Content for the First Exam
The Squeeze Theorem
The Squeeze theorem allows us to exchange difficult functions for easy functions.
Could it be anything?
Two young mathematicians investigate the arithmetic of large and small numbers.
Limits of the form zero over zero
We want to evaluate limits where the Limit Laws do not directly apply.
We explore functions that ‘‘shoot to infinity’’ at certain points in their domain.
We explore functions that behave like horizontal lines as the input grows without bound.
Content for the Second Exam
The definition of the derivative
We compute the instantaneous growth rate by computing the limit of average growth rates.
The derivative as a function
Here we study the derivative of a function, as a function, in its own right.
Differentiability implies continuity
We see that if a function is differentiable at a point, then it must be continuous at that point.
The derivative of the natural exponential function
We derive the derivative of the natural exponential function.
Derivatives of products are tricky
Two young mathematicians discuss derivatives of products and products of derivatives.
The Product rule and quotient rule
Here we compute derivatives of products and quotients of functions
Derivatives of trigonometric functions
We use the chain rule to unleash the derivatives of the trigonometric functions.
Rates of rates
Two young mathematicians look at graph of a function, its first derivative, and its second derivative.
Position, velocity, and acceleration
Here we discuss how position, velocity, and acceleration relate to higher derivatives.
In this section we differentiate equations without expressing them in terms of a single variable.
Derivatives of inverse exponential functions
We derive the derivatives of inverse exponential functions using implicit differentiation.
Derivatives of inverse trigonometric functions
We derive the derivatives of inverse trigonometric functions using implicit differentiation.
The Inverse Function Theorem
We see the theoretical underpinning of finding the derivative of an inverse function at a point.
Computations for graphing functions
We will give some general guidelines for sketching the plot of a function.
Content for the Third Exam
We use a method called ‘‘linear approximation’’ to estimate the value of a (complicated) function at a given point.
Approximating area with rectangles
We introduce the basic idea of using rectangles to approximate the area under a curve.
Relating velocity and position, antiderivatives and areas
We see a connection between approximating antiderivatives and approximating areas.
The First Fundamental Theorem of Calculus
The rate that accumulated area under a curve grows is described identically by that curve.
The Second Fundamental Theorem of Calculus
The accumulation of a rate is given by the change in the amount.
Additional content for the Final Exam
The idea of substitution
We learn a new technique, called substitution, to help us solve problems involving integration.