The Quantity Z Can Be Expressed as a Function of X

question 33

Essay

The quantity z can be expressed as a function of x and y as follows: z = f(x, y).Now x and y are themselves functions of r and $\theta$ , as follows: $x = g ( r , \theta )$ and $y = h ( r , \theta )$ Suppose you know that g(1, $\pi$ /2)= -1, and h(1, $\pi$ /2)= 1.In addition, you are told that $$\begin{array} { l } \frac { \partial f } { \partial x } ( - 1,1 ) = 1 , \quad \frac { \partial f } { \partial y } ( - 1,1 ) = 6 , \quad \frac { \partial g } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 7 \\\frac { \partial g } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 7 , \quad \frac { \partial h } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 6 , \quad \frac { \partial h } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 4\end{array}$$ Find $\frac { \partial z } { \partial r } ( 1 , \pi / 2 )$

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Definitions:

Antibodies

Proteins produced by the immune system to neutralize or destroy foreign substances like bacteria and viruses.

Immune Function

The activities performed by the immune system to protect the body against disease-causing organisms and substances.

Antigens

Substances that the immune system recognizes as foreign or dangerous, eliciting an immune response, often found on pathogens or produced by them.

Leukocytes

White blood cells, part of the immune system, involved in protecting the body against both infectious disease and foreign invaders.