Sample problems for MVT/Integral proofs (elementary level calc)

[DrWahoo]

Note the code requires the appropriate package in latex given here; just don't forget to include you document class in the preamble of the document. View attachment 7595

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\begin{document}
\begin{center}
	{\LARGE Integral Proofs and MVT Practice\\[0.25em]  Practice Problems} \\[1em]
	{\large  Don't use the Internet, use your textbook to help. }	
\end{center}
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\begin{problem}[1.]
Note that if $f$ is positive and continuous on $[a,b]$, then there exists at 
least one $c \in [a,b]$ such that
\[ 
	\int_a^b f(t) \ dt = f(c)(b-a).		
\]
Prove that this conclusion continues to hold for any differentiable function $f$ by applying the 
mean value theorem  on the interval $[a,b]$ to the function $A(x)$ defined by
\[	
	A(x) = \int_a^x f(t) \  dt. 
\]
\end{problem}
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\begin{problem}[2.]  
Given that $f$ has a continuous derivative, does
\[	\frac{d}{dx} \left [ \int_a^x f(t) \ dt \right ] 
			=  \int_a^x \frac{d}{dt} [f(t)]\ dt \ \text{\Large ?}    \]
Explain why or why not.  (Note that the above statement says that, if $f \in \mathcal{C}^1$, 
then we can change the order of the derivative and integral, i.e., we can bring the derivative 
inside the integral.)
\end{problem}
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\begin{problem}[3.]
Let $\displaystyle F(x) = \int_1^{x^2} \cos t \, dt$. Note that $F(x) = (f \circ g)(x)$ where 
$g(x) = x^2 $ and $f(x) = \displaystyle \int_1^x \cos t \, dt$.  Use the chain rule to compute 
$\displaystyle \frac{d}{dx}F(x)$.
\end{problem} 
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\begin{problem}[4.]  
Given that $f$ is a continuous function, let 
\[	F(x) = \int_0^x xf(t) \ dt.	\]
Find $F'(x)$.	 \emph{Hint:}  The answer is not $xf(x)$.
\end{problem}
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\begin{problem}[5.]
Evaluate without doing any algebraic computations.  (Consider any relevant symmetry or 
geometry related to the function being integrated or the region over which we are integrating.)
\begin{enumerate}
	 \item $\displaystyle \int_{-1}^1 x^3 \sqrt{1-x^2} \, dx$
	 \item $\displaystyle \int_{-1}^1 (x^5 + 3) \sqrt{1-x^2} \, dx$
\end{enumerate}
\end{problem} 
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\begin{problem}[6.]
 Suppose $u$ and $v$ are differentiable and $f$ is continuous.  Show that
\[
	\frac{d}{dx} \left ( \int_{u(x)}^{v(x)} f(t)\,dt \right ) = f(v(x))v'(x) - f(u(x))u'(x)
\]
\emph{Hint:} Break the integral up into  $\int_{u(x)}^{c} f(t) \ dt + \int_{c}^{v(x)} f(t) \ dt$.
\end{problem}
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