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\line{\hskip 2.5in \bf NAME: \hrulefill}
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\line{\bf Math 180, Calculus I \hfil Hour Exam Two}
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\line{\bf 10:00 am Lecture \hfil November 3, 1995}
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\item{1.} The function $f(x)$ is increasing.  Some of its values are given in the table:
$$\matrix{ x   &  0  & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 \cr
          f(x) & 1.0 & 1.2 & 1.6 & 2.0 & 2.4 & 2.6 & 3.2 & 3.4 \cr}$$
\itemitem{(i)} Compute the left and right Riemann sums with two subdivisions ($n=2$) for the integral $$\int_2^3 f(x)\,dx.$$
\itemitem{(ii)} How do these sums compare with the value of the integral?
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\item{2.}
Find the derivatives of the following functions.  Please do {\it not} simplify your answers.
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\itemitem{(i)}$\displaystyle{x^{1642}-2x^5+x^{1/3}+\pi}$ \smallskip
\itemitem{(ii)}$\displaystyle{{{x}\over{x^3+1}}}$ \smallskip
\itemitem{(iii)}$\displaystyle{e^{-x}\sin x}$ 
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\item{3.} \hskip1.5mm (i)\hskip5pt Find a function whose derivative is $4x+1$. 
\smallskip\itemitem{(ii)} Evaluate: $\displaystyle{\int_1^b (4x+1) \,dx}$.
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\item{4.} The temperature of a pie in a $325^\circ$ oven is given by
$$f(t)=325 -255e^{-0.1t}$$
where $t$ is the time (in minutes) the pie has been in the oven.
\itemitem{(i)} Write a formula using an integral for the average temperature of the pie during the first 30 minutes, $0 \le t \le 30$.  \itemitem{(ii)} Calculate this average temperature with an error of at most $5^\circ$.
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\item{5.} For the curve $xy + 3y^2 = 18$, 
\itemitem{(i)} Find $\displaystyle{{dy}\over{dx}}$ in terms of $x$ and $y$.
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\itemitem{(ii)} Write the equation of the tangent line at the point $(3,2)$.
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\item{6.} Differentiate the following functions and show your work.
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\itemitem{(i)} $\displaystyle{\ln(x+x^3)}$\smallskip
\itemitem{(ii)} $\displaystyle{e^{\cos(\sqrt{x})}}$ \smallskip
\itemitem{(iii)} $\displaystyle{\tan(3x-2)}$.
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