Please write legibly, answer only the ones you know, you dont have to answer all of them.

Give an example of a function that is defined for all real numbers. is continuous everywhere. but its derivative does not exist at the point x = 3. Consider the function f(x)=[x|. Prove that this function is not differentiable at the point x=O. (HINT: You may want to think about limits from the left and right in the definition of the derivative.) Consider the function {xl=x2+2x . Using the limit definition of the derivative. show that for any real number a. the derivative of this function exists at x=a and is equal to 23+2. Consider the function g(x)={3x2-5x)(2x-3}{2x3+5x-6)(4x5-3x2+2x-8). Find the equation of the line tangent to the graph of this function at the point where x=-1. Show your work. If function fix) is not continuous at the value x = a then the derivative of f does not exist at x = a. Consider the function f{x)=x" where n is a positive integer. Use induction to prove that for any value x=a. f' (a)=n |I(a)]] ("'1) for for all positive integers n=1.2,3