Limits & Continuity
A function has a limit as x --> c. What must be true?
The left-hand and right-hand limits must be equal.
What does a derivative represent?
The slope of the tangent line
What does a critical point tell you?
It is a point where f'(x)=0 or f'(x) does not exist and may indicate a local extremum.
What is the true meaning of a definite integral?
Net accumulation (area under a curve)
What does the Mean Value Theorem guarantee?
At some point, the instantaneous rate of change equals the average rate of change.
A function is continuous at x=c. Name the three conditions that must be satisfied.
If f'(x)>0 on an interval, what does this tell you about f(x)?
The function is increasing on that interval.
TRUE OR FALSE: If f and g are differentiable functions of x and h(x)=f(g(x)), then h'(x)=f'(g(x))g'(x)
TRUE
If f(x) is negative on an interval, what effect does that have on the value of a definite integral?
It contributes negatively to the integral.
What condition must be true for the Mean Value Theorem to apply?
The function must be continuous on [a,b] and differentiable on (a,b).
If a function has a jump discontinuity at x=4, can it be made continuous by redefining f(4)?
No
If f'(c)=0, does that guarantee a local maximum or minimum at x=c?
No. It could be a local max, local min, or neither (like a flat point or inflection point).
A particle’s velocity changes from negative to positive. What happens to its motion?
It changes direction from moving left/backward to moving right/forward.
According to the FTC, what is the relationship between differentiation and integration?
They are inverse operations.
If a function satisfies Rolle’s Theorem conditions, what must exist inside the interval?
At least one point where f'(x)=0
Can a function be discontinuous at a point and still be differentiable there? Why or why not?
No. Differentiability implies continuity, so a function must be continuous before it can be differentiable.
A function is continuous everywhere but not differentiable at x=3. Describe two graph features that could cause this.
A corner/cusp or a vertical tangent.
A function has a local maximum at x=c. What must be true about the sign of f'(x) around c?
f'(x) changes from positive to negative.
What is the summation notation for both a left and right Riemann Sum?
right: n sum k=1 f(xk )deltax
left: n-1 sum k=0 f(xk)deltax
A function is continuous on [a,b], differentiable on (a,b), and f(a)=f(b). What does Rolle’s Theorem guarantee, and what is the strongest possible conclusion about the graph?
At least one c in(a,b) where f'(c)=0; the graph must have a flat tangent somewhere inside the interval (a turning point or stationary inflection).