a) Evaluate using Cauchy's integral formula \( \int_{c} \frac{5^{2}}{\left(7-z^{2}\right)(x+20} d z \quad \) where \( C:|z|=4 \) b) Prove that the function below satisfy Cauchy integral theorem. \[ f(z)=2 z^{2}+z+5 \text { where } \mathrm{C} \text { is a circle }|x|=5 \text {. } \] c) Evaluate \( \int_{(0,6)}^{(3,7)}\left(-5 y+2 x^{2}\right) d x+(6 x-2 y) d x \) along the parabola \( x=3 t, y=t^{2}+6 \) d) Evaluate \( \$ \frac{e^{4 z}}{(z-1)^{2}} d z \) along \( c:|z|=3 \) by utilizing Cauchy integral formula. e) Show that the function \( U(x, y)=x^{3}-3 x y^{2}+3 x^{2}-3 y^{2}+1 \) is harmonic. Hence or otherwise find \( v \) such that \( f=u+i v \)

Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) A. \( x=\square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around. c) Find the \( y \)-intercept by computing \( f(0) \). (0) \( =\square \) d) Determine the symmetry of the graph. Odd; origin symmetry

Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) c) Find the \( y \)-intercept by computing \( f(0) \). \( f(0)= \) d) Determine the symmetry of the graph. Odd; origin symmetry Even; \( y \)-axis symmetry Neither

\( \lim _ { x \rightarrow 0 } ( \frac { a - \cos ( \underline { 3 } x ) } { x - \sin ( b x ) } ) = ? \)

(2) \( f(x)=x^{3}-10 x^{2}-5 x+20 \) a) \( \int_{4}^{7} f(b)-f(a) \) b) \( 1^{\text {ra }} \) derivada c) Velocidad \( [5,8] \quad V_{m}=\frac{f\left(x_{f}\right)-f\left(x_{i}\right)}{V_{f}-x_{i}} \)

II. Dadas las funcione \( \mathrm{f}(\mathrm{x})=5 \mathrm{x}^{2}+4 \mathrm{x}-1 \) \( \mathrm{~g}(\mathrm{x})=\mathrm{x}^{2}-6 \) Encontra a) \( \operatorname{Lim}_{x \rightarrow 1}\left[\frac{f(x)}{g(x)}\right] \) b) \( \operatorname{Lim}_{x \rightarrow 0}[f(x) \cdot g(x)] \)