This is the process of sketching a curve, based on information about the function. We look at different parts of the function such as its derivative and second derivative.
Step 1. Consider any information given about the type of function. For example if the graph is polynomial or rational. Whatever type of function it is will determine its domain. The domain is very important when sketching the graph.
Step 2. Mark any asymptotes, which are straight lines approached by a given curve as one of the variables in the equation of the curve approaches infinity.
- If the limit of f(x) as x approaches negative infinity or positive infinity is equal to k, then there is a horizontal asymptote of y=k
- If the limit of f(x) as x approaches c is negative infinity or positive infinity, then there is a vertical asymptote of x=c.
Step 3. Plot all given points
- These are any known points on the graph such as x-intercept and y-intercept.
Step 4. Consider first derivative sign chart. The first derivative sign chart tells you where the graph is increasing or decreasing at a certain point. The points where the graph changes from increasing to decreasing is considered a critical value point on the graph.The critical values are where the second derivative is equal to zero. If the first derivative is positive at a point then it is increasing at that point and if the first derivative is negative at a point it is decreasing at that point.
- Mark horizontal tangent lines whenever f'(x)=0
- Mark f'(x) is not defined at a point, determine whether it has a discontinuity, vertical tangent line, or sharp bend and mark according
- Mark all local minimums and maximums. (remember to pay attention to whether it’s a curvy max/min or sharp bend.
Step 5. Consider the second derivative sign chart. The second derivative tells you if the graph is concave up or concave down at a certain point. The second derivative can tell us the inflection points of the graph which are points where the graph changes concavity. If the second derivative is negative at a point it is concave down at that certain point and if the second derivative is positive at a point it is concave up at that certain point. Sketch in the curve one piece at a time taking the intervals from the second derivative sign chart, paying attention to the concavity and also pay attention to the notes you have already marked on the graph.
Step 6. You should be able to sketch the graph.
When the derivative does not exist:
There are 3 reasons the derivative may not exist:
1. Different types of discontinuities
- Point Discontinuities
- Jump Discontinuities
- Vertical Asymptotes
2. Vertical Tangent Lines
3. Sharp Bend
Once you make the sign charts for the first derivative and the second derivative, sketching the graph is easy. You have to sketch the graph one piece at a time. It is important to focus on key points of the graph and not try to drawl the whole graph at once. This will lead to errors and simple mistakes. Sketching graphs is not a speedy process. It takes time, patience, and diligence.
I HOPE EVERYONE HAD AN AWESOME SPRING BREAK!
Have you every thought about where all the symbols come from in math. I have always thought that maybe they were just made up randomly, but I now know that they actually have a reason behind their shape, size , and form. Most symbols in math actually give away their own meaning, which can help you on a test if you are stumped.
The symbol for zero is 0. If you look at the symbol for zero, it is a circle that has nothing in the middle. This could resemble a hole that a zero amount of dirt in it.
The symbol for one is a simple and single little line, 1. There is only one line, like a person standing alone with no one around.
The symbol for infinity is an endless amount of loops, . The symbol reminds me of a track that you could keep following continuously without every have to stop. You can also draw and trace the symbol an infinite amount of times without even picking up the pencil. No matter where you start drawing you are forced to follow loop after loop.
The equal sign is just two lines that are parallel to each other, =. This is perfect because parallel lines always stay an EQUAL distance apart. The originator of the symbol, Robert Recorde, explains that “no two things can be more equal.”
The addition symbol, +, makes sense because it is two single line placed together. When adding you, use the two numbers next to each other and add them together to make one number just like they added two single lines together to make one symbol, a cross.
In calculus the integral symbol is used a lot, ! This symbol was chosen wisely because in mathematics the integral symbol is used to express the most enchanting harmonies of mathematics and in music it is used similarly as the symbols for the f-hole of a violin, also known as a clef. The symbols relate and connect music and math together.
Most of all the other symbols in math come from greek symbols. I am not as familiar with greek symbols but I was thinking maybe I should join a sorority because members are forced to learn the Greek alphabet. ( there is one good thing or perk for being in a sorority, I guess)
I never have tried to figure out the reason behind math symbols before and now find it pretty interesting. If any of y’all know about other symbols just comment on my blog and let me know. Knowing the history behind symbols can be very important in our understanding about math. It can also help up if we are stumped on question and have no idea what a symbol means. We look at the symbols context clues and let the symbol reveal its own meaning to us.
I also wanted to post this video about math symbols I found. It is quit interesting and possibly more helpful and fun than my writing. It talks about the counting numbers I did write about.
The above is a graph of and its first and second derivatives, to show how critical points, points of inflection, and maxima and minima are related among functions and their derivatives.
Here is some background on the relationship between functions and their first and second derivatives; the information is generalizable to any number of derivatives down the line:
Where a function is decreasing, its first derivative will be negative, i.e. below the x axis. Where the function is increasing, its first derivative will be positive, i.e. above the x axis. Where a function hits a plateau, say it’s increasing then hits a plateau then starts decreasing, or vice versa, that’s a maximum or minimum: where that happens, the first derivative will be zero, as it is either changing from positive to negative, or negative to positive, and thus has to cross the x axis.
Next we should look at the concavity of the function, and later learn about the second derivative test, otherwise known as the concavity test. Basically, if the function at a point is resting above the tangent line to that point, like a letter U, that’s concave up. If the function at a point is resting below the tangent line to that point, like an upside-down letter U, that’s concave down.
Where a function changes concavity is called a “point of inflection/point of inflexion.” So for example, think of a function may be increasing, and thus has a positive first derivative; where that increasing function changes concavity and is still increasing, the first derivative will still be positive, but will change direction, and thus have either a maximum or minimum at that point. On the graph above of , the function is decreasing but concave up, but then switches to concave down going from negative x and crossing zero to positive x. Since it is a decreasing function there, the first derivative is negative.
Where the concave was up, the first derivative was negative but increasing; then where the concavity switches to concave down and the function is still decreasing, the first derivative hits a maximum, then starts decreasing, and is still negative. The first derivative continues to be negative until the function switches concavity to concave up, hits a minimum, and starts increasing, which it does at x=3. Now, we know that where a function has a maximum or minimum, its first derivative will be zero. That’s because the tangent line at that maximum or minimum point is flat, horizontal; for any change in x there is no change in y because it is perfectly flat.
The second derivative test, otherwise known as the concavity test, is important: where a function is concave up, its second derivative is positive, and where a function is concave down, its second derivative is negative. If the second derivative is positive, meaning the function is concave up like a U, and the first derivative is zero, that means there is a minimum at that point on the function. If the second derivative is negative, meaning the function is concave down like an upside-down U, and the first derivative is a zero, that means there is a maximum at the point on the function. If the second derivative is equal to zero, and the first derivative is a zero, there may be a maximum, minimum, or point of inflection at that point on the function. See section 8.2 in the book Mathematics for Economists (read the first edition for free on Google Books!) for more information.
Now, where a function has a point of inflection, its first derivative will be a maximum or minimum, but also the value of the derivative may or may not be zero, i.e. it may or may not be a critical point, depending on if the tangent line at that point of inflection is horizontal or not. For more information on points of inflection being critical or non-critical, see footnote 2 on this page in the book Mathematics for Economists and the following information from Wikipedia:
Points of inflection can also be categorised according to whether f’(x) is zero or not zero.
- if f’(x) is not zero, the point is a non-stationary point of inflection
An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.
A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.
While we are looking at graphing, here are some other vital points to help understand graphing polynomials. (See Algebra II for Dummies for a quick, useful reference that brings together a lot of stuff your teacher may have neglected to cover, as usual, in your high school algebra/precalculus classes or that you simply didn’t pay attention to at the time. Lack of a thorough grasp of Algebra II probably accounts for 80% of the difficulty anyone ever has learning calculus, IMHO.)
First, an important thing to remember is that critical points are points that will equal zero in the function’s first derivative; zeros of the function itself are called roots, zeros, solutions, x-intercepts.
For polynomials, the maximum number of roots/zeroes/x-intercepts/solutions the polynomial can have is determined by the highest degree/the highest power of the polynomial: so for an equation like , there could be at most 8 x-intercepts.
Turning points are where the function has a maximum or minimum, changing directions from up to down or vice versa. A polynomial can have one less turning point than its highest degree, so in the equation , there could be at most 7 turning points.
I FOUND THIS ARTICLE AND COPIED IT OVER SO Y’ALL COULD READ IT! HERE IS THE ORIGINAL!
IN CLASS WE HAVE BEEN TALKING ABOUT INCREASING AND DECREASING FUNCTIONS. HERE ARE SOME NOTES THAT I HAVE
When we talk about increasing and decreasing, we mean whether the graph goes up or down as it moves from LEFT to RIGHT.
When a function is increasing at a particular point, the tangent line at that point is increasing. This means that the slop of the tangent line is positive, so the derivative of the function at that point is positive.
When a function is decreasing at a particular point, the tangent line at that point is decreasing. This means that the slope of the tangent line is negative, which means that the derivative at that point is negative.
When trying to find if the graph is increasing or decreasing we have to first find the derivative and set it to zero to find the critical values.
The critical values are x-values in the domain of the function where the derivative of the function is equal to zero or does not exist. Once you find your critical values you can pick points on either side of the critical values to find if the derivative is negative or positive. If the derivative is negative then the function is decreasing between those intervals and if the derivative is positive it is increasing between those intervals.
When a function changes direction from either increasing to decreasing or form decreasing to increasing, we have a turning point.
If the turning point is changing from decreasing to increasing it a relative minimum. If the turning point is changing from increasing to decreasing then it is the relative maximum.
CONCAVITY AND SECOND DERIVATIVE
It is frequently useful to analyze the derivative of the derivative of a function. This is called the second derivative of the function.
If the original functions gives you an objects position, the first derivative gives you velocity. The second derivative gives you acceleration.
If the second derivative of a function is positive at a particular point, then the original function is concave downward a that point.
If the second derivative of a function is negative at a particular point, then the original function is concave downward at that point.
There are inflection points on graphs, which is where the second derivative changes sign. ( Remember, a function can change sign at points where it is zero, or at points where there is a discontinuity)
To find if the graph is concave down or up, you need to find the second derivative and set it equal to zero. This will give you the critical values for the second derivative. Pick numbers to the left and right of the critical values to find if the graph is concave up ro down between those intervals.
This sums up the notes I have on increasing and decreasing functions and concavity and second derivative. We had a test yesterday on some of this, I hope I did good. I am so ready for spring though so I can have time to relax. I wish everyone a wonderful and safe spring break. :]]]
Since this calculus class is Calc for Life Sciences, I figured a lot of y’all have the functional biology test tomorrow too. Here is a clever song to help you remember cell respiration. Don’t forget we also have a caculus test tomorrow! GOOD LUCK TO EVERYONE!