Calculus Chapter 2

Rates of Change

100

The slope of the curve f(x) at x = c is given by this

f'(c)

Instantaneous rate of change at x = c

y'(c)

The derivative of f(x) evaluated at x = c

200

Graphically, the average rate of change of f(x) on [a,b] represents this

slope of the secant line through (a, f(a)) and (b, f(b))

300

Write down the directions that ask for this work:

[f(4)-f(2)]/(4-2)

Find the average rate of change of f(x) on [2,4].

400

Graphically, f'(c) represents this

The slope of the tangent line at x = c

Derivatives

100

d/dx[f(g(x))] = f'(g(x)) * g'(x)

Chain Rule

200

The derivative of y = 3/x²

y' = -6/x³

300

d/dx[cos(3x⁵)]

-15x⁴sin(3x⁵)

400

If f(x) = tan³(sin(4x² + 7x)), f'(x) equals this

f'(x) = 3(8x + 7)tan²(sin(4x² + 7x))sec²(4x² + 7x)cos(4x² + 7x)

Related Rates

100

Formulas for volume of a cone and volume of a sphere

V = 1/3(pi)r²h and V = 4/3(pi)r³

200

If V(r) gives the volume of a sphere, dV/dt equals this

dV/dt = 4(pi)r²(dr/dt)

300

If the radius of a circle increases at a rate of 4 cm/min, the rate of change of the area of the circle when the radius is 12 cm

dA/dt = 96(pi) cm²/min

400

If an object's velocity and acceleration are both positive or both negative, what can you say about the object's speed?

The object's speed is increasing

500

If F'(x) = -3csc²(3x)e^cot(3x) then F(x) equals this

F(x) = e^cot(3x)

Pos/Vel/Acc

100

Speed

l Velocity l

200

If s(t) gives the position of an object at time t, s''(4) represents this

acceleration of the object at t = 4

300

If s(t) represents the position of an object at time t, then [s(8)-s(3)]/[8-3] represents this

The average velocity of the object over on the interval [3, 8]

400

If an object's position graph is a parabola, its acceleration graph will look like this

A horizontal line

Misc.

100

The definition of the derivative of f(x)

lim h->0 [f(x+h)-f(x)]/h

200

For f(x) to be continuous at x = c, these two statements must be true.

lim x->c f(x) exists

lim x->c f(x) = f(c)

300

Theorem that says if f(x) is continuous on [a,b] and differentiable on (a,b), then f(x) takes on every value between f(a) and f(b) at some point on (a,b).

Intermediate Value Theorem

400

Theorem that says if f(x) <= g(x) <= h(x) and lim x->c f(x) = lim x->c h(x) = L, then lim x-> c g(x) = L

Squeeze Theorem or Sandwich Theorem