Content for the First Exam
4.2Linear approximation
We use a method called “linear approximation” to estimate the value of a
(complicated) function at a given point.
4.3Explanation of the product and chain rules
We give explanation for the product rule and chain rule.
5.1What’s the graph look like?
Two young mathematicians discuss how to sketch the graphs of functions.
6.1Wanted: graphing procedure
Two young mathematicians discuss how to sketch the graphs of functions.
6.2Computations for graphing functions
We will give some general guidelines for sketching the plot of a function.
7.2Implicit differentiation
In this section we differentiate equations that contain more than one variable on one
side.
7.3Finding dx dy
In this section we differentiate equations without expressing them in terms of a single
variable.
Content for the Second Exam
10.1Derivatives of inverse trigonometric functions BreakGround
Two young mathematicians think about the plots of functions.
10.2Derivatives of inverse trigonometric functions
We derive the derivatives of inverse trigonometric functions using implicit
differentiation.
10.3The Inverse Function Theorem
We see the theoretical underpinning of finding the derivative of an inverse function at
a point.
13.1A limitless dialogue
Two young mathematicians consider a way to compute limits using derivatives.
14.1Indeterminate mutterings
Two young mathematicians consider a way to compute limits using derivatives.
14.2L’Hopital’s rule for other forms
We use derivatives to give us a “short-cut” for computing limits.
Content for the Third Exam
16.2Approximating area with rectangles
We introduce the basic idea of using rectangles to approximate the area under a
curve.
18.2The definite integral
Definite integrals arise as the limits of Riemann sums, and compute net
areas.
20.2The First Fundamental Theorem of Calculus
The rate that accumulated area under a curve grows is described identically by that
curve.
21.2The Second Fundamental Theorem of Calculus
The accumulation of a rate is given by the change in the amount.
22.1What could it represent?
Two young mathematicians discuss whether integrals are defined properly.