Fun with Functions
It is important that everyone understand functions and their graphs in order to communicate trends or advance beyond basic mathematics. This book has several interactive applets that illustrate functions. It starts with set of common functions where the independent, [math]x[/math], value can be adjusted and arrows show how a dependent, [math]y[/math], value can be obtained. Generally a different function can be chosen by clicking a next button or a user function can be entered. The questions focus on the domain (all possible [math]x[/math] values ) and range ( all possible [math]y[/math] values ). It would help, but is not necessary, that you know the definitions of the basic trigonometric, exponential and logarithmic functions. The applets illustrate basic functions, piecewise functions, graphs that are not functions, addition, subtraction , multiplication and division of functions, shifting and scaling functions, and composite functions. These are all concepts that should be understood for the study of calculus. You could have hours of enjoyment playing with different functions in this book.
Table of Contents
- Functions and Relations
- Introduction to Functions
- Piecewise Functions
- Function Fun, Relations
- Function Fun, Addition
- Function Fun, Shifting
- Function Fun, Composites
- Inverse Function
- Preliminary Calculus Concepts
- Trip to the Big Game
- Average Rate of Change
- Limits
- Sandwich Theoerm
- Formal Limit Definition
- Formal Continuity of Functions
- Intermediate Value Theorem
- Slope of Line
- Lines in a Plane
- The Derivative
- Slope of a Curve
- Derivative as a Function
- Derivative with Function Math
- Linear Approximation
- Chain Rule with Lines
- Chain Rule for Derivative of sin(2 x)
- Identifying Derivatives Quiz
- Derivative of Trigonometric Functions
- Angle Sum Formulas for Sine and Cosine
- Simple Harmonic Motion
- Related Rates
- Light and Shadow
- Separating Ships
- Airplane Angle Related Rate
- Min and Max
- Boundary Min and Max
- Absolute Min and Max
- Mean Value Theorem
- Rolle's Theorem
- Exploring Derivatives of Functions
- Toy Cannon
- Extrema Quiz
- Parametric Functions, Surfaces and Volumes
- Parametric Curve in the Plane
- Function of Two Variables
- 3D Coordinate Systems
- Newton Raphson Method and Taylor Series
- Newton Method
- Error Bounds on Taylor Series
- Integration
- Area Under a Curve
- Riemann Sum Value Locations
- Integral and Area
- First Fundamental Theorem of Calculus
- Simpson's Rule
- Summation of Area ( Riemann Sums)
- Three Dimensions
- Quadric Surfaces
- General Quadric Surface
- Double Integral on Rectangular Region
- General Region Double Integral
Introduction to Functions
This applet illustrates the relation between the independent variable [math](x)[/math] and the dependent variable [math](y)[/math] for functions.[br][br]The drop down list allows you to chose a function to explore. The last function can be edited in the box that appears.[br][br]You can move the dot on the [math]x[/math] axis to change the [math]x[/math] value. The output [math]y[/math] value is shown along with arrows indicating the flow of information.[br][br]The middle mouse or shift mouse allows you to move the graph. The mouse wheel scales the graph. You scale the axes independently with the middle mouse over the axis.
Choose a function from the dropdown box. Choosing the last function in the list will allow you to enter your own function in the entry box. You can use some of the suggestions.[br][br]Are there [math]x[/math] values where the [math]y[/math] value is undefined or does not exist?[br][br]Are there [math]y[/math] values you cannot get with any [math]x[/math] value?[br][br]Are there more than one[math]y[/math] values for any [math]x[/math] values?[br][br]Why is the [math]x[/math] value called the independent variable?[br][br]If the [math]y[/math] value is undefined, then the corresponding [math]x[/math] value cannot be in the [u]domain[/u] of the function.[br]If a [math]y[/math] value cannot be obtained with any [math]x[/math] value ( in the domain ) it is not in the [u]range[/u] of the function.
Trip to the Big Game
This is a graph of a trip to the big game. It starts with a planned route that includes some time at a party before the game starts.[br][br][list=1][br][*] Click the advance button (double right arrow ). To see the average rate for this trip. The planned average speed is distance divided by time or 50 km/hr.[br][/*][*] Unfortunately the traveler did not leave at the planned time and was caught in a traffic jam. So they used a revised route and skipped the party . Click advance[br][/*][*] The traveler was stopped by a police officer. Click advance.[br][/*][*] The traveler showed the officer that their average speed was the same as for the original planned route of 50 km/hr. Click advance. The officer however insisted on using the instantaneous speed that they measured of 200 km/hr and gave the traveler a speeding ticket.[br][/*][/list]
This story is about the difference between average rate or average speed and instantaneous speed. The officer was not interested in the average speed of the traveler. The ticket was based on the instantaneous speed. The officer actually did not measure the instantaneous speed but measured the speed based on a very small distance over a very small time which was very close to the instantaneous speed.
Slope of a Curve
This applet has two graphs. The left graph is of a function that can be modified with the black points. A line between two points on the curve is shown. The horizontal distance of the points from a point on the curve can be controlled with a slider and check boxes. The line can be to the right, to the left or centered on the [math]x[/math] point. The line between two points on a curve is called the secant line, which is not related the secant function.[br]Clicking the "Animate h" button will cause the 'h' value to decrease repeatedly. This will leave a path on the right graph showing how the slope varies with 'h'. Selecting "Show trace" will show all points on this path.
Set the [math]x[/math] location where you would like to get the slope of the curve.[br]How does the slope vary as the distance between points is decreased?[br]What happens when the distance between points is zero?[br]How does slope vary with different combinations of left and right check boxes?[br]Does the value near h=0 vary with different combinations of left and right check boxes?[br]What happens when one of the points on the curve is outside the domain of the function?[br][br]Mathematicians have solved the problem of an undefined slope by defining the slope at a point as the slope of the tangent line which is calculated as the limit as h -> 0 of the slope of the secant line. Checking the tangent line box shows the tangent line. Selecting show trace will show all values of the secant line slope for 0
Angle Sum Formulas for Sine and Cosine
The angle sum formulas for sine and cosine are developed here. Use the advance button at the bottom to show the steps in the development.[br][list=1][br][*] Two adjustable sliders allow you to change the angles in the ranges [math]0^\circ \le \alpha \le 180^\circ [/math] and [math]-180^\circ \le \beta \le 180^\circ [/math] [br][*] Draw a right triangle with [math]\alpha[/math] at the origin[br][*] Draw another right triangle along the hypotenuse of the first triangle so the angles add[br][*] Set a scale to the triangles by setting the hypotenuse of the second triangle to 1[br][*] The adjacent side is then [math]\cos \beta[/math][br][*] The opposite side is then [math]\sin \beta[/math][br][*] Similarly for the first triangle, the adjacent side is then [math]\cos \alpha[/math] times the hypotenuse[br][*] The opposite side is then [math]\sin \alpha[/math] times the hypotenuse[br][*] Draw a third right triangle by extending the opposite side of the first triangle[br][*] Note that the first angle is [math]\alpha[/math][br][*] Calculate the adjacent side length[br][*] Calculate the opposite side length[br][*] Add another right triangle to complete the rectangle[br][*] Note the angle that is [math]\alpha + \beta[/math] because of the opposite angles between parallel lines[br][*] The adjacent side is on top[br][*] The opposite side is on the side[br][/list][br]Note: The angles [math]\beta[/math] and [math]( \alpha +\beta )[/math]change colors if they are negative.
With the side lengths shown it can be seen that[br][math]\sin(\alpha + \beta) = \sin \alpha \:\cos \beta + \cos \alpha \:\sin \beta[/math][br]and[br][math]\cos( \alpha+\beta ) = \cos \alpha \:\cos \beta - \sin \alpha \:\sin \beta[/math].[br][br]Note that [math]\beta[/math] can be negative resulting in the angle difference formulas [math]\sin ( \alpha - \beta) [/math] and [math]\cos ( \alpha - \beta) [/math]. Include the relations [math]\sin\left(-\beta\right)=-\sin\left(\beta\right)\text{ and }\cos\left(-\beta\right)=\cos\left(\beta\right)[/math] from odd and even symmetry respectively.
Light and Shadow
The classic related rate problem of a shadow for a person walking away from a lamppost.[br]The lamp is 6 m tall and the person is 2 m tall.[br]The speed of the person (dx/dt) can be varied ( 1 to 3 recommended range ).[br]The second graph optionally shows the Distance of the person from the lamppost, the length of the Shadow and the distance of the shadow tip from the lamppost.[br]The play button in the lower left animates the time.
Light and Shadow
Derive an equation for the length of the shadow from the distance.[br][br]Take the derivative with respect to time.[br][br]Solve for the rate of increase of the shadow length from the speed of the person.[br][br]Does this equation match what you see in the lower graph?
Boundary Min and Max
Description
This is an illustration of why an open boundary cannot be a minimum or maximum value. The curve [math]f\left(x\right)=x^2,\text{ for }1\le x<3[/math] with a closed boundary at [math]x=1[/math] and an open boundary at [math]x=3[/math]. [br][br]Approaching the closed boundary the [math]y[/math] value decreases until you get to the boundary point where [math]y=1[/math]. Therefore the boundary point is a minimum value.[br][br]Approaching the open boundary, the [math]y[/math] value increases but when you reach the boundary point the [math]y[/math] value is undefined. For any point close to the open boundary there exist a point closer to the boundary with a greater [math]y[/math] value. Therefor you cannot define a point where the [math]y[/math] value is a maximum.
Instructions
Move the orange plus symbol to move the [math]x[/math] value towards 1. Note how the [math]y=f(x)[/math] value changes as you approach and get to [math]x=1[/math]. Left of 1 the [math]y[/math] value is undefined. [br][br]Move the [math]x[/math] value towards 3. Note that when you get to [math]x=3[/math] the [math]y[/math] value is undefined so [math]f(3)[/math] is not a maximum. In order to see points very close to [math]x=3[/math] check the AutoZoom checkbox. This will enlarge the scale as [math]x[/math] approaches 3. Note that within computer accuracy you do not reach a maximum value for [math]y[/math]. The curve continues upwards to the right of the point. ( If you get to close computer roundoff will cause the curve to not be shown. ) Note the coordinates of the top left corner and the bottom right corner of the graph to indicate the scale.
Parametric Curve in the Plane
Description
A parametric curve has the coordinates given as a function of another variable which is often time.[br]As [math]t[/math] changes the position changes. This applet illustrates a two dimensional parametric function where the functions can be edited. The time [math]t[/math] is shown on a slider and the resulting position is shown as a point [br][math](x(t),y(t))[/math][br][br]Controls[br][b]t slider [/b]: shows and adjust the time [math]\left\{\text{time }|0\le t<2\pi\right\}[/math][br][b]x(t) [/b]: the the horizontal position function[br][b]y(t)[/b] : the vertical position function[br][b]Show Velocity [/b]: Will show the velocity vector on the right and left graphs.[br][b]Show Trace [/b]: Will plot a point or the position at each time[br][b]Show Curve [/b]: Will show a curve all of the positions for the time interval and a velocity curve if Show Velocity is on[br][b]Run : [/b]will continuously move the time from 0 to 2[math]\pi[/math], after clicking run a play/pause control will appear which can be used to pause and restart the time motion.[br][b]Show Acceleration[/b] : will replace the velocity curve with an acceleration point and curve[br]If Show Velocity or Show Acceleration are checked numerical values for position,velocity and acceleration in magnitude and angle form.[br]
Basically try the controls and a few different functions. The original function is periodic but what you enter does not need to be periodic. [br][br]With the original functions:[br]What are the maximum values of Position, Velocity and Acceleration?[br]How are these related to the 3 in the [math]x[/math] function?[br][br]Try the position functions x(t)=t/2 and y(t)= 2t-t^2/2.[br]What does this look like?
Newton Method
Derivation of the Method
If you want to find an [math]x[/math] value where a function, [math]f(x)=0[/math] and the derivative can be calculated, Newton's Method is a good approach. This applet graphically steps through what is happening with the method.[br]Step 1 : The function where we want to find the root. In this case the root is [math]\sqrt{2}[/math][br]Step 2 : Solve for the derivative[br]Step 3 : Choose an [math]x[/math] value to start. Here we chose [math]x_0=2[/math]. This is an art with Newton's method.[br]Step 4 : Evaluate the function and the derivative at the [math]x[/math] value. Draw the point [math](x,f(x))[/math].[br]Step 5 : The length of the segment shown is [math]f(x)[/math]. Note: it may be negative so it is not really a length.[br]Step 6 : Using the derivative, draw the tangent line. We will approximate the value of [math]f(x)=0[/math] by finding where this line is [math]0[/math]. [br]Step 7 : The length of the red vector can be calculated from the lines slope.[br]The slope of the line, [math]f'(x)=2x=4[/math], is the rise over the run or [math]f'(x)=\frac{\text{rise}}{\text{run}}=\frac{f\left(x\right)}{\text{- red vector}}[/math] . Solving for the red vector gives [math]\text{red vector = - \frac{f\left(x\right)}{f'\left(x\right)}}[/math] [br]Step 8 : An alternate method is to find the point on the line, [math](x_{next},0)[/math] using the slope of the line through two points. [math]f'\left(x\right)=m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-f\left(x\right)}{x_{next}-x}[/math] . Solving for [math]x_{next}=x-\frac{f\left(x\right)}{f'\left(x\right)}[/math] gives the Newton method. This is the same as adding the red vector to the initial [math]x[/math].[br]Step 9: Looking at the graph of the function it can be seen that the new [math]x[/math] values is close to the desired x value where [math]f(x)=0[/math] .[br]Step 10: Repeat the operation using this new [math]x[/math] value to get a better guess of the desired [math]x[/math] value.
Illustration of the Newton Method
In this applet you can enter a function in the entry box and provide an initial guess of [math]x_0[/math] by moving the point on the x-axis.
Newton Method for Finding Roots of a Function
Activities
For the following functions experiment with different values of the initial guess.[br][math]x^2-2[/math] = x^2 - 2[br][math]\frac{1}{2}+\sin\left(x\right)[/math] = 1/2 + sin(x)[br][math]\frac{x}{2}-\cos x[/math] = x - cos(x)[br][math]\frac{1}{2}-\sqrt[3]{x}[/math] = 1/2 - nroot(x,3)[br]If[ x< 0 , sqrt( -x) , sqrt(x)]
Area Under a Curve
The goal of finding the area under a curve is illustrated with this applet.[br]This applet shows the sum of rectangle areas as the number of rectangles is increased. Different values of the function can be used to set the height of the rectangles. This sum should approximate the area between the function and the x axis. The options for choosing the height of the rectangles are:[br][list][br][*]The upper sum is the sum of rectangles using the maximum value of the curve within the rectangle range.[br][*]The left sum uses the value of the function on the left side of the rectangle.[br][*]The midpoint sum uses the value of the function in the middle of the rectangles range.[br][*]The right sum uses the value of the function on the right side of the rectangle.[br][*]The lower sum is the sum of rectangles using the minimum value of the curve within the rectangle range.[br][/list][br]The number of rectangles can be changed with the slider or animated with the play button.[br]Left,midpoint and right sums can be turned on or off.[br]The right graph shows how the area varies as the number of rectangles is increased. The trace can be cleared with ctrl-F.[br]The bounds of the sum can be adjusted by moving points A and B.[br]The function can be changed by moving the small points on the function.[br]The true area under the curve will be between the upper sum and lower sum.
Area Under a Curve
What happens as the number of rectangles is increased?[br]Are the other sums always between the upper and lower sums?[br]How many rectangles are required to obtain an exact value?[br]What do you think the thin dashed line on the right graph represents?[br]Which sum uses the fewest rectangles to obtain a good approximation of the area?[br]If the number of rectangles approached infinity would any of the sums give different values?
Quadric Surfaces
Description
3 dimensional view of basic quadric surfaces[br][br]Check a box to view a quadric surface and its equation.[br][br]The view can be rotated and translated, see [url=https://wiki.geogebra.orgen/3D_Graphics_View]GeoGebra Manual 3D View[/url] ( scroll down to [size=150]Display of Mathematical Objects[/size])[br][br]The sliders can be used to vary the coefficients. Note: for Hyperbolic Paraboloid [math]c=0.1[/math] works best.