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\begin{document} \begin{frame} Calculate the following integrals using Part II of the Fundamental Theorem of Calculus. \begin{alignat*}{2} &a)\;\displaystyle\int_{-1}^2(x^3-4x)\,dx &\qquad\qquad\qquad\qquad &b)\;\displaystyle\int_4^9 \sqrt{x}\,dx \\ &c)\;\displaystyle\int_{\frac{\pi}{6}}^{\pi} \sin(\theta)\,d\theta &\qquad\qquad\qquad\qquad &d)\;\displaystyle\int_0^1 (x+3)(x-6)\,dx \\ &e)\;\displaystyle\int_1^16 \dfrac{x-3}{\sqrt{x}}\,dx &\qquad\qquad\qquad\qquad &f)\;\displaystyle\int_{-2}^1 x^{-4}\,dx \end{alignat*} \end{frame} \begin{frame} You are traveling with velocity $v(t)$ that varies continuously over the interval $[a, b]$ and your position at time $t$ is given by $s(t)$. Which of the following represent your average velocity for that time interval: \vskip 5pt \begin{itemize} \item[\bf (I)] $\dfrac{1}{b-a}\displaystyle\int_a^b v(t)\,dt$ \vskip 5pt \item[\bf (II)] $\dfrac{s(b)-s(a)}{b-a}$ \vskip 5pt \item[\bf (III)] $v(c)$ for at least one $c$ between $a$ and $b$. \end{itemize} \vskip 15pt \begin{enumerate}[a)] \item I, II, and III \item I only \item I and II only \end{enumerate} \end{frame} \begin{frame} Below is the graph of a function $f$. \begin{center} \includegraphics[height=1.6in]{FTC_Graph.png} \end{center} Let $g(x) = \displaystyle\int_0^x f(t)\,dt$. Then for $0 < x < 2$, $g(x)$ is \begin{itemize} \item[\bf (a)] increasing and concave up. \item[\bf (b)] increasing and concave down. \item[\bf (c)] decreasing and concave up. \item[\bf (d)] decreasing and concave down. \end{itemize} \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} If $f$ is continuous on the interval $[a,b]$, then $$ \dfrac{d}{dx}\left(\displaystyle\int_a^b f(x)\,dx\right) = f(x) $$ \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} Let $f$ be continuous on the interval $[a,b]$. There exist two constants $m$ and $M$, such that $$ m(b-a) \leq \int_a^b f(x)\,dx \leq M(b-a) $$ \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} If $f'(x) = g'(x)$, then $f(x)=g(x)$. \end{frame} \begin{frame} Calculate the following derivatives using Part I of the Fundamental Theorem of Calculus. \begin{alignat*}{2} &a)\;\frac{d}{dx}{\displaystyle\int_0^x {\frac{\text{dt} }{ 1+t^2}}} & \qquad\qquad\qquad\qquad &b)\;\frac{d}{dx}{\displaystyle\int_0^{x^2}{\frac{\text{dt} }{ 1+t^2}}} \\ &c)\;\frac{d}{dx}{\displaystyle\int_{{-x^2}}^{x^2}{\frac{\text{dt} }{ 1+t^2}}} & \qquad\qquad\qquad\qquad &d)\;\frac{d^2 }{ dx^2}{\displaystyle\int_0^x {\frac{\text{dt} }{ 1+t^2}}} \\ &e)\;\frac{d}{dx}{\displaystyle\int_{1}^{{{\tan x}}}{{t^{10}\cos t}}\;dt} & \qquad\qquad\qquad\qquad &f)\;\frac{d}{dx}{\displaystyle\int_{{{x^3}}}^{{{x^5 +1}}}{\frac{1}{ t}}\;dt} \end{alignat*} \end{frame} \begin{frame} If $f$ is continuous and $f(x) < 0$ for all $x$ in the interval $[a,b]$, then $\displaystyle\int_a^b f(x)\,dx$ \begin{itemize} \item[\bf (a)] must be negative. \item[\bf (b)] might be zero. \item[\bf (c)] not enough information. \end{itemize} \end{frame} \begin{frame} If $f$ is a differentiable function, then $\displaystyle\int_0^x f'(t)\,dt = f(x)$ \begin{itemize} \item[\bf (a)] Always. \item[\bf (b)] Sometimes. \item[\bf (c)] Never. \end{itemize} \end{frame} \begin{frame} A sprinter practices by running various distances back and forth along a straight line. Her velocity at $t$ seconds is given by the function $v(t)$. What does $\displaystyle\int_0^{60} |v(t)|\,dt$ represent? \begin{itemize} \item[\bf (a)] The total distance the sprinter ran in one minute. \item[\bf (b)] The sprinter's average velocity in one minute. \item[\bf (c)] The sprinter's distance from the starting point after one minute. \item[\bf (d)] None of the above. \end{itemize} \end{frame} \begin{frame} Water is pouring out of a pipe at the rate of $f(t)$ gallons per minute. You collect the water that flows from the pipe between $t=2$ and $t=4$ minutes. The amount of water you collect can be represented by \begin{itemize} \item[\bf (a)] $\displaystyle\int_2^4 f(x)\,dx$ \item[\bf (b)] $f(4)-f(2)$ \item[\bf (c)] $(4-2)f(4)$ \item[\bf (d)] the average of $f(4)$ and $f(2)$ times the amount of time the elapsed. \end{itemize} \end{frame} \begin{frame} If $f$ is continuous on the interval $[a,b]$ then \begin{itemize} \item[\bf (i)] $\displaystyle\int_a^b f(x)\,dx$ is the area bounded by the graph of $f$, the $x$-axis, and the lines $x=a$ and $x=b$. \item[\bf (ii)] $\displaystyle\int_a^b f(x)\,dx$ is a number. \item[\bf (iii)] $\displaystyle\int_a^b f(x)\,dx$ is an antiderivative of $f(x)$. \item[\bf (iv)] $\displaystyle\int_a^b f(x)\,dx$ may not exist. \end{itemize} \end{frame} \begin{frame} If \( \displaystyle\int_{a}^{b} f(x) \; dx = b^{3} - a^{3} \) for all numbers \( a \) and \( b \) , what is \( \displaystyle\int_{a}^{b} f'(x) \; dx \) ? \vskip 30pt If \( \frac{d}{dx} \left( \displaystyle\int_{a}^{x} f(t) \; dt \right) = x^{3} - 1 \) , what is \( \displaystyle\int_{a}^{b } f'(x) \; dx \) ? \vskip 30pt If \( \displaystyle\int_{a}^{b} f(u(x)) u'(x) \; dx = (2/3)(b^{2}+1)^{3/2 }- (2/3)(a^{2}+1)^{3/2} \) for all numbers \( a \) and \( b \) , what might \( f(x) \) and \( u(x) \) be? Are they unique? \end{frame} \end{document}