High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 2 Textbook > Multivariable Functions

Differentials

Tangent plane & Linear ApproximationPrevious Section
Taylor Series in Two Variables [Coming Soon!]Next Section
Popular Courses
Find My Course
0:00 / 0:00

Differentials

Given a function z=f(x, y)z=f\left(x,\ y\right), if (x,y)\left(x,y\right) changes from (a1, b1)\left(a_1,\ b_1\right) to (a2, b2)\left(a_2,\ b_2\right), then we can calculate the change in z using Δz=f(a2, b2)−f(a1, b1)\Delta z=f\left(a_2,\ b_2\right)-f\left(a_1,\ b_1\right)

This calculation is sometimes more complicated/tedious, so instead, we can use the total differential to approxiate this change in zz using dz=fx(a1,b1) dx+fy(a1,b1)dydz=f_x\left(a_1,b_1\right)\ dx+f_y\left(a_1,b_1\right)dy, where dx=a2−a1dx=a_2-a_1 and dy=b2−b1d_y=b_2-b_1

Example
Given the function z=f(x,y)=x2y+3x−y2z=f\left(x,y\right)=x^2y+3x-y^2, if xx changes from 0 to 0.01 and yy changes from 1 to 0.98, find Δz\Delta z and dzdz.
Change in z:
Δz=f(0.01, 0.98)−f(0, 1)\Delta z=f\left(0.01,\ 0.98\right)-f\left(0,\ 1\right)
Δz=[(0.01)2(0.98)+3(0.01)−(0.98)2]−[(0+0−1)]\Delta z=\left[\left(0.01\right)^2\left(0.98\right)+3\left(0.01\right)-\left(0.98\right)^2\right]-\left[\left(0+0-1\right)\right]
Δz=0.069698\Delta z=0.069698

Total Differential:
dz=(2xy+3)dx+(x2−2y)dydz=\left(2xy+3\right)dx+\left(x^2-2y\right)dy
dz=(2(0)(1)+3)(0.01)+(0−2(1))(−0.02)dz=\left(2\left(0\right)\left(1\right)+3\right)\left(0.01\right)+\left(0-2\left(1\right)\right)\left(-0.02\right)
dz=0.03+0.04=0.07dz=0.03+0.04=0.07

Practice: Differentials

Suppose z = kxmyn for constants k, m, n. Approximate the percent change in z if x changes by r% and y changes by s%.

Practice: Differentials

The escape velocity for a massive body is v=2GMR\displaystyle v = \sqrt{\frac{2GM}R} where M is the mass of the object escaping and R is the distance from the centre of mass of the body to the object, and G is the universal gravitational constant. Estimate the percent change in the escape velocity if the mass of the object increases by 3.2% and the distance to the centre of mass decreases by 0.8%.