

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{1} \frac{1}{\sqrt{4-x^2}} \,\mathrm{d}x}

To integrate this function, we use a substitution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 2 \sin{t}} . Observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} changes from $0$ to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} changes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{6}} ):

${\begin{aligned}\int _{0}^{1}{\frac {1}{\sqrt {4-x^{2}}}}\,\mathrm {d} x&=\int _{0}^{\frac {\pi }{6}}{\frac {1}{\sqrt {4-(2\sin {t})^{2}}}}\,\mathrm {d} (2\sin {t})\\&=\int _{0}^{\frac {\pi }{6}}{\frac {\left(2\sin {t}\right)}{\sqrt {4-4\sin ^{2}{t}}}}\,\mathrm {d} t\\&=\int _{0}^{\frac {\pi }{6}}{\frac {2\cos {t}}{\sqrt {4\cos ^{2}{t}}}}\,\mathrm {d} t\\&=\int _{0}^{\frac {\pi }{6}}1\,\mathrm {d} x\\&={\frac {\pi }{6}}\end{aligned}}$

Bonus Problem 5

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p : \mathbb{R} \to \mathbb{R}} is locally a polynomial, which means that for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in \mathbb{R}} , there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} coincides with a polynomial on the interval $\left(c-\epsilon ,c+\epsilon \right)$. Prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is a polynomial.

Proof. For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in \mathbb{R}} let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c} denote a polynomial and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_c} denote a positive number such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = p_c(x)} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in (c-\epsilon_c, c + \epsilon_c)} . Consider two sets:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} E_+ &= \left\{ x > 0 ~\mid~ p(x) \ne p_0(x) \right\} \\ E_- &= \left\{ x < 0 ~\mid~ p(x) \ne p_0(x) \right\} \end{align}}

We are going to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+ = E_- = \emptyset} .

Assume that the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+} is not empty. Clearly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+} is bounded below, hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \inf{E_+}} is a well-defined real number. Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+ \subset \left[ \epsilon_0, \infty \right)} Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \ge \epsilon_0 > 0} .

Observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = p_0(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in (0,d)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = p_d(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \left( d - \epsilon_d, d + \epsilon_d \right)} . The interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,d)} overlaps with the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( d - \epsilon_d, d + \epsilon_d \right)} . Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_d} coincides with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_0} on the intersection Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,d) \cap (d - \epsilon_d, d + \epsilon_d)} . Equivalently, the difference Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_d - p_0} is zero on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,d) \cap (d - \epsilon_d, d + \epsilon_d)} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_d - p_0} is a polynomial and any nonzero polynomial has only finitely many roots, we conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_d - p_0} is identically 0. Then the polynomials Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_d} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_0} are the same. It follows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = p_0(x)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in (0,d+\epsilon_d)} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \ne \inf{E_+}} , a contradiction.

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+ = \emptyset} . Similarly, we prove that the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_-} is empty as well. Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+ = E_- = \emptyset} , the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} coincides with the polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_0} everywhere.

Bonus Problem 6

Show that a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} \mathrm{e}^{-\frac{1}{1-x^2}} & \left| x \right| < 1 \\ 0 & \left| x \right| \ge 1 \end{cases}}

is infinitely differentiable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} .

Observe that the integral of this function goes from 0 to 1 smoothly. Constant functions and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{f(x)}} are infinitely differentiable.