# AP Calculus Flash Cards

## 75 terms · AP Calculus AB, calculus terms and theorems

### Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

### Alternative Definition of a Derivative

f '(x) is the limit of the following difference quotient as x approaches c

### Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

### Critical Number

If f'(c)=0 or does not exist, and c is in the domain of f, then c is a critical number. (Derivative is 0 or undefined)

### Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

### Mean Value Theorem

The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.

### Combo Test for local extrema

If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c.
If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.

### Fundamental Theorem of Calculus #1

The definite integral of a rate of change is the total change in the original function.

### -ln(cosx)+C = ln(secx)+C

hint: tanu = sinu/cosu

### Formula for Disk Method

Axis of rotation is a boundary of the region.

### Formula for Washer Method

Axis of rotation is not a boundary of the region.

D: (-∞,+∞)
R: (-∞,+∞)

D: (-∞,+∞)
R: (o,+∞)

D: (-∞,+∞)
R: (-∞,+∞)

### Reciprocal function

D: (-∞,+∞) x can't be zero
R: (-∞,+∞) y can't be zero

D: (0,+∞)
R: (0,+∞)

D: (-∞,+∞)
R: (0,+∞)

D: (0,+∞)
R: (-∞,+∞)

D: (-∞,+∞)
R: [-1,1]

D: (-∞,+∞)
R: [-1,1]

D: (-∞,+∞)
R: [0,+∞)

D: (-∞,+∞)
R: (-∞,+∞)

D: (-∞,+∞)
R: (0, 1)

### Given f(x): Is f continuous @ C Is f' continuous @ C

Yes lim+=lim-=f(c)
No, f'(c) doesn't exist because of cusp

### Given f'(x): Is f continuous @ c? Is there an inflection point on f @ C?

This is a graph of f'(x). Since f'(C) exists, differentiability implies continuouity, so Yes.
Yes f' decreases on X<C so f''<0
f' increases on X>C so f''>0
A point of inflection happens on a sign change at f''