$3\mathrm{.}2$ A function $z=f(x$;$t)=e^{-{}^2{}^{2}t}sin(x)$

where, $z$ denotes temperature at $x$ position and $t$ time is said to satisfy the heat equation. If ${}^2{f}_{}=f_t,$ where $$ is a constant, show that : $f(x$;$t)=e^{-{}^2{}^{2}t}sin(x)$ satisfies the heat equation.

OUESTION $4[15]$

$4\mathrm{.}1$ Use partial derivatives to find $\frac{dy}{dx}$ if $sinhx+cosy=e^{xy}$

$4\mathrm{.}2$ To calculate the moment of inertia of a solid rectangular section, the following formula is used: $I=\frac{b{d}^3}{12},$ where $b$ is the breadth and $d$ is the depth of the section. If the possible error in the measurement of $d$ is $+-2\%$ and the possible error in the measurement of I is $+-3\%,$ determine the maximum possible error in $b$ by making use of the total deferential.

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EMA$21$P$1,$ MAT$21$P$1,$ MAT$25$P$1$ SUPPLEMENTARY EXAM JULY $2022$

$4\mathrm{.}3$ Use a tree diagram to write the chain rule formula for $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}$

if : $u=f(v),v=g(w,x,y),w=h(t),x=f(t,z)$ and $y=q(t,z)$