Main Calculus Theorems

In order to fully grasp calculus, one must first have a strong understanding of the many calculus theorems. It is not only important to commit these formulas and theorems to memory, but also to have some knowledge on how the theorems came to be and why they make sense. In my calculus class we make an effort to practice these theorems as much as possible and base our homework and practice AP problems around them. We also take cumulative tests on the theorems that we have learned since the beginning of the school year. I have found this very helpful and recommend having a distinct place in your calculus binder where you can reference all of your theorems.

Concepts I Have Learned Thus Far:

*A function is even if f(-x)=f(x) and will be symmmetric with the y-axis.
*A function is odd if f(-x)=-f(x) and will be symmmetric with respect to the origin.

*lim x-->0 sin(x) ⁄ x = 1

*A function f(x) has a limit as x approaches "a" if and only if the right hand limit and the left hand limit at "a" exist and are equal. when: lim x--> a+ f(x)= L and lim x--> a-f(x)= L, Then lim x--> a f(x)= L

*Continuity: If c is a point in the domain of (a,b), then the function is continuous at x=c if and only if lim x--> c+ f(x)= lim x--> c- f(x)= f(c)

*A function is continuous on an interval if and only if it is continuous at every point on that interval.

*Intermediate Value Theorem for Continuous Functions: If y=f(x) is continuous on [a,b] then f(x) takes on every output value between f(a) and f(b) at least once.

*Average Rate of Change: y2 - y1 ⁄ x2 - x1

*Difference Quotient: lim h-->0 f(x+h) - f(x) ⁄ h

*Intermediate Value Theorem for Derivatives: If f is differentiable on [a,b], then f | (x) takes on all values between f | (a) and f | (b).

*Product Rule: If y=u*v, then y | = u | (x) * v(x) + v | (x) * u(x)

*Quotient Rule: If y= u(x) ⁄ v(x), then y | = u | (x) * v(x) - v | (x) * u(x) ⁄ [v(x)] 2

*Basic Derivatives:
  • f(x) = sinx, f |(x) = cosx
  • f(x) = cosx, f |(x) = -sinx
  • f(x) = tanx, f |(x) = sec2x
  • f(x) = secx, f |(x) = secx * tanx
  • f(x) = cscx, f |(x) = -cscx * cotx
  • f(x) = cotx, f |(x) = -csc2x

    • *Chain Rule is a method for finding derivatives for composition functions. If f is differentiable at the point u=g(x) and g is differentiable at x, then the composite function f(g(x)) is differentiable at x and dy ⁄ dx = f | (g(x)) * g | (x)

    * Numeric Property Relationship: If f is differentiable at every point on an interval I and df ⁄ dx is never zero on I, then f has an inverse and the inverse is differentiable at every point on the interval f(I). If f(x) meets the above conditions and point (a,b) on f(x) has an IROC of c ⁄ d , then on f -1 (x) must have an IROC of d ⁄ c .

    *Extreme Value Theorem: If a function is continuous on a closed interval [a,b], then the function has BOTH a maximum value and a minimum value on the interval.

    *Critical points are points in the interior of the domain of a function where f |= 0 or f | DNE (Candidates for extreme values)

    *Mean Value Theorem: If f(x) is a continuous function at every point of a closed interval [a,b], and differentiable at every point of its interior (a,b) then there is at least one special point c on the interval (a,b) where IROC =ARC; f | (c) = f(b) - f(a) ⁄ b-a

    *The graph of a differentiable function y=f(x) is concave up on an open interval I if y | is increasing on I (y || > 0), or concave down on an open interval I if y | is decreasing on I (y || < 0).

    *Point of inflection is a special point where a graph has a tangent line and the concavity changes.

    *To have a minimum, the first derivative must change from negative to positive, or the second derivative must be greater than zero.

    *To have a maximum, the first derivative must change from positive to negative, or the second derivative must be less than zero.

    References: http://www.math.csi.cuny.edu/, My Calculus Notes