Chapter 3 Theory

Explain differentiability.

• If f’(x) exists at a particular x, we say f is differentiable at x.

• If f’(x) exists at every point in the domain of f, we say that f is differentiable.

Explain the tangent line and give its equation in point-slope form.

• The slope of the tangent line is given by the derivative; this is the geometric interpretation.

• Its equation in point-slope form is y - f(a) = f’(a)(x - a)

Explain the relationship between differentiability and continuity.

• If f is differentiable at a point, then it is necessarily continuous at that point.

• If f is not continuous at x = c, then it is not differentiable at x = c.

In what conditions might differentiability fail?

• When f is not continuous at c

• When f has a corner at c (ex. an absolute value function)

• When f has a vertical tangent line (ex. cube root functions)

List the number of days per month of the calendar.

• January: 31

• February: 28

• March: 31

• April: 30

• May: 31

• June: 30

• July: 31

• August: 31

• September: 30

• October: 31

• November: 30

• December: 31