Formulas and Tables

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.
Equation of a tangent and a normal
  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.
Local extrema and inflection point
  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
    \(\lim\limits_{x \to c} {\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} = \lim\limits_{x \to c} {\large\frac{{f’\left( x \right)}}{{g’\left( x \right)}}\normalsize},\) \(\text { if }\) \(\lim\limits_{x \to c} f\left( x \right) =\) \(\lim\limits_{x \to c} g\left( x \right) = \left[
    {\begin{array}{*{20}{c}}
    0\\
    \infty
    \end{array}}
    \right..\)