MATH 221 Midterm 1 Exam Questions with 100% Correct Answers

Secant Line Correct Answer The line connecting any two points on a curve. Tangent Line Correct Answer The line that best approximates a function near a point A. It's the limit of secant lines connecting points A & B as B approaches A. Squeeze Theorem Correct Answer If f(x)≤g(x)≤h(x) and if the top and bottom functions both have the same limits, we can conclude that the middle function must have the same limit. Continuity Correct Answer A function, f, is continuous at x=a if the limit of f(x) as x approaches a exists (not infinity), f(a) is defined, and the limit of f(x) as x approaches a is equal to f(a). Intermediate Value Theorem Correct Answer If f is continuous on [a, b] and N is a number that exists between f(a) and f(b), then there is a c in (a, b) with f(c)=N. Question: The Derivative of f(x)= x*|x| at x=0 is... - 0 - DNE because |x| isn't differentiable at x=0 - DNE because f(x) is defined piecewise - DNE because the left and right limits don't agree Correct Answer The derivative of f(x)=x*|x| at x=0 is 0. Question: Given a function f(x), if f'(a) exists, then the limit of f(x) as x approaches a... -Must exist, but there isn't enough info to determine it exactly. - Equals f'(a) - Equals f(a)