What is the mathematical definition of a derivative?
lim{h→0}( [f(x+h) - f(x)] / h )
How do you write a linear function in point-slope form?
y - y1 = m(x - x1)
How do you write a linear function in slope-intercept form?
y = mx + b
How do you write a linear function in standard form?
ax + by = c
What are the four ways to write a derivative?
f’(a)
ḟ(a)
dy/dx | x=a
df(x)/dx ] x=a
If f(x) is differentiable at some point A, must it always be continuous at that point A?
True
If f(x) is continuous at some point A, must it always be differentiable at that point A?
False
What are the three reasons that could cause f’(a) to not exist?
- f(a) is undefined
- Tangent line at f(a) is perfectly vertical
- More than one tangent line exists at f(a)
What three conditions must be satisfied for f(x) to be continuous at x=c?
- f(c) exists
- lim{x→c}[ f(x) ] exists
- lim{x→c}[ f(x) ] = f(c)
If the endpoint of a function has a defined one-sided limit, does this satisfy the “limit exists” condition for continuity?
True
What are the three types of discontinuities?
Nonremovable Jump
Nonremovable Asymptote
Removable Hole
What are the two limit equations to find a derivative?
f’(a) = lim{h→0}[( f(a+h) - f(a) ) ÷ h ]
f’(a) = lim{x→a}[( f(x) - f(a) ) ÷ (x - a) ]
What is a tangent line?
A line that intersects a curve at a point such that the line’s slope is the same as the curve’s instantaneous slope at that point
What is a normal line?
A line that intersects a curve at a point such that the line’s slope is perpendicular to the curve’s instantaneous slope at that point
If g(x) is continuous at x=a and f(x) is continuous at g(a), what can be said of the composite function f(g(x))?
f(g(x)) is continuous at x=a
What is an explicit function?
A function where the dependent variable is isolated on one side of the equation [ y = f(x) ]
What is an implicit function?
A function where the dependent variable is not isolated on one side of the equation [ f(x, y) = g(x, y) ]
What are “related rates”?
Two distinct rates of change that are both based on the change of a single variable
How do you determine the derivative of one variable given the derivative of another?
- Find the equation that relates the two variables
- Take the derivative of both sides
- Solve the equation for the desired derivative
What is the equation for the line tangent to a function f(x) at the point (a, f(a))? [Also known as a linearization/linear approximation]
y = L(x) = f(a) + f’(a) * (x - a)
dy = ?
dy/dx * dx
f’(x) * dx
f’(a) * dx
a = known input of f(x) that’s close to x
dx = ?
x - a
a = known input of f(x) that’s close to x
f(a + dx) ≈ ?
a = known input of f(x) that’s close to x
f(a) + dy
a = known input of f(x) that’s close to x
