Calculus One

The calculus one topics focus on how to quantify change. They include the following:

  • Average Rate of Change

  • Instantaneous Rate of Change

    • The Limit
    • The Definition of the Derivative
    • Derivative Rules

  • Change Over An Interval

    • The Mean Value Theorem
    • Increasing/Decreasing Over an Interval

  • Total Change

    • Total Change
    • Riemann Sums
    • The Definite Integral
    • The Fundamental Theorem of Calculus

The goals for each day are given below.

Kinematics

Goals for activity 1

Students should be able to do each of the following:

  • Determine if the change in position is positive or negative given
    the velocity.
  • Make rough sketches of the position given piecewise constant
    velocities.
  • Relate how the sign of the velocity impacts the change in position.
  • Given a piecewise linear position determine the velocity of an
    object.
  • Calculate estimates for the change in position given the velocity
    over a given set of time intervals.
  • Sketch a plot of the graph of the position or the velocity given a
    written description of a situation.

Goals for activity 2

Students should be able to do each of the following:

  • Determine the displacement of an object given its velocity.
  • Identify the difference between distance an object travels and the
    object’s displacement from a given position.
  • Determine local minimums and maximums of an object’s displacement
    given the velocity.
  • Identify that local minimum and maximums occur when an object’s
    velocity changes sign.
  • Be able to calculate the average rate of change of a function over a
    given interval.
  • Identify the difference between total change and average rate of
    change.
  • Determine the displacement over a time interval given the average
    rate of change.
  • Determine whether or not displacement is negative or positive given
    the average rate of change.
  • Determine whether or not the average rate of change is positive or
    negative given the displacement.
  • Relate the average rate of change with the slope of the secant line.
  • Determine a sequence of average rates of changes over smaller
    intervals chosen by the student.

Goals for activity 3

Students should be able to do each of the following:

  • Compute the average rate of change of a function over an arbitary
    interval.
  • Construct a sequence of intervals whose width gets close to zero and
    approximate the average rate of change over each interval.
  • State the difference between the total change and average rate of
    change over an interval.
  • Demonstrate that the average rate of change gets close to a specific
    value for a low order polynomial when the width of the interval gets closer to zero.
  • Calculate the instantaneous rate of change of a low order polynomial
    at an arbitrary value of time.
  • Identify the relationship between the instantaneous rate of change
    and the slope of the tangent line.

Goals for activity 4

Students should be able to do each of the following:

  • Determine the derivative of a low order polynomial using the
    definition of the derivative.
  • Determine the derivative of the square root using the definition of
    the derivative and the conjugate of the numerator.
  • Determine the slope of the tangent line of a low order polynomial.

Goals for activity 5

Students should be able to do each of the following:

  • Make a rough sketch of an object’s position given the graph of its velocity.
  • Make a rough sketch of an object’s velocity given the graph of its
    position.
  • Recognize that an object’s position decreases when the velocity is
    negative.
  • Recognize that an object’s position increases then the velocity is
    positive.
  • Recognize that a constant velocity results in a linear position
    graph.
  • Identify local minima and maxima based on the change in sign of the
    velocity.
  • Explain why knowing that the velocity is zero does not necessarily
    imply that the position is at a local minimum or local maximum.
  • Recognize that if you are given the velocity you need to be given
    the position at one point in order to reconstruct the position of an object.

Goals for activity 6

Students should be able to do each of the following:

  • Use the power rool to determine the derivative of a function.
  • Use the poer rule to determine the anti-derivative of a function.
  • Recognize that the anti-derivative cannot be uniquely determined.
  • Determine the anti-derivative of a function given one value of the
    anti-derivative at a pre-determined point. (Solve for the constant that results from determining the anti-derivative.)

Newton’s Second Law

Goals for activity 7

Students should be able to do each of the following:

  • Determine the acceleration of an object given its position.
  • Determine the position of an object given its acceleration, initial
    velocity, and initial position.
  • Make a sketch of the free body diagram of an object when the forces
    are in simple orientations such as along the primary axes.
  • Derive the equation of motion for an object using Newton’s Second
    Law.
  • Determine the displacement of an object given its acceleration and
    initial velocity.

Goals for activity 8

Students should be able to do each of the following:

  • Determine the position of an object given a piecewise defined
    velocity and an initial position.
  • Determine when the local minima or local maxima occur for the position
    function given a velocity function based on where the velocity changes signs.
  • Given a simple constrained maximization problem, make a sketch of
    the constraint.
  • Given a simple constrained maximization problem, make a sketch of
    the family of costs functions.
  • Given a simple constrained maximization problem determine the
    solution graphically by examining the constraint and a family of solutions to the cost function and estimating where the cost function touches the constraint at a single point.

Goals for activity 9

Students should be able to do each of the following:

  • Determine the position of an object given the forces acting on it
    and the necessary initial conditions.
  • Determine the position of an object given the forces acting on it
    and the necessary initial conditions where the forces are piecewise defined, nonconstant functions.

Goals for activity 10

Students should be able to do each of the following:

  • Determine the inverse of a function given the function in tabular,
    graphical, or algebraic form.
  • Derive the product rule for differentiation.
  • Use the product rule to determine the derivative of the product of
    two functions.
  • Use the product rule to determine the derivative of the product of
    more than two functions.

Goals for activity 11

Students should be able to do each of the following:

  • Evaluate the composition of two functions given the functions in
    tabular, graphical, or algebraic forms.
  • Derive the chain rule for differentiation.
  • Use the chain rule to determine the derivative of the composition of
    two functions.
  • Use the chain rule and the product rule to determine the derivative
    of functions that are products and compositions of multiple functions.

Goals for activity 12

Students should be able to do each of the following:

  • Determine the derivative of relationships using implicit
    differentiation.
  • Determine the equation of the tangent line to relationships that are
    not functions using implicit differentiation.

Goals for activity 13

Students should be able to do each of the following:

  • Determine the parameterized path for objects moving in a straight line.
  • Determine the parameterized path for objects moving along a circle.
  • Determine the velocity of an object given its parameterized path as a function of time.
  • Determine a vector tangent to the graph of a parameterized path given the formula for the path as a function of time.

Goals for activity 14

Students should be able to do each of the following:

  • Calculate the factorial of any positive integer.
  • Derive the differential equation for the velocity of an object where the only forces acting on the object are air friction and a possible normal force.
  • Derive the average and instantaneous rates of change for a simple exponential function.
  • Provide a numerical argument as to why the derivative of $e^t$ is $e^t$.
  • Recognize that $e$ is a number and is approximately 2.718.
  • Determine the derivative of a simple exponential function when the base is $e$.
  • Recognize that polynomial functions do not grow as fast as exponential functions with positive rates.
  • Recognize that simple exponential functions can either decay or grow, and correctly draw the basic shape of the graph of a simple exponential function.

Goals for activity 15

Students should be able to do each of the following:

  • Determine the inverse of a simple exponential function.
  • Show that the derivative of \(\ln(t)\) is \(\frac{1}{t}\).
  • Determine the anti-derivative of \(\frac{1}{t}\).
  • Use the chain rule to determine the anti-derivative of \(\frac{f'(t)}{f(t)}\) where \(f(t)\) is a differential function and not zero.
  • Perform basic algebraic manipulations of exponential and logarithmic functions.

Goals for activity 16

Students should be able to do each of the following:

  • Derive the equations of motion for a moving object taking into consideration air friction.
  • Determine a solution to a simple linear differential equation using an appropriate guess for the general form of the solution.
  • Determine a solution to a simple linear differential equation using the method of separation of variables.

Goals for activity 17

Students should be able to do each of the following:

  • Approximate the solution to a given constrained optimization problem graphically.
  • Determine the solution to a given constrained optimization problem analytically.
  • Derive the constraint and cost function for a constrained optimization problem given in a written form.

Goals for activity 18

Students should be able to do each of the following:

  • Sketch the graph of the derivative of the sine and the cosine function given a graph of the original function.
  • Provide an argument that the derivative of the sine function is the cosine function based on the graphs of the functions.
  • Provide an argument that the derivative of the cosine function is the negative sine function based on the graphs of the functions.
  • Provide an argument that the derivative of the sine function is the cosine function based on numerical approximations of the limit definition of the derivative.
  • Provide an argument that the derivative of the cosine function is the negative cosine function based on numerical approximations of the limit definition of the derivative.

Goals for activity 19

Students should be able to do each of the following:

  • Determine vector components in the x and y directions given the magnitude and an angle associated with a vector.
  • Derive the equations of motion for an object when one of the forces is rotating at a constant rate.
  • Determine the anti-derivative of a general sine or cosine function both graphically and analytically.
  • Determine the anti-derivative of a general sine or cosine function.
  • Determine the position of an object given an acceleration with a general since or cosine function and given the initial position and velocity of the object.
  • Determine the derivative of any function that includes powers, exponentials, logarithms, or trigonometric functions using the power rule, product rule, and/or the chain rule.

Work/Energy

Goals for activity 20

Students should be able to do each of the following:

  • Given a physical situation discretize the motion in time by breaking the time up into discrete intervals.
  • Construct a sum that represents an approximation to the motion of an object after discretizing the movement in time.

Goals for activity 21

Students should be able to do each of the following:

  • Construct a Riemann sum to approximate the work done on an object given the force as a function of its position.
  • Determine if an approximation for the work will be high or low given the graph of the force as a function of its position.
  • Use correct notation to express the Riemann sum representing the approximation of the work done on an object given the force as a function of its position for an arbitrary number of intervals.
  • Recognize that increasing the number of intervals for a Riemann sum will result in a better approximation of the work done on an object.

Fundamental Theorem of Calculus

Goals for activity 22

Students should be able to do each of the following:

  • Use Riemann sums to construct an approximation for the signed area under a curve given the function, an interval, and the number of sub-intervals.
  • Determine an approximation for the displacement of an object over a given time interval using Riemann sums given the velocity of the object.
  • Use correct notation to express the Riemann sum used to approximate the displacement of an object given its velocity and an arbitrary number of sub-intervals.

Goals for activity 23

Students should be able to do each of the following:

  • Recognize the relationship between the signed area function and the integral notation.
  • Determine an exact analytic value for a simple definite integral.
  • Use u-substitution to determine the general anti-derivative of a function.

Goals for activity 24

Students should be able to do each of the following:

  • Determine the change in momentum of an object given formulas for the forces as functions of time.
  • Derive the impulse-momentum theorem from Newton’s Second Law.
  • Determine the work done by a force given the force as a function of the object’s position.

Goals for activity 25

Students should be able to do each of the following:

  • Determine the area between to curves.
  • Determine the mass of an two dimensional object given its density and the bounds on the object. (Students should be able to derive integrals in either the horizontal or vertical directions.)

Goals for activity 26

Students should be able to do each of the following:

  • Determine the density of a one dimensional object whose density varies.

  • Determine the first moment of a one dimensional object whose density varies.

  • Determine the center of mass of a one dimensional object whose density varies.

  • Given a two dimensional object with a description of the bounds the

    student should be able to do the following:

    • Sketch a picture of the object.
    • Estimate the center of mass from the graphical depiction.
    • Determine the center of mass analytically.

Goals for activity 27

Students should be able to do each of the following:

  • Draw a free body diagram for an object with multiple forces acting in two dimensions.
  • Determine the differential equations describing the motion of an object with forces acting in two dimensions.
  • Construct an approximation to the solution of a differential equation by dividing the time in to sub-intervals and constructing a Riemann sum to determine the change in momentum of the object.
  • Use the approximation to estimate the work done by a force on an object.

Goals for activity 28

Students should be able to do each of the following:

  • Use the product and quotient rules to derive the derivative formulas for all trigonometric functions based on the derivatives of the sine and cosine functions.
  • Use the derivative rules for other trigonometric functions to determine anti-derivatives. For example, students should be able to determine the anti-derivative of \(\sec^2(t)\).

Goals for activity 29

Students should be able to do each of the following:

  • Determine the position of an object given an acceleration as a function of time and appropriate initial conditions.
  • Evaluate the stability of a solution with respect to growth, decay, and oscillatory behaviour.
  • For solutions that grow in time identify the growth rate based on the dominant term in a solution.
  • For solutions that decay in time identify the decay rate based on the dominant term in a solution.
  • Use Newton’s Second Law to derive the differential equation describing the motion of a spring-mass system without friction.
  • Determine the position of an object that is part of a spring-mass system without friction given appropriate initial conditions.

Goals for activity 30

Students should be able to do each of the following:

  • Use Newton’s Second Law to derive the differential equation describing the motion of a spring-mass system without friction where some parameters for the system are not explicitly defined numerically.
  • Based on a written description of a spring-mass system without friction, determine the cost function and constraint associated with the system.

Goals for activity 31

Students should be able to do each of the following:

  • Recognize that the linearization of a function and the tangent line at the same point are the same functions.
  • Determine the linearization of a given function at a point.
  • Use the linearization of a function at a point to construct an approximation to the original function at a nearby point.
  • Use a coherent strategy to determine which point to use for a linearization based on the context of a written question.
  • Convert an expression into an equivalent root finding problem.
  • Use the linearization to approximate the zero of a function.

Goals for activity 32

Students should be able to do each of the following:

  • Calculate the dot product of two vectors.
  • Derive the formula for the dot product.

Goals for activity 33

Students should be able to do each of the following:

  • Calculate the cross product of two vectors.
  • Derive the formula for the cross product.

Goals for activity 34

Students should be able to do each of the following:

  • Determine the formula for the path of an object moving around a circle of a given radius centered at the origin.
  • Determine the angular velocity of an object moving in a circle given the formula for the path as a function of time.
  • Determine the speed and acceleration of an object moving in a circle given the formula for the path as a function of time.

Goals for activity 35

Students should be able to do each of the following:

  • Determine the center of mass of a one-dimensional rod whose density is given as a function of position along the rod.
  • Determine the moment of inertia of a one-dimensional rod whose density is given as a function of position along the rod.

Goals for activity 36

Students should be able to do each of the following:

  • Determine the tangent vector given a formula for an object’s position that is moving along a circle of a given radius centered at the origin.
  • Determine the general formula for the kinetic energy of an object moving along a circle with a constant angular velocity.
  • Show that the velocity and position vectors for an object moving along a circle with a constant angular velocity are perpendicular.

Goals for activity 37

Students should be able to do each of the following:

  • Show that if an object is moving around a circle with a constant angular velocity then \(\|\vec{a}\|=\frac{\|\vec{v}\|^2}{\|\vec{r}\|}\).
  • Show that if an object is moving around a circle with a constant angular velocity then the acceleration is perpendicular to the velocity vector and parallel to the position vector.
  • Show that if an object is moving around a circle but not at a constant angular velocity then the identity \(\|\vec{a}\|=\frac{\|\vec{v}\|^2}{\|\vec{r}\|}\) is not necessarily true.
  • Determine the angular and radial components of the acceleration for an object that is moving around a circle.

Goals for activity 38

Students should be able to do each of the following:

  • Define a limitation on the domain of a basic trigonometric function that allows an inverse function to be defined on the restricted domain.
  • Determine the derivative of the inverse of a basic trigonometric function using implicit differentiation.
  • Derive the general relationship of the derivative of an inverse function in terms of the original function.