Review for MAT210 Test 2 

Covers sections 11.2-12.3

Know the product rule:  

Know the quotient rule:

Know the chain rule:  

General power rule: 

Know that if you have an equation with "x's" and "y's" and you can't get "y" by itself then you have to do Implicit Differentiation to find dy/dx. 

Know how to do implicit differentiation: 

  1) Take d/dx of both sides of the equation.  Remember that   

      all the x-derivatives are regular derivatives and that the y-derivatives  

      must be multiplied by y' or dy/dx. 

  2) Get all the y' terms on one side and everything else on the other. 

  3) Factor out y' and divide both sides by the other factor. 

Know the equation for a tangent line for the function f(x) at x=a: 

   y = mx + b, where m = f '(a) and you can solve for "b" by plugging in x = a, y = f(a), m = f '(a) 

    or  

    y - y₁ = m(x - x₁), where x₁=a and y₁=f(a)

Know the derivatives for exponentials and logarithms: 

• Know that logarithmic differentiation is used to find dy/dx when y = f(x) and "x" is in both the base and the exponent.

• Know how to find dy/dx with logarithmic differentiation

   1) Take "ln" of both sides. 

   2) Simplify using the power rule:  

  3) Find dy/dx by taking "d/dx" of both sides. 

  4) Replace y with f(x). 

• Know how to find absolute min.'s and max.'s of y = f(x) on a closed interval [a,b]:

   1) Find the critical numbers of y = f(x) by solving for x's in the domain such that f '(x)=0 and f '(x) DNE. 

   2) Evaluate f(a) = ___ 

                      f(b) = ___ 

                      f(c) = ___   for all critical numbers, c, from step (1) on the interval [a,b] 

 3) From step (2), largest = absolute max. 

                             smallest = absolute min. 

• Know how to find absolute max.'s and min.'s of y = f(x) on an open interval (or half-open interval) by finding the critical numbers of f(x) and applying the first or second derivative test.

• Know that if f '(x) > 0, then f(x) is increasing.  If f ' (x) < 0, then f(x) is decreasing

• Know the 1st derivative test:

     +              -                             -              +            

<----------|--------->                                     <----------|---------> 

                c (critical number)                                            c (critical number) 

means f(c) is a relative maximum             means f(c) is a relative minimum 

If f '(x) does not change sign at x = c, then there is neither a relative max. nor a rel. min. 

• Know that:

         (1) If f ''(x) > 0, then f(x) is CU (concave up) 

         (2) If f ''(x) < 0, then f(x) is CD (concave down) 

        (3) If f ''(x) changes sign at x = c and f(x) is continuous at x = c, then (c, f(c)) is an inflection point. 

• Know the 2nd derivative test:

         (1) If f ''(c) > 0 and c is a critical number, then f(c) is an relative minimum. 

         (2) If f ''(c) < 0 and c is a critical number, then f(c) is an relative maximum. 

• Be able to do word problems involving one given equation and one function to optimize by putting the function in terms of 1 variable, finding the critical numbers, and applying the first or second derivative test.