1. The slope of a function at the point of its local or global maxima is zero. Explain why using an example.
The slope of a function is zero at the point of its local or global maxima because of the fact that it is the point where the function is horizontal, thus the slope is really zero. For example, given a function f(x) =-x^2. The first derivative of f is -2x and equating it to zero will yield to solution x=0 which is our candidate for maximum or minimum point. Furthermore, we apply the second derivative test. The second derivative is -2 thus 0 is a local maximum. Accidentally 0 is the only local maximum thus 0 is also the global maximum of the function. At x=0, f(x) =0, which has slope of 0.
2. Show how the derivative of the function f(x) = (2x^4) (3x+2)2 can be obtained with out using the product rule.
We can differentiate the given function 2x^4 (3x+2)2 without using the product rule by just simply distributing (2x^4) to the term (3x+2) giving you 6x^5+ 4x^4. Afterwards, multiply it with the constant 2, thus you have 12x^5+8x^5. Now you can solve the derivative using the simple idea of getting the derivative of function. Hence you have (12)(5)(x^5-1)+(8)(5)(x^4-1) yielding you to 60x^4+40x^3 which is the derivative of the function f.
3. Provide a discussion showing that the limit of the function, f(x) =2x^4 / (x-2) does not exist at x=2.
It is possible that the limit of a given function doest not exist at a particular point. In the problem, to show that the limit of f(x) as x approaches 2 does not exist we need to get the right hand side and left hand side limit of f(x). The right hand side limit of f is positive infinity while the left hand side limit of f is negative infinity. Since they are not equal, we are forced to conclude that the limit of f(x) does not exist.
Reference:
What the Derivative Tells Us About a Function. Retrieved October 12 2007 from http://www.ugrad.math.ubc.ca/coursedoc/math102/keshet.notes/chapter5