Definition [Linear Approximation]. Linear Approximation of \( \Delta f \) If \( f \) is differentiable at \( x = a \) and \( \Delta x \) is small, then \[ \Delta f \approx f'(a) \Delta x \]
Remark. \(\Delta f \) is the actual change in \( f \) and \( f'(a) \Delta x \) is the estimated change.
\[ \frac{d}{dx} (\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} \]
\[ \frac{d}{dx} (\tan^{-1} x) = \frac{1}{1 + x^2} \]
\[ \frac{d}{dx} (\sec^{-1} x) = \frac{1}{|x| \sqrt{x^2 - 1}} \]
\[ \frac{d}{dx} (\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \]
\[ \frac{d}{dx} (\cot^{-1} x) = -\frac{1}{1 + x^2} \]
\[ \frac{d}{dx} (\csc^{-1} x) = -\frac{1}{|x| \sqrt{x^2 - 1}} \]
To approximate the function \( f \) itself rather than the change \( \Delta f \), we use the linearization \( L(x) \) centered at \( x = a \), defined by \[ L(x) = f'(a)(x - a) + f(a) \] Notice that \( y = L(x) \) is the equation of the tangent line at \( x = a \).
Let \( f \) be a function on an interval \( I \) and let \( a \in I \). We say that \( f(a) \) is the
A continuous function \( f \) on a closed (bounded) interval \( I = [a, b] \) takes on both a minimum and a maximum value on \( I \).
We say that \( f(c) \) is a
A number \( c \) in the domain of \( f \) is called a critical point if either \( f'(c) = 0 \) or \( f'(c) \) does not exist.
If \( f \) has a local maximum or minimum at \( x = c \), then \( c \) is a critical point of \( f \).
Suppose that \( f \) is continuous on an open interval containing \( c \) and that \( f'(x) \) exists for all \( x \) in the interval except possibly at \( x = c \). Then,