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Math 151-copyright Joe Kahlig, 13C
Section 3.11: Differentials; Linear and Quadratic Approximations
Definition 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 the values of x = 2 and dx = 0.3.
y = x3 + 2x + 7
Example: Find dy and evaluate dy for the values of x = 1 and dx = 0.4.
y=
√
x2 + 3
f(x)
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Math 151-copyright Joe Kahlig, 13C
Example: Use differentials to estimate
√
4
16.1.
Example: Use differentials to estimate cos(59o )
Example: The edge of a cube is measured to be 20inches with an error of ±0.1 inches. What is the
error in the volume?
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Math 151-copyright Joe Kahlig, 13C
Linear Approximation
Definition The formula L(x) = f (a)+f ′ (a)(x−a) is called the linear approximation or linearization of f (x) at x = a.
Example: Find the linearization at a = 16 for y =
Example: Use y =
x.
√
x + 7 to answer these questions.
A) Find the linearization at a = 2
B) Evaluate
√
4
√
√
9.06 and 11
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Math 151-copyright Joe Kahlig, 13C
Definition The quadratic approximation at x = a for f (x) is
P (x) = f (a) + f ′ (a)(x − a) +
f ′′ (a)
(x − a)2
2!
Example: Find the quadratic approximation of f (x) = sin(x) at a = 0.
Example: Find the quadratic approximation of y = x4 + 3x3 + 7x2 + 5 at a = 1.