AP Calculus AB - Chapter 2

• change in position/change in time

• delta s/delta t

• [s(t2) - s(t1)]/t2 - t1

• change in y/change in x

• delta y/delta x

• [y(x2) - y(x1)]/x2 - x1

[f(x) - f(c)]/x -c

Average Rate of Change (passes through two points)

[f(b) - f(a)]/b - a

(passes through one point)

[f(a) - f(a)]/a - a = 0/0

The limit of [f(x+h) - f(x)]/h as h→0 is called the derivative of f(x) (f’(x), dy/dx, y’, y’(x), df(x)/dx, d(x^n)/dx), the slope of the tangent line, and/or the instantaneous rate of change.

f’(x) = lim of [f(x+h) - f(x)]/h as h→0 = a function

f’(c) = lim of [f(c+h) - f(c)]/h as h→0 = a number

f’(c) = lim of [f(x) - f(c)]/x - c = a number

d(x^n)/dx = nx^n-1

(d/dx)(uv) = (u)(derivative of v) + (v)(derivative of u)

(d/dx)(u/v) = [(v)(derivative of u) - (u)(derivative of v)]/(v)²

*Based on composite functions

(d/dx)(f(g(x))) = f’(g(x)) * g’(x)

1. Pretend inner function is not there and work on the outer function. Find its derivative

2. Then find derivative of inner function

3. Expand/simplify if needed

Think levels.

1. Determine f(x)

2. Determine f’(x)

3. Determine f’(x) at given point

4. Write equation in point-slope form (y - y1 = m(x - x1)

1. Determine c and type of derivative form (standard, modified, or alternative?)

2. Determine function

3. Determine derivative of function

4. Find derivative at x=c

To find the normal line, rewrite the tangent line except the slope (m) becomes the negative reciprocal.

Derivatives of one-sided limits must exist and be equal to each other, differentiable on closed interval [a,b].

A function is not differentiable at a point when the graph has:

• Sharp turn/cusp

• Vertical tangent line

• Discontinuity

1. Find intervals where function is increasing and/or decreasing

2. Determine slope (is m > or < 0?)

3. Determine horizontal tangent line

4. Determine position of f’(x) (will the slope be above or below the x-axis?)

5. Graph derivative based on identifications

1. Find f’(x)

2. Determine zero(es)

3. Plug x-value(s) into function f(x)

4. Write the point(s)

*Remember that vertical tangent lines are when the slope is undefined

1. Find f’(x)

2. Find x in terms of y (if derivative is a fraction, make denominator = 0 and find x)

3. Plug x into original function and solve for y

4. Plug y into original equation and solve for x

5. Write point

If needed/possible

1. Simplify expression

2. Express everything w/ cosine and sine

3. Factor

4. Rewrite after applying trigonometry identities

1. Continuity Test (equate)

2. Use differentiability to find a

3. Go back to the equated expression and plug in a, solve for b

1. Differentiate the two functions given in piecewise function and equate them

2. Plug in x if given and see if the values are equal to each other. If not, then it’s not differentiable.

When the variables aren’t the same, continue to apply any rules when applicable (Power, Product, Quotient, Chain) and make it explicit. (dy/dx)

1. Plug x into original function to find value(s) for y

2. Use implicit differentiation on the function to find the slope(s)

3. Write the equation of tangent/normal line(s)

1. Find the point, x-value

2. Find the derivative of the function

3. Plug x-value into derivative

4. Find the derivative of the inverse function using reciprocal

g’(x) = 1/f’(g(x))