Musterklausur 19

Give the domain of $f(x)$ and examine the function for zeros, poles and extreme values.

Examine the behavior of $f(x)$ for $x → +∞$ and $x → ∞$ with the rules of l'Hospital.

Sketch the graph of the function on the interval [−2,4] (rough sketch).

Calculate the Taylor polynomial of the 2nd order with respect to the point $x_0 = 3$.

Show that for $x > 1$ the function $F(x) = \frac{4}{x-1} + 4 ln(x − 1)$ is an antiderivative of $f(x)$ and determine $\int_2^{e^5+1} f(x)dx$.

Determine the interval [0,T], in which $R(t) ≥ 0$ for $t = [0,T]$ applies.

Determine the continuous interest rate $p$ equivalent to $p$.

Determine the present value of the payment stream, which flows from $t = 0$ to $t = T$, where $T$ is the size determined under (a).

Show that the point $(4, ½)$ lies on the isoquant to the level $z = 6$.

Show by implicit differentiation that $h'(x) = \frac{-2xy^2 - y}{2x^2y + x}$ and calculate $h'(4)$.

Calculate the tangent that lies on the graph of the isoquant to the level $z = 6$ at the point $(4, 1)$.

Determine the points where $f$ takes on local extrema using the method of Lagrange multipliers, where $x, y$ must satisfy the constraint $2x - y = 2$. (The proof of the sufficient condition is omitted.)

By how much does the optimal objective function value change approximately if the right-hand side of the constraint in (a) is reduced from 2 to 1.8 by 0.2? (The Lagrange multiplier is to be used!)

Determine the gradient of $f(x, y)$ at the point $(xº, yº) = (0,3)$.

Determine the directional derivative of $f(x, y)$ at the point $(xº, yº) = (0,3)$ for the direction vector $r = \begin{pmatrix} 2 \ 1 \end{pmatrix}$.

Determine the total differential $f(x, y)$ at the point $(xº, yº) = (0,3)$ for $dx = 0, 1$ and $dy = −0, 01$.

Determine the partial elasticity $e_{f,y} (0,3)$ of $ƒ$ with respect to $y$ at the point $(x, y) = (0,3)$.

Determine the points where $f(x, y)$ has local extrema or saddle points, and determine what is the case.

By $0 = f(x, y) – 1$ in a sufficiently small neighborhood of $x = 0$ a function $y = h(x)$ is implicitly defined, for which $h(0) = 3$ applies. Determine the derivative $h'(0)$.

Determine the solution of (1) for the initial conditions $y_0 = 10$ and $y_1 = 17$.

For the general solution $y_t$ of (1) determine the limit $lim_{t→∞} y_t$.