MATH2019 PROBLEM 1-PARTIAL DIFFERENTIATION, MULTIVARIABLE TAYLOR SERIES AND LEIBNIZ’ RULE Solution

EXAMPLES1
PARTIAL DIFFERENTIATION, MULTIVARIABLE TAYLOR SERIES
AND LEIBNIZ’ RULE
−x2+y2 and x = r cosθ, y = r sinθ. Calculate ∂f and evaluate ∂f when 1. Given f(x,y) = e
∂θ ∂θ
x = 1, y = 0.
1998 2. For what values of n does
f(x,y,z) = sin(3x)cos(4y)e−nz
satisfy the Lapace equation = 0?
1997 3. Let f and g be twice-differentiable functions of a single variable. Show by direct substitution into the partial differential equation that
w(x,t) = f(x + t) + g(x t) −
is a solution of the wave equation
∂2w ∂2w
2 = 2 .
∂t ∂x
4. Show that if w = f(u,v) satisfies the Laplace equation

and if and v = xy then w satisfies the Laplace equation
.
Multivariable Taylor Series

1994 5. Calculate the Taylor series expansion up to and including second order terms of the function
z = F(x,y) = lnxcosy ,
about the point (1,π/4). Use your result to estimate F(1.1,π/4).
2011, S1
6. Expand f(x,y) = ey sinx about (0,1) up to and including second-order terms, using Taylor series for functions of two variables.
2014, S1 8. A cone with radius r and perpendicular height h has volume
Determine the maximum error in calculating V given that r = 4 cm and h = 3 cm to the nearest millimetre.
2014, S2 9. The pressure P of a gas in a reactor is given by
P = rρT,
where ρ is the density, T is the temperature, and r is a constant. If the pressure in the reactor decreases by 5% and the temperature increases by 7%, what is the percentage change in the density of the gas inside the reactor? [Note that you do not need to know the value of r.]
,
V = πr2h,
where r is the radius of the cylinder and h is the height of the cylinder. Small variations in the manufacturing process can result in errors in the cylinder radius of 1% and the cylinder height of 2%. What is the maximum percentage error in the volume of the cylinder?
by V = πr2h. Use a linear approximation to estimate the maximum percentage error in calculating V given that r = 30 metres and h = 20 metres, to the nearest metre.
Leibniz’ Rule for Differentiation of Integrals

given that
.
2014, S1 14. Use Leibniz’ rule to find

2014, S2 15. Use Leibniz’ rule to find
given that
.
2015, S1 16. Use Leibniz’ rule to calculate


.
Use Leibniz’ rule to evaluate

.
Use Leibniz’ rule to evaluate

,
with initial condition . A student solves this ordinary differential equation and writes the solution in an integral form, i.e.,

i) Verify that this function y satisfies the initial condition.
ii) Use Leibniz’ rule to verify that y satisfies the differential equation.
.
Use Leibniz’ rule to find the following integral in terms of α
∞ 1
2 + x2)2 dx. (α
0