Content deleted Content added
rmv non-WP:RS : user generated material / open content platform |
m Reverted Edit(s) by 2409:40e3:2096:fd0c:ecb8:8aff:feb2:e8b5 (Talk) (AV) |
||
| (2 intermediate revisions by 2 users not shown) | |||
Line 11:
This is a good approximation when <math>x</math> is close enough to {{nowrap|<math>a</math>;}} since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the [[tangent line]] to the graph of <math>f</math> at <math>(a,f(a))</math>. For this reason, this process is also called the '''tangent line approximation'''. Linear approximations in this case are further improved when the [[second derivative]] of a, <math> f''(a) </math>, is sufficiently small (close to zero) (i.e., at or near an [[inflection point]]).
If <math>f</math> is [[concave down]] in the interval between <math>x</math> and <math>a</math>, the approximation will be an overestimate (since the derivative is decreasing in that interval). If <math>f</math> is [[concave up]], the approximation will be an underestimate.<ref>{{cite web |title=12.1 Estimating a Function Value Using the Linear Approximation |
Linear approximations for [[vector (geometric)|vector]] functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the [[Jacobian matrix and determinant|Jacobian]] matrix. For example, given a differentiable function <math>f(x, y)</math> with real values, one can approximate <math>f(x, y)</math> for <math>(x, y)</math> close to <math>(a, b)</math> by the formula
| |||