- Calculus Lab
- On Thursday you'll have a short (ungraded, but very important) test, which we'll use to predict your success in this course (based on your preparation).
- You have a new assignment:
- Go to on-line homework website, IMathAS (http://www.norsemathology.org/homework), log-in and try the Homework Problems for section 2.1 (due next Tuesday night, at 11:59!)
- Read section 2.2 for next time.
- Problems to hand in (next Thursday, 1/21): pp. 48--, #17, 20, 21, 22
Functions
If you don't understand this concept, you're toast.... Don't be toast!
- Access
- Sample homework problems
- Three Problems with One Solution:
- velocity
- slope
- rate
- Calculus consists of two parts, the derivative calculus and the integral calculus. Both parts were developed to solve problems of interest to mathematicians and natural scientists (usually the same people during the 16th century.)
- Derivative Calculus
- Finds the slope of a curve at any point
- If the position of an object is known (and possibly quite complicated), the derivative finds the velocity of the object.
- The (second) derivative is related to the force on an object. This makes it easy to set up an equation - a differential equation - whose solution is a function giving the motion of an object under various forces. This is the topic of mat 325.
- Finds the marginal of a function (used extensively in economics.)
- Finds the rate of change of almost any function y=f(x). That is, it tells you how fast y changes when x is changed.
- Integral Calculus
- Finds the area of odd-shaped regions.
- Given the velocity of an object, it finds the position of the object.
- Given the rate of change of a quantity, it finds the total amount by which the quantity changes over time.
- Is used to calculate masses, volumes, times, etc. of objects.
- Derivative Calculus
- While these two operations constitute the greatest part of calculus, there are several other important things to learn.
- Limits - this is the basic concept on which all the rest of calculus is built.
- Fundamental theorem - this establishes a relationship between the differential calculus and the integral calculus. This relationship is inverse - like addition and subtraction, or squares and square roots.
- Chain Rule - this is one of the formal 'rules' of calculus. But it also encompasses important ideas which occur and reoccur throughout calculus.
- Reading of the historical perspective, p. 42
- Informal limits and functions whose graphs are lines
- Linear functions
- Affine functions
- Rate of change:
- velocity as a rate of change of position with respect to time
- captured in the famous "dirt" formula:
, or rather
- Important note (sidebar of p. 42): speed and velocity are different.
- Average velocity is an approximation to the instantaneous velocity, given as a divided difference:
- Exercise #1, p. 47.
- A secant line demo, showing how the secant line tends to a tangent line (which illustrates the geometric idea of an average velocity turning into an instantaneous velocity).
- Exercise #8, p. 47.
- Unfortunately, with real data we often can't let points get arbitrarily close. So we're stuck with the averages.
- Case study: The Keeling curve, and data
- Keeling Curve
- How would you go about getting that "Annual Cycle" plot?
- Here's the data
- How can we apply the idea of rate of change to this data set?
