Calculus

cos(3π/2)

0

cos(π/2)

0

sin(π)

0

cos(2π)

1

sin(π/2)

1

cos(5π/3)

1/2

cos(π/3)

1/2

sin(5π/6)

1/2

sin(π/6)

1/2

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.

Cubing function

D: (-∞,+∞) R: (-∞,+∞)

Identity function

D: (-∞,+∞) R: (-∞,+∞)

Logistic function

D: (-∞,+∞) R: (0, 1)

Exponential function

D: (-∞,+∞) R: (0,+∞)

Squaring function

D: (-∞,+∞) R: (o,+∞)

Cosine function

D: (-∞,+∞) R: [-1,1]

Sine function

D: (-∞,+∞) R: [-1,1]

Absolute value function

D: (-∞,+∞) R: [0,+∞)

Reciprocal function

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

Natural log function

D: (0,+∞) R: (-∞,+∞)

Square root function

D: (0,+∞) R: (0,+∞)

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

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.

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.

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).

Fundamental Theorem of Calculus #1

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

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.

Alternative Definition of a Derivative

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

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

hint: tanu = sinu/cosu

cos(π)

−1

sin(3π/2)

−1

cos(2π/3)

−1/2

cos(4π/3)

−1/2

sin(7π/6)

−1/2

cos(3π/4)

−√2/2

cos(5π/4)

−√2/2

sin(5π/4)

−√2/2

cos(5π/6)

−√3/2

cos(7π/6)

−√3/2

sin(4π/3)

−√3/2

cos(7π/4)

√2/2

cos(π/4)

√2/2

sin(3π/4)

√2/2

sin(π/4)

√2/2

cos(11π/6)

√3/2

cos(π/6)

√3/2

sin(2π/3)

√3/2

sin(π/3)

√3/2