Calculus I Review
Keep
in mind that you should understand key concepts in a variety of ways:
- graphically (draw a picture
which explains it)
- algebraically (write a
formula for it)
- numerically (compute a value
for it)
- in words (write a description
of it)
Chapter 2 – Limits, Average and
Instantaneous Speed
- Limits
- Average
speed on an interval [a,b]
- Instantaneous
speed at a particular time t = a
- Definition
of the derivative of a function at a point
- Methods for estimating
the derivative of a function at a point
- The derivative function
- Practical
interpretation of derivatives (including units!)
- Second derivatives
(relationships between function, first derivative and second derivative)
- Continuity
- Differentiability
- Local linearity and
tangent line approximations
Other odds & ends
- Interpretation and estimation
of second derivatives
- Derivatives of inverse
functions
- Using log plots to fit data
for exponential and power functions
Some other problems on limits,
continuity and differentiability
- Let f(x) = sin(1/x) if x is
not 0, and 0 if x is 0.
Let g(x) = x sin(1/x) if x is not 0, and 0 if x is 0.
Let h(x) = x2 sin(1/x) if x is not 0, and 0 if x is 0.
Find the limit as x tends toward 0 of each function. [DNE, 0, 0]
Determine whether or not each function is continuous at x=0. [no, yes,
yes]
Find the derivative of each function at x=0, if it exists. [DNE, DNE, 0]
Chapter 3 - Shortcuts for
differentiation
- Rules for constant multiple,
sum, difference
- Derivative of xn
- Derivative of exponentials (ax,
in particular, ex)
- Product and Quotient Rules
- Chain Rule