Math 151
Section 3.11
Differentials
Differential Let y = f(x), where f is a differentiable function. Then the differential dx is an
independent variable; that is dx can be given the value of any real number. The differential dy is
then defined in terms of dx by the equation dy = f !( x) dx .
Example: Find dy and evaluate dy for x = 2 and dx = 0.3.
y = x 3 + 2x + 7
Example: Find dy and evaluate dy for x = 1 and dx = 0.4.
y = x2 + 3
Graphical Interpretation
Math 151
Example: Use differentials to estimate 4 16.1 .
Example: Use differentials to estimate cos59° .
Example: The edge of a cube is measured to be 20 inches with an error of at most 0.1 inches. What
is the error in the volume?
Math 151
Linear Approximation L( x) = f ( a) + f !( a)( x " a) is the linear approximation or linearization
of f(x) at (x − a).
Example: Find the linearization at a = 16 for f ( x ) = 4 x .
Example: . y = x + 7 .
A. Find the linearization at a = 2.
B. Evaluate
9.06 and 11 .
C. Find the values of x where the approximation is accurate to within 0.5.
Math 151
Quadratic Approximation P ( x) = f ( a) + f !( a)( x " a) +
f !!( a)
2
( x " a)
2
is the quadratic
approximation of f(x) at (x − a).
Example: Find the quadratic approximation at a = 0 for y = sin x .
Example: Find the quadratic approximation at a = 1 for y = x 4 + 3x 3 + 7x 2 + 5 .