Unit 1
What are two other names for the word "derivative"?
Instantaneous Rate of Change
Slope of Tangent Line
f(x) = 3x5
Find f'(1).
15
What are the conditions needed to prove a function is differentiable on an interval?
1. function is continuous
2. limx-cf'(x) exists
What conditions must be satisfied to use Mean Value Theorem (MVT)?
1. Function is continuous on interval
2. Function is differentiable on interval
Left Riemann sum includes which endpoint in it's equation?
What about Right Riemann sum?
Left Riemann sum includes left endpoint
Right Riemann sum included right endpoint
What is the difference between a secant line and a tangent line?
Secant line goes through two points on the curve.
Tangent line only goes through one
h(x) = g(x)f(x)
Find h'(x).
h'(x) = g(x)f'(x) + g'(x)f(x)
Product rule!
What is a cusp? What information do we know from a cusp?
A cusp is a sharp point on the graph. A function is not differentiable at a cusp point.
What does MVT guarantee?
Some value c between a and b such that
f'(c) = f(b)-f(a)/ (b-a)
How do you find the delta x in Riemann sum problems?
Delta x = (b-a)/# of shapes
How can we tell if a limit exists?
If the right hand limit value and left hand limit value both exist and equal each other
f(x) = sec(x) + ex
What is f'(x)?
f'(x) = sec(x)tan(x) + ex
Note:
[sec(x)]' = sec(x)tan(x)
[tan(x)]' = sec2(x)
If f is increasing, what do we know about f' ?
Hint: use FUN chart
f' is positive
What are the conditions for using Extreme Value Theorem (EVT)?
Function is continuous on closed interval
What is the fundamental theorem of calculus formula?
integral from a to b of f'(x) = f(b)-f(a)
How do we prove a function is continuous at x=c? (3 steps)
1. f(c) exists
2. limx-cf(x) exists
3. limx-cf(x) = f(c)
f(x) = ln(3x)
Find f'(1).
f'(x) = 1/(3x) * 3
f'(1) = 1
If there is a zero at the derivative, what must we have at the original function?
A critical point (max, min, saddle)
When we use EVT, what are we typically solving for?
Absolute max and absolute min
(Global extrema)
What is the derivative of velocity called?
What is the integral of velocity called?
Derivative = acceleration
Integral = position
What condition must be met in order to use the intermediate value theorem (IVT)?
Function is continuous on given interval [a,b]
[arctan(x)]' = ?
Note: arctan(x) = tan-1(x)
1/(1+x2)
If speed is increasing, what do we know about acceleration and velocity?
What if speed is decreasing?
Speed increasing: velocity and acceleration have the same sign
Speed decreasing: velocity and acceleration have different signs
Why can't MVT be applied to f(x) = tan(x) on the interval [-pi,pi]?
Our function is not continuous on this interval. Vertical asymptotes
What is the formula for displacement?
What is the formula for total distance traveled?
Displacement: integral from a to b of velocity
Total distance traveled: integral from a to b of the ABSOLUTE VALUE of velocity (|v|)