MAC 2311 Complete Course
The Fundamental Theorem of Calculus [Test-3]
*KEY ∫(a->x) [f(x)] STANDS FOR INTEGRAL OF 'f(x)' OVER THE INTERVAL 'a' TO 'x'* We assume that a function 'f' is continuous on the interval [a,b]. 1) Then the function 'g' is defined by: g(x) = ∫(a->x) [f(t) * dt] -g(x) is differentiable on [a,b] AND g'(x) = f(x) 2)If 'F' is an anti-derivative of 'f' then ∫(a->b) [f(t) * dt] = F(b) - F(a)
Implicit Differentiation Steps [Test-2]
1) Differentiate both sides of the 'function' with respect to x [d/dx]. **Note: If there is a 'y' variable on its own in the function it becomes dy/dx** 2)Isolate dy/dx to solve for the implicit derivative. !ITS THAT EASY! **REMEMBER USE ALL CALCULUS RULES (identities, product, & quotient) ALONG WITH ALGEBRA RULES (remember the basics when isolating its easy to get tripped up here)
Related Rate Questions AKA Applicable Differentiation Steps[Test-2]
1) Read the problem and create an illustration. 2.) Write down any formulas that will be needed to prove a relationship. (try to think how can you connect each value given). 3) Express that relationship using the differential operator (what derivatives, or changes, connect to other values). 4) Plug-in the given values and check the relationship makes common sense. 5) Use implicit differentiation to an equation that has the missing information expressed as a variable or implicit derivative. 5) Isolate that missing information. **Note: there will be a question like this on the final** **Problems include: objects in two directions, cone volume, spotlight, & ladders**
Derivative Operator [Test-1]
1) The first derivative operator is a way to shorthand define the derivative. The operator is (dy) / (dx). You read this as: "The derivative of function 'y' in respect to 'x'." 2) The second derivative operator is another shorthand way to define the derivative. The operator is the apostrophe. It is used as f'(x) and this is read as: "The derivate of the function 'f' in respect to 'x'."
Continuity [Test-1]
A function is continuous at x = a if LIM(x->a) f(x) = f(a) or when the left and right limit In other words a function is continuous if there is no value of x that is undefined over a given interval [a,b].
Discontinuity & Types [Test-1]
A function is discontinuous if that function is undefined for a value x in a given interval [a,b]. Types... 1) Removable = when LIM(x->a^-) EQUALS LIM(x->a^+) [but f(x) does not exist] 2) Jump = when LIM(x->a^-) DOESN'T EQUAL LIM(x->a^+) 3) Infinite = when either the left or right limit is equal to positive or negative infinity [LIM(x->a^(+||-)) = +||- Inf.]
Limit [Test-1]
A limit is explained by coming from the left or right. If the limit coming from the left and right are equal then a limit exists.
Anti-Derivatives [Test-3]
An anti-derivative is the "inverse" derivative (think of trig and how the inverse will tell you the angel while the regular function gives you the value). A function F(x) is differentiable for all values of 'x' on a given interval. That functions derivative is f(x). Using the derv. operator F'(x) = f(x). However, when going from f(x)-> F(x) we must remember that the derivative of any constant is 0. So when given f(x) [which is F'(x)] we must add a constant to the end that takes this into account (we use capital C). So when we find F(x) from f(x) our final answer will be F(x) + C.
Limit Riemann Sum Problems [Test-3]
Another Riemann Sums problem type is asking you to convert an equation over a given interval as a limit. In this question types the equation will be some function f(x) and the interval will be some interval [a,b]. In these questions you will almost always be expressing as a Right Riemann Sums. In this case you use the Riemann notation LIM(n->inf.) E [f(xi) * D(x)]. Except they want you to express the equation in this notation. So you must plug in f(xi) into the sum notation. This is done by first plugging in 'xi' into the equation. 'xi' in a right Riemann is 'a + [ i * (b-a) / 2]). To completely extend this you will substitute the f(xi) for: f(a + [ i * (b-a)/2]) **note this is only for a right riemann sums these are the only type we have been asked to put into limit form in previous test, quizzes, and hws**
Speeding Up and Slowing Down Problem Steps [Test 2]
Before you can use these steps you need to have a function f(x) and find f'(x) you must find the critical values for this derivative. These are values 'c' that satisfy f'(x) = 0 || UNDEFINED when x = c. 1) Find f'(x) and its critical values. [for this question type only where f'(x) = 0] 2) Tell whether f(x) is speeding up [when it is moving away from f'(x) = 0.] or slowing down when it is approaching f'(x) = 0. **Note: These can be graphing problems and have been used as both in exams and quizzes. When it is a graph go straight to step #2. Remember f'(x) is when the tangent line's slope is zero. More simply where there is a max or min.**
Increasing and Decreasing Test [Test-2&3]
Before you can use this you need to have a function f(x) and find f'(x) you must find the critical values for this derivative. These are values 'c' that satisfy f'(x) = 0 || UNDEFINED when x = c. After finding f'(x) you can tell where a function is increasing or decreasing over a given interval. If f'(x) > 0 on the interval [a,b] it is increasing on that interval. If f'(x) < 0 on the interval [a,b] it is decreasing on that interval.
Rolle's Theorem [Test-2]
IF there is a function 'f(x)' that satisfies the three rules: 1) 'f' is continuous on the interval [a,b]. 2) 'f' is differentiable on the interval [a,b]. 3) f(a) = f(b) THEN there is a number 'c' in the interval [a,b] that satisfies f'(c) = 0
Mean Value Theorem [Test-2]
IF there is a function 'f(x)' that satisfies the two rules: 1) 'f' is continuous on the interval [a,b]. 2) 'f' is differentiable on the interval [a,b]. THEN there is a number 'c' in the interval [a,b] that satisfies the formula: (b - a) * f'(c) = f(b) - f(a)
Squeeze Theorem [Test-1]
IF: Functions f(x), g(x), and h(x) relate in a way such as f(x) LTE g(x) GTE h(x) THEN: Their limits relate in the similar way such as that LIM(x->a) of f(x) [LTE] LIM(x->a) of g(x) [GTE] LIM(x->a) of h(x) **You can use the limit laws to simplify these types of problems and find your answer. Remember the rules what happens to one side happens to all and when dividing by -1 change signs.**
Indeterminate Value Theorem [Test-1]
If a function f(x) is continuous from [a,b] & N is a number between [f(a),f(b)] then there is a number 'c' from [a,b] that satisfies f(c) = N
Implicit Differentiation Definition and Explanation [Test-2]
Implicit differentiation is taking the derivative of a 'function' in respect to multiple variables. This is needed when it is not possible to isolate one of the 'function's' variables. Note that this only happens when the 'function' is not really a function since it's graph doesn't pass the vertical line test such as a circle. Also note that implicit differentiation is simple and is just an application of the chain rule. Since 'y' (the variable we usually isolate) cannot be isolated we have to write the 'function' as "y in terms of x". In other words we write 'y' as y = y(x). To apply the differential operator: d(y(x)) / dy. Read as: "The derivative of the function of 'y of x' in terms of 'y'.
L'Hospital's Rule[Test-2]
Is a rule that gives us a way to find the limit of a function that is indeterminate in the two forms 0/0 OR +||- Inf./Inf. The rule states: LIM(x->a) of [f(x)/g(x)] = LIM(x->a) of [f'(x)/g'(x)] = LIM(x->a) of [f''(x)/g''(x)] and on and on. **Note: the rule only works for the two indeterminate forms 0/0 and +||- Inf./Inf. If it is not in either of these forms use algebra and calculus to change them. There are great videos in the modules section on canvas**
Asymptotes [Test-1]
Just like in college algebra asymptotes are when a function f(x) is undefined for a specific 'x' value. We can find these by setting a denominator equal to zero (vert.) and knowing the rules for horizontal asymptotes or knowing when a specific function (such as logs) are undefined. They can be explained in Calculus by these formulas: Vertical Asymptote IF: LIM(x->a) of f(x) = +||- Inf. LIM(x->a^-) of f(x) = +||- Inf. LIM(x->a^+) of f(x) = +||- Inf. Horizontal Asymptote IF: LIM(x-> +||- Inf.) of f(x) = L then the value L is the horizontal asymptote.
Limit Notation [Test-1]
Left Limit: LIM(x->a^(-)) Right Limit: LIM(x->a^(+))
Position(Speed) , Velocity, & Acceleration [Tests-1&2]
Position is almost always given as a function "s(x)". This is the base function that we use to find the velocity "v(x)" and acceleration "a(x)". NOTE: 1) Position = s(x) 2) s'(x) [first derivative of s] = v(x) 3) s''(x) [second derivative of s] = v'(x) = a(x)
Approximate Reinmann Sums Problems [Test-3]
Reinmann problems will come in one of two ways. The first will ask to find the sums of data given in a chart. The second is by giving you an equation, a number of intervals, a general interval, and which end points to use. Chart Problems: You will have to interpret what the chart is giving you then take the average of the values the question specifies. Then convert that data into whatever the question is asking for. Note: most of these will be asking for the approximation of the specified data. Equation Problems: The problem will give you the equation, number of subintervals, general interval and specify which points to use. Then you will take the Reimann Sum using that information. To evaluate this you must use the specified end points and find their value by plugging them into the equation. Then add each of those values and multiply by the width of the subintervals. Note: most of these ask for the approximation of the area under a curve.
Reinmann Sums Explanation [Test-3]
Reinmann sums are a way to add a series of sub intervals together in order to find the sum of a general interval. Note when E is used the number of times the sum is done is denoted by 'n' and the sum starts at the variable x sub 0. Also note D(x) is being used for delta x; in this case the size of the sub interval. Reimann notation is: LIM(n->inf.) E [f(xi) * D(x)]
Equations of a Line [pre-req]
Slope-Intercept: y = mx + b Point-Slope: y - Y = m(x - X) Vertical Line: x = X (Slope = 0) Horizontal Line: y = Y (Slope = undef.)
Anti-Derivative Rules [Test-3]
These are the rules that explain how to go from f(x) to F(x). In these everything to the left is f(x) [what is given] and to the right is its equivalent F(x) value. 1) k (any constant) => kx 2) c * f(x) (where c is any constant) => c * F(x) 3) f(x) + g(x) => F(x) + G(x) 4)x^n (any variable raised to the n'th power where n is not equal to 1) => 1/(n + 1) * x^(n + 1) 5) x^(-1) OR 1/x (where x is any variable) => ln(|x|) 6) e^x => e^x 7) b^x (where b>0 and not equal to 1) => [b^(x)] / [ln(b)] 8) sin(x) => -cos(x) 9) cos(x) => sin(x) 11) sec^2(x) => tan(x) 12) sec(x)tan(x) => sec(x) 13) 1 / (1 + x^2) => arctan(x)
Derivative [Test-1]
The derivative is the instantaneous rate of change of a function f(x) for any value of 'x'. It is defined as the slope of the tangent line or the formula for the IRC.
Differential [Test-2]
The differential is another way of taking the derivative of an entire function. This is a form rather than formula and should be used as such. The differential of y = f(x) is... dy = f'(x) dx
Linearization [Test-2]
The linearization of function 'f' at the x value of 'a' is the tangent line of the function at that value of x = a. The formula for linearization is... L(x) = f'(a) * (x - a) + f(a)
Limit Laws [Test-1]
There are 9 limit laws that are "common sense laws." These explain the possibility to add, subtract, multiply, divide, etc limits. These can be found on google and in the Topic Review. [sorry I'm cheap and didn't want to pay for quizlet]
Infinite Limit Laws [Test-1]
There are a total of ten Inf. Limit Laws. Unlike the Limit Laws these are not common sense. These are extremely important to know and showed up multiple times on exams. These can be found on google and in the Topic Review. [sorry I'm cheap and didn't want to pay for quizlet]
Integrating [Test-3]
There are a totals of 24 integration rules. These are used along with the FTC to convert integrals into equations that can be evaluated. I will not put these all here. These can be found on google and in the Topic Review. [sorry I'm cheap and didn't want to pay for quizlet]. *NOTE YOU MUST BE ABLE TO INTEGRATE FOR ALL OF THE TEST-4 SEGMENT OF THE FINAL. YOU MUST KNOW THESE*
Trigonometric (Including Inverse) & Logarithmic Derivative Identities [All]
These are absolutely necessary and are needed to pass any test; however I cannot put them all on this card. I would advising being familiar (but don't kill yourself) with the hyperbolic identities; while not tested you never know. These can be found on google and in the Topic Review. [sorry I'm cheap and didn't want to pay for quizlet]
Graphing Problem Steps[Tests-2&3]
These are extremely easy once you break them down and see that they are just asking you to do a few easy steps. Follow these rules to graph a function f(x) that is twice differentiable. Note before you begin find f'(x) and f''(x). [I found this to be the most efficient way] 1)Using Algebra Find: -Domain & Range -X & Y Intercepts -Asymptotes **Plot these on the graph** 2)Using f'(x) Find: -f'(x) critical values -Intervals of Inc./Dcr. **Plot these on a separate number line** 3) Using f''(x) Find: -f''(x) critical points -Intervals of concavity **Plot these on a separate number line** 4) Plot the f'(x) & f''(x) number lines on the graph. **REMEMBER TO LABEL EVERYTHING** **Note: where the domain is undefined will be critical points for f'(x) & f''(x). Also horizontal asymptotes will tell you the end behavior of your graph.**
Product & Quotient Rules of the Derivative [All]
These are formulas that explain how to find the derivative of two functions being multiplied or divided. Product: d/dx [ f(x) * g(x) ] => [ f'(x) * g(x) ] + [ f(x) * g'(x) ] Quotient: d/dx [ f(x) / g(x) ] => ( [ f'(x) * g(x) ] - [ f(x) * g'(x) ] ) / (g(x))^2
Indeterminate Limit Forms [Test-1]
These are the Indeterminate limit forms. They are used throughout the course. 1) 0/0 2) Inf./Inf. 3) Inf. * 0 4) Inf. - Inf. 5) 0^(0) 6) Inf.^(0) 7) 1^(Inf.) Exceptions: 0^(Inf.) = 0 & Const./Inf. = 0 & Const./0 = Inf. [This is only for infinite limits]
Extrema Explanation and Steps[Test-3]
These are the maximum and minimum values. Calculus allows us to find these no matter what according to Fermat's theorem (don't need to memorize that here just in case you want to further explanation). We can find local extremas always! Yet we can only find absolute extremas on closed intervals! These are 'critical values' of the function 'f(x)'.These are values 'c' that satisfy f'(x) = 0 || UNDEFINED when x = c. The steps are: 1) Find the derivative of the function f(x). 2) Find critical values of the derivative by setting f'(x) = 0 or noting when f'(x) is undefined. When setting f'(x) = 0 algebraically finding the values that satisfy the equation.
First and Second Derivative Tests Explained [Tests-2&3]
These exact rules can be found on google but here is an explanation. Before you can use either you need to have a function f(x) and find f'(x) and f''(x) you must find the critical values for each derivative. These are values 'c' that satisfy f'(x) = 0 || UNDEFINED when x = c. First Derv. Test: Proves where there is a minimum or maximum. Second Derv. Test: Proves where there is concavity and what that concavity is.
Derivative Identities [All]
These like the Limit Laws are crucial to the course and MUST be known by heart. There are five that are used (not including the trig identities here there is another card): 1) d/dx of 'c' (any constant) => 0 2) d/dx of 'x' (any variable) => 1 3) d/dx of 'x^n' (any variable raised to the n'th power) => n*x^(n-1) 4) d/dx of 'c/x' (any constant divided by a variable) => -(c) / (x^2) 5) d/dx of 'a^x' (any number to a power including a variable) => a^(x) * ln(a)
Optimization Problem Steps [Test-3]
These problems can utilize the inc./dcr., first, & second derivative tests. However, they do not come in till the last steps. 1) Read the problem and draw an illustration of what it is asking mark on your illustration what is being asked for and label it Q. 2) Label everything else that is given. Express any value that changes as a variable and the equation that proves it. 3) Using your illustration and equations create a function for Q. [Think what information can be used to define Q] 4) Finding Q' and Q'' (the derivatives of the function you create) then find their critical values. 5) Using those values find which is the min/max and use this to find your answer. **Sometimes require to put the min/max back in your original equation. Other times it will just ask for that min/max value. I can't find a way to generalize this.**
Average Rate of Change & Secant Line Slope & Average Velocity & Average Acceleration[Test-1]
These two are equal to one another and are found by the below function. This finds the slope of a line when given two points. This is simply the slope formula. Avg. Rate of Change & Others (given points (a,f(a)) & (b,f(b)) = ( f(b) - f(a) ) / ( b - a )
Disk Method for Volume [Test-4]
This can theorem can be used to find the volume of a shape whose cross-section lies between x = a & x = b has an area given by pi[f(x)]^2. IF a shape has a cross-section that lies between [a,b] and area is given by f(x) THEN its volume can be calculated: *about the x-axis by* V = ∫(a->b)[pi(f(x))^2* dx] *about the y-axis by* V = ∫(a->b)[pi(f(y))^2* dy]
Washer Method for Volume [Test-4]
This can theorem can be used to find the volume of a solid whose cross-section lies between x = a & x = b and whose area is between two curves and has an area given by pi[f(x)]^2 - pi[g(x)]^2. IF a shape has a cross-section that lies between [a,b] and area is given by f(x) THEN its volume can be calculated: *about the x-axis by* V = ∫(a->b)[pi[f(x)]^2 - pi[g(x)]^2* dx] *about the y-axis by* V = ∫(a->b)[pi[f(y)]^2 - pi[g(y)]^2 * dy]
Instantaneous Rate of Change & Tangent Line Slope[Test-1]
This is a massive blunder of our course these are the same thing and both are equal to the derivative. They actually define the derivative. Inst. Rate of Change (x=a) = LIM(x->a) of ( f(x) - f(a) ) / (x - a) **OR** LIM(h->0) of ( f(a+h) - f(a) ) /h
KEY (READ FIRST)
This set is best used in the full window form. These are common shorthands I use in these cards. - || is the 'or' operator (so +||- is the same as + or -) - GTE & LTE = abbreviations for greater than or equal to and less than or equal to respectively. - When I explain a limit as (x->a^-) that means from the left converting to regular text is a bit ambiguous
Area Between Two Curve Theorem [Test-4]
This theorem can be used to find the area between any two curves. However, the graph of f(x) must always be above g(x). The theorem is: IF f(x) GTE g(x) in the interval [a,b] then the area between the two graphs is equal to the integral: *about the x axis by*: ∫(a->b) [ |f(x) - g(x)| dx ] *about the y axis by*: ∫(a->b) [ |f(y) - g(y)| dy ] *Note |f(x) - g(x)| is f(x) - g(x) in absolute value brackets*
Shell Method for Volume [Test-4]
This theorem can be used to find the volume of a cylindrical shell whose circumference and height are in the same terms (same axis x/y). While the width of the region is a change in the same term. This term is the axis of rotation. While that same variable is within the interval [a,b]. The two formulas are: *about the x-axis & radius (given by f(x))*: ∫(a->b) 2pi [f(x)(ht) dx] *about the y-axis & radius (given by f(y))*: ∫(a->b) 2pi [f(x)(ht) dx]
Questions on the tangent and secant lines.
We will be asked questions on these two lines. They can be defined as the 'normal' line of another function which means it is perpendicular at a given point. It is important to remember when you find these using their respective formulas you are actually just finding the slope. You have to use the Point-Slope formula to find the actual function of that line.
Integrals '∫' [Test-3]
Where Riemann Sums are simply an approximation (due to the use of points) definite integrals (I will just say integrals for convenience) are a way to find the exact sum of a given equation. However, integrals are defined by Riemann Sums. The definite integral of a function 'f' over [a,b] is defined as: ∫(a->b) [f(x)*dx] = LIM(max Dx -> 0) E [f(xi)Dx]
Newton's Method
While this was not quizzed or tested maybe just take a look to be safe.
Exponentials & Logarithms Rules [pre-req]
u^[w] = e^[w*ln(u)] ln(u*v) = ln(u) + ln(v) ln(u/v) = ln(u) - ln(v) ln(u^v) = v*ln(u)
