Calculus for Economics, Exercises (Dec. 25,2017)
- I
Find the value of the following integrals.
-
(1)
$\int^1_0 e^{-2t}dt$,
(2)
$\int^{\frac {\pi}2}_0 \sin tdt$,
(2)
$\int^{\frac {\pi}2}_0 \sin 2tdt$,
(3)
$\int^2_1 \frac 1{x^3}dx$,
(4)
$\int^8_1 x^{\frac 13}dx$,
(5)
$\int^2_1 (x-1)^4dx$,
(6)
$\int^2_1 \frac 1{2x+1}dx$
- II
Find the value of the following integrals.
-
(1) $\int^{\frac {\pi}2}_0 t\sin tdt$,
(2) $\int^1_{-1} \frac 1{\sqrt{x+2}}dx$,
(3) $\int^1_0 x(x-1)^3dx$,
(4) $\int^6_{0} \left(\frac 13x-1\right)^4dx$,
-
(5) $\int^{-1}_{-3} \frac 1{(2x+1)^3}dx$,
(6) $\int^{1}_{0} (x+1)e^xdx$,
(7) $\int^{1}_{-1} (x+1)^3(x-1)dx$ iby integration by partsj
- III
Find the value of the following integrals.
-
(1) $\int^2_{-1}\frac x{\sqrt{3-x}}dx$,
(2) $\int^1_{0}\frac {x-1}{(2-x)^2}dx$,
(3) $\int^2_{1}x{\sqrt{2-x}}dx$,
(4) $\int^6_{0}\left(\frac x3-1\right)^4dx$,
-
(5) $\int^2_{1}\frac {e^x}{e^x+1}dx$,
(6) $\int^2_{1}\frac {e^x}{(e^x+1)^2}dx$,
(7) $\int^e_{1}\frac {(\log x)^2}{x}dx$,
(8) $\int^1_{0}\sqrt{3-2x}dx$
- IV
- We consider $z=f(x,y)=x+2y$ subject to the constraint
$g(x,y)=1-xy=0$. Find the stationary points and check whether
they are maximal or minimal.
- V
-
We solve the equation
$g(x,y)=x^2-y^2-1=0$ in a neighborhood of $(2,\sqrt 3)$ to get
\begin{equation}
\varphi(x)=\sqrt{x^2-1}
\end{equation}
Find $\varphi''(2)$ by the partial derivative of $g$ of the first and the second order.
- VI
-
We consider $z=f(x,y)=xy$ subject to the constraint
$g(x,y)=x+2y-1=0$. Find the stationary points and check whether
they are maximal or minimal.