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Calculus

# Basic Applications of the Derivative

Functions: $$f$$, $$g$$, $$y$$
Argument (independent variable): $$x$$
Point coordinates: $${x_0},$$ $${y_0},$$ $${x_1},$$ $${x_2},$$ $${x_3}$$
Real numbers: $$a$$, $$b$$, $$c$$

Position of an object: $$s$$
Velocity: $$v$$
Acceleration: $$w$$
Time: $$t$$

1. Velocity and acceleration
Suppose that the function $$s\left( t \right)$$ describes the position of an object in a coordinate system at time $$t$$. Then the first derivative of the function $$s\left( t \right)$$ is the instantaneous velocity of the object:
$$v = s^{\,\prime} = f'{\left( t \right)}$$
The second derivative of the function $$s\left( t \right)$$ is the instantaneous acceleration of the object:
$$w = v^{\,\prime} = s^{\,\prime\prime} =$$ $$f^{\prime\prime}{\left( t \right)}$$
2. Equation of a tangent
$$y – {y_0} =$$ $$f^\prime {\left( {x_0} \right)} {\left( x – {x_0}\right)},$$
where $$\left( {x_0},{y_0} \right)$$ are the coordinates of the point of tangency, $$f^\prime {\left( {x_0} \right)}$$ is the value of the derivative of the function $$f\left( x \right)$$ at the point of tangency.
1. Equation of a normal
$$y – {y_0} =$$ $$– {\large\frac{1}{{f’\left( {{x_0}} \right)}}\normalsize} \left( {x – {x_0}} \right),$$
where $$\left( {x_0},{y_0} \right)$$ are the coordinates of the point of intersection, $$f^\prime {\left( {x_0} \right)}$$ is the value of the derivative of the function $$f\left( x \right)$$ at this point.
2. Increasing and decreasing functions
If $$f’\left( {{x_0}} \right) \gt 0$$, then the function $$f\left( x \right)$$ is increasing at $${x_0}$$. In the figure below, the function increases at $$x \lt {x_1}$$ and $$x \gt {x_2}$$.
If $$f’\left( {{x_0}} \right) \lt 0$$, then the function $$f\left( x \right)$$ is decreasing at $${x_0}$$ (the interval $$\left.{{x_1} \lt x \lt {x_2}}\right).$$
If $$f’\left( {{x_0}} \right) = 0$$ or the derivative does not exist, then the test fails.
1. Local extrema of a function
A function $$f\left( x \right)$$ has a local maximum at a point $${x_1}$$ if and only if there exists some interval containing $${x_1}$$ such that $$f\left( {{x_1}} \right) \ge f\left( x \right)$$ for all $$x$$ in this interval.
Similarly, a function $$f\left( x \right)$$ has a local minimum at a point $${x_2}$$ if and only if there exists some interval containing $${x_2}$$ such that $$f\left( {{x_2}} \right) \le f\left( x \right)$$ for all $$x$$ in this interval.
2. Critical points
A point $${x_0}$$ is called a critical point of the function $$f\left( x \right)$$ if and only if either $$f’\left( {x_0} \right)$$ is zero or does not exist.
3. First derivative test for local extrema
If $$f\left( x \right)$$ is increasing ($$f’\left( x \right) \gt 0$$) for all $$x$$ in some interval $$\left( {a,{x_1}} \right]$$ and decreasing $$\left({f’\left( x \right) \lt 0}\right)$$ for all $$x$$ in some interval $$\left[ {{x_1},b} \right)$$, then the function $$f\left( x \right)$$ has a local maximum at the point $${x_1}$$.
Similarly, if $$f\left( x \right)$$ is decreasing $$\left({f’\left( x \right) \lt 0}\right)$$ for all $$x$$ in some interval $$\left( {a,{x_2}} \right]$$ and increasing $$\left({f’\left( x \right) \gt 0}\right)$$ for all $$x$$ in some interval $$\left[ {{x_2},b} \right)$$, then the function $$f\left( x \right)$$ has a local minimum at the point $${x_2}$$.
4. Second derivative test for local extrema
If $$f’\left( {{x_1}} \right) = 0$$ and $$f^{\prime\prime}\left( {{x_1}} \right) \lt 0,$$ then the function $$f\left( x \right)$$ attains a local maximum at the point $${x_1}.$$
If $$f’\left( {{x_2}} \right) = 0$$ and $$f^{\prime\prime}\left( {{x_2}} \right) \gt 0,$$ then the function $$f\left( x \right)$$ attains a local minimum at the point $${x_2}.$$
5. Concavity of a function. First derivative test for concavity
A function $$f\left( x \right)$$ is concave upward at a point $${x_0}$$ if and only if the derivative $$f’\left( x \right)$$ is increasing at this point (the interval $$x \lt {x_3}$$ in the figure above).
Similarly, a function $$f\left( x \right)$$ is concave downward at a point $${x_0}$$ if and only if the derivative $$f’\left( x \right)$$ is decreasing at this point (the interval $$x \gt {x_3}$$).
6. Second derivative test for concavity
If $$f^{\prime\prime}\left( {{x_0}} \right) \gt 0$$, then the function $$f\left( x \right)$$ is concave upward at $${x_0}$$.
If $$f^{\prime\prime}\left( {{x_0}} \right) \lt 0$$, then the function $$f\left( x \right)$$ is concave downward at $${x_0}$$.
If $$f^{\prime\prime}\left( {{x_0}} \right) = 0$$ or does not exist at $${x_0},$$ then the test fails at this point.
7. Inflection point
If the first derivative$$f’\left( {x_3} \right)$$ exists at a point $${x_3}$$ and the second derivative $$f^{\prime\prime}\left( {x_3} \right)$$ changes sign at $$x = {x_3}$$, then the point $$\left( {{x_3},f\left( {{x_3}} \right)} \right)$$ is called an inflection point of the graph of the function $$f\left( x \right)$$. If the second derivative $$f^{\prime\prime}\left( {x_3} \right)$$ exists at the inflection point, then $$f^{\prime\prime}\left( {x_3} \right) = 0$$.
8. L’Hopital’s rule
$$\lim\limits_{x \to c} {\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} = \lim\limits_{x \to c} {\large\frac{{f’\left( x \right)}}{{g’\left( x \right)}}\normalsize},$$ $$\text { if }$$ $$\lim\limits_{x \to c} f\left( x \right) =$$ $$\lim\limits_{x \to c} g\left( x \right) = \left[ {\begin{array}{*{20}{c}} 0\\ \infty \end{array}} \right..$$