Calculus 1

What is the relation between limits and discontinuity?

A limit can exist where there is certain type of discontinuity such as a whole, so they aren't the same thing although they share an overlap in conditions.

What does local extremum mean?

It means a function at which the function is either at its relative lowest or highest point within a given interval.

Very simply explain the concept of continuity and discontinuity on a graph.

Continuity refers to a line on a graph that is not broken and discontinuity is its opposite in which the graph varies.

What is the limit proof verbatim and the following steps? (Repeat 40x)

"For every positive number epsilon > 0 there is a positive number delta > 0 such that if the distance between 0 < abs(x - a) < delta then abs(f(x) - L) < epsilon." Step 1: Start with the delta equation "If 0 < abs(x - a) < delta then". Step 2: Solve for the abs(f(x) - L) < epsilon equation. Step 3: Set epsilon = delta by using the epsilon that was just solved for "For any epsilon > 0 there exists something delta = something epsilon". Step 4: Reverse the proof to prove it by plugging in the delta variable into the abs(f(x) - L) equation "If 0 < abs(x - a) < delta then abs(f(x) - L) = math that eventually equals epsilon all by itself". Step 5: "Therefore by the delta-epsilon proof the (insert original limit equation)".

What does the intermediate value theorem state?

"If f(x) is continuous and on a closed interval (a1, a2) where the endpoints f(a) does not equal f(b) and N is any number within the interval such that f(c) = N where f(a) < N < f(b)." "By the intermediate value theorem, there exists at least one c in the interval (a1, a2) such that f(x) = c."

What is the official definition of the intermediate value theorem verbatim, explain in the simplest of terms what the intermediate value theorem, and the conditions that must be met in order for it to be utilized.

"The intermediate value states if function f(x) is continuous on the closed interval [a, b] and k is a number between f(a) and f(b), then there exists one number c that exists in the interval (a, b) such that f(c) = k." In other words, the intermediate value theorem requires that the closed interval be continuous, and the y-values of the endpoints must have different signs such as f(a) > 0 and f(b) < 0 or visa-versa, then it can be said that there exists a number c within the interval (a, b) such that f(c) = k.

What is the rational sin property?

It is sin(theta)/(theta), check if correct and will likely be on test.

What is linear approximation, how is it accomplished, and what is it used to determine?

1. Select the point on the graph that must be approximated and it is (a, f(a)). 2. Find the slope of the tangent line by solving for the derivative of f'(a). 3. Use the point-slope form equation of y−f(a)=f′(a)(x−a), this equation represents the tangent line to the graphs function at the point (a, f(a)). 4. It now becomes possible to approximate the function for x values that are close to a. Plugging these values into the equation to find the corresponding y-values, which are approximations of the values functions.

What is the equation for the area of a circle?

A = π x r^2

What equation will be on the test with regard to related rates?

A = π x r^2, the Pythagorean theorem, the volume of a sphere, and the volume of a cone will be used on the test for related rates, as well as many other equations that will be given.

How are derivatives related to rates?

A derivative gives a rate of change y' = dy/dx = (change in y)/(change in x).

What are the absolute max/min and relative max/min, how can they be determined, and what are they collectively referred to as?

Absolute max/min is the highest and lowest points of the function within the specified interval. To determine the absolute maximum and minimum of a function over a closed interval: Find critical points by setting the derivative equal to zero and solve. Evaluate the function at the critical points and at the endpoints of the interval. The largest function value is the absolute maximum, and the smallest is the absolute minimum. Relative max/min are points that are high or low relative to the function but aren't the highest or lowest points. Absolute max/min and relative max/min points are collectively referred to as extrema.

What are the three trig values on the unit circle?

All the values of the unit circle will be some variation of 1/2, sqrt(2)/2, or sqrt(3)/2.

What is an arc trig function?

An arc trig function is essentially a trig function in reverse where the input becomes the output and visa-versa.

Why do the differentiation rules of trigonometric functions work?

Because the differentiation of the trigonometric functions follows the horizontal lines of f(x) being the same as f'(x).

What is different about solving for the slope of a derivative as opposed to solving for a normal derivative?

Both the x-value and y-value must be entered in order for the slope to be found.

How can you determine if a limit does or doesn't exist?

Check the limit from both the left and right side approach and the limit exists if they are the same value and visa-versa.

What is the implicit differentiation of x^3 + y^3 = 6xy -1? (Finish later)

Find the differentiation of the entire equation using the power rule and put a d/dx next to the y variables after using the power rule.

How to find the slope of a tangent line if only an x-value or y-value and the function is given?

Find the other value by plugging in the known value and solving the equation.

What does it mean to find in terms of x and y with regard to implicit differentiation, and what does this mean in a general sense with regard to other variables?

Finding x and y means finding the differential of an equation by having both the x and y values present in the final derivative with the notation dy/dx. This is true with respect to all other variables as well, as finding any variable with regard to that variable means to have only it remaining in the final derivative.

Where are functions also not differentiable?

Functions are not differentiable at sharp points.

Question 2:

Graphing the function and finding the limit.

How to find where the tangent line of a function is horizontal or vertical?

Horizontal tangent line: Find derivative of function, set y=0, solve. Vertical tangent line: Find derivative of function, set x=0, solve.

What will happen if the absolute max/min value of a function is at an open point?

If the absolute max/min value of a given point is at an open point then there is no known absolute max/min value, as the point becomes a never-ending decimal like -3.9999999999 or something else.

When and how would you solve a problem using implicit differentiation?

Implicit differentiation is usually used in the context of solving an equation that is not solvable through simple differentiation. The general idea behind implicit differentiation is that a dy/dx should multiplied or stacked next to any y term that was differentiated and then the equation is solved for dy/dx.

When finding a derivative, what should almost always be attempted first?

It is best to try and turn the fraction, two polynomials, etc into one polynomial total, so the power rule can be utilized.

What does it mean to use implicit differentiation to solve a certain equation in terms of only a certain variable?

It means to leave only that variable remaining and enter the values of every other variable.

What is the implicit differentiation of cot(y) and why is this the case?

It would be -csc(y) x dy/dx because the y-value is differentiated.

What is the derivative of pi^2?

It's just 0 because pi^2 is just a number.

What is the basic idea behind linear approximation and what are each of the variables?

Linear approximation is basically when a whole number is used to approximate another number. The tangent line to this point is then determined to find the point is then determined with the L(x) equation. Then plug the original value that it was asking for into the equation. So essentially, the nearest whole number is used to create the equation of the tangent line, and the original value is then entered into the equation. The variable a is the whole number estimation.

Remember to do what just to be thorough when solving equations of any kind and why?

Make sure to write out the subtraction or addition sign as well as the term itself, as this is the cause of many errors and potentially many missed points (it really doesn't take much longer than usual).

What should you do when using the product or quotient rule with regard to implicit differentiation? Remember to add what? (Repeat 40x)

Make sure to write out the x and y as well as the x' and y' to avoid errors and confusion. Remember to add a dy/dx at the end of whatever the y' is.

Are negative exponent derivatives allowed?

No, write the exponent in the denominator if it is negative.

How to graph piecewise functions?

Piecewise functions should start at whatever the x > # or x < # and enter the x into the function to achieve the y-values. That was you get the coordinates for the starting values of each function.

What are the two common methods of finding the area under a curve?

Reimann sums and definite integrals (Double check)

Remember what with regard to related rates?

Remember to add either positive or negative to related rates.

How to solve to find the y coordinate for a point on a differentiated function?

Solve by plugging in the x-value for the coordinate into the original function and not the differentiated one.

How to use intermediate value theorem to show that there exists a root to the equation x^3 - x^2 + x -2 = 0?

Start by determining if the function is continuous or not, and since the function is a polynomial it will always be continuous. Write: "Since f(x) = x^3 - x^2 + x - 2 = 0 is continuous on interval (0, 3) and" Next, drop the "= 0" portion and plug in the x-values (0, 3) to determine if f(a) > 0 and f(b) < 0 or visa-versa and write: "(0)^3 - (0)^2 + (0) - 2 < 0" "(3)^3 - (3)^2 + (3) - 2 > 0" Write: "Then by the intermediate value theorem, there is a root on (0, 3)."

What is the extrema of a function?

The absolute max/min and relative max/min (double check)

What is considered the absolute max/min value of a function and what is not?

The absolute max/min value of a function is a point at which the value is the highest or lowest point within a given interval. A value can not be the absolute max/min value if the point is a hollow circle and not filled in.

What is average velocity and the average velocity equation?

The average velocity is the change in position between two points over a given interval. The basic synopsis of the equation is that the t-value is the values the represent the endpoints of the interval and the f(t) represents the original function.

What must be done when differentiating a trigonometric function that has a coefficient such as with sin(2x)?

The chain rule must be used in conjunction with the standard equations for finding the derivative of a trigonometric function.

How to find the derivative of a composite function?

The derivative of a composite function can be found by using the chain rule, as the chain rule if basically a composite function.

What is the derivative of f(x) = sin^5(x)? Why is this the case?

The derivative of f(x) = sin^5(x) is 5sin^4(cos(x)).

What is the derivative of f(x) = tan(2x)? Why is this the case?

The derivative of f(x) = tan(2x) would be f'(x) = 2sec^2(x). This is because the function must be solved using the chain rule whereby tan(2x) = sec^2(x) and 2x = 2, so the derivative would be 2sec^2(x).

What are the implicit differentiation rules?

The derivative of xy should use the power rule will use the power rule and become (x)(1)dy/dx + (1)(y) and the chain rule remains the same.

What is the derivative of sinx^5 and what is it not?

The derivative would be cos(x) x 5x^4 because the exponent is on the x and not the sinx.

Why is h -> 0 or h becomes zero in the difference formula and what are the implications of this when solving using the difference quotient?

The difference formula models a tangent line or a line that represents the slope of a function at that given x coordinate. The h variable represents (x2 - x1) or the difference between the two intervals, so when h -> 0 the secant line will become a tangent line because the point is smack dab in the middle of the two intervals. This means that when solving for the derivative using the difference quotient, the h becomes a 0 once the expression has been simplified.

How to find the difference quotient?

The difference quotient is simply the (f(x+h) - f(x))/h. Plug the x + h and x values into the function and perform the following operations to get the answer.

How to determine the domain of a rational function and what does it correspond to?

The domain of a rational function is determined by setting the denominator equal to zero and solving for x. This makes sense when the denominator is equivalent to zero then there is a vertical asymptote that has been reached.

Why in the derivative of xy is it (x)(1)dy/dx + (1)(y) with the dy/dx at the place it is?

The dy/dx is placed in the equation whenever the y is set to its derivative, so the dy/dx is at that spot because the y was solved for that derivative.

What to do if you need to find the derivative of two polynomials?

The easiest way to solve this is to foil the two polynomials into one polynomial and then use the power rule to find the derivative.

How is the Reimann sum formation limit formula actually solved?

The equation will boil down to something like: summation notation(i - 1) at which point you will need to separate the i and - 1 and then use the summation notation rules to separate and solve and the i rules to turn i into an expression containing n. After this has been completed you should be left with an expression that consists of some numbers and n. This will usually consist of zeroing out the n terms because #/infinity is just 0.

What is the derivative of sin^2(sinx) and why is this the case? (Repeat 20x)

The exponent becomes the outside of the function even though it is notated on the inside. Derivative becomes 2sin(sinx) x cos(sinx) x cosx. Basically it becomes a triple chain rule problem.

How to find the given limit of a piecewise function and why does this make sense?

The limit of a piecewise function is usually denoted as either x-> 4+ or x-> 4-, and this represents the limit approaching from the right in the case of +4 or the limit approaching from the left in the case of -4. Solve for the x > # if approaching from the right and x < 4 if approaching from the left. This is because approaching from the right means that the limit is x > # or whatever the # is and approaching from the left means that the limit is x < # or whatever the # is essentially.

One of the proofs I think.

The limit theta -> 0 (sin(theta))/(theta) = 1 The limit theta -> 0 (theta)/(sin(theta)) = 1

What should be remembered about the linear approximation formula?

The linear approximation formula is the same as the point-slope formula. The f(a) is simply moved to the left of the equation to get from linear approximation from to point-slope form.

How to determine if f(x) is indeed a function and why is this the case?

The vertical line test is used to determine if f(x) is indeed a function because if the is points for one x-value that means that the function is indeed not a function because a function states that one y can be multiple x's but one x can't be multiple y's.

What are the derivatives of each trigonometric function and when are the only times that these rules apply?

These rules only apply when the is only an x with no coefficient present in the trig function. For example, sin(x) is fine but sin(2x) is not.

How to evaluate the slope of a tangent line of f(x) = x^4 at point (2, 16)? What do you not do and why does this make sense?

To find the slope of the tangent line of f(x) = x^4, you need to take the first-order derivative and then plug the slope into point slope form. Don't use any higher derivative than the first order. This is because the first-order derivative solves for the slope of the tangent line no matter if the starting power is something like x^10. The point-slope form is what actually solves for the tangent line.

What is the easiest way to find a trigonometric value from an angle or radian and visa-versa?

To go from a trigonometric value to radians or degrees, use the calculator to find the arc of the trigonometric function with the answer set to degrees. Then use the equation of pi/180 to find the radians.

What's the first thing to do before starting the calculus exam? (Repeat 40x)

To write your name on the exam.

Remember what about answers that involve trigonometric functions?

Trigonometric function answers can have more than one answer and they usually have a + 2 pi n at the end of the answer.

How to get the derivative of a fraction without using the quotient rule?

Use the "adding fractions with the same denominator rule" but in reverse. For example, f(x) = (x^2 + x + 3)/x can be turned into individual fractions x^2/x + x/x + 3/x. The fractions can then be solved individually using the power rule on each.

How to differentiate the function g(x) = (x-1)^4/(x^2+2x)^5?

Use the quotient rule which will then also involve the chain rule as well.

When is the derivative of a function officially defined?

When y is a function of x, but the equation is not solved for y explicitly, then y is implicitly defined.

What is an easy way to remember what variable reflect across which axis?

Y flips across the x-axis. X flips across the y-axis.

What is the derivative sin(sinx)?

cos(sinx) x cosx

What is the derivative constant rule?

d/dx(constant) = 0

Each order of derivative are known as what?

f'(x) is a derivative of f(x) f''(x) is a derivative of f'(x) f'''(x) is a derivative of f''(x) f''''(x) is a derivative of f'''(x) f'''''(x) is a derivative of f''''(x)

How do you convert radians to degrees and visa-versa?

pi/180 or 180/pi.

What is the general idea behind finding limits and the two possible outcomes?

Finding limits is all about how a function behaves as x approaches a. The two possible outcomes are either the limit does or doesn't exist when x approaches a.

What does the differentiation/continuity theorem state?

If something is differentiable then it is continuous but just because something is continuous doesn't mean it's differentiable.

Given the functions y = (3x)^2 and y = (1/3x)^2, which one is vertically stretched and which is vertically shrunk?

In the equation y = (3x)^2, the graph is vertically shrunk, and of course the corollary of that is that y = (1/3x)^2 is vertically stretched.

Simply explain what the squeeze theorem is and how it is used.

(THE < > SIGNS REPRESENT LESS THAN OR EQUAL TO AND GREATER THAN OR EQUAL TO SIGNS FOR THIS NOTECARD) The Squeeze Theorem essentially states that if f(x) < g(x) < h(x) and the lim x -> c then the limit of g(x) can be deduced by plugging in the limit c for both f(x) and h(x). In practice, this works by first setting up an inequality from the given equation, which is usually some variant of -1 < sin(theta) < 1 multiplied by the remaining number or expression that's not a part of sin(theta) because that's the limit of a unit circle. Next, set everything equal to each other and solve for one side for every function in the "compound inequality", but mainly the middle function g(x) which is most pertinent.

What are the five general methods to exhaust when evaluating limits?

1. Direct Substitution using limit laws. 2. If Direction Substitution yields an Indiscriminate Form of 0/0, then: Factor, Cancel, Try Again Rationalize, Simplify, Try Again Simplify, Cancel, Retry 3. #/0 form then it is a horizontal asymptote with DNE or negative infinity or positive infinity as the answer. 4. If Piecewise Function then use one-sided limits. 5. Finally, should nothing else work then use the Squeeze Theorem.

Where on a graph is a function not differentiable and why?

A function is not differentiable at the holes, jumps, any kind of discontinuity, or on vertical lines. Not vertical lines in particular because differential functions measure the rate of change and there is no rate of change on a vertical line much like there is no rate of change at a point of discontinuity.

Simply explain what asymptotic discontinuity is, when it occurs, and how it differs from infinite discontinuity.

Asymptotic Discontinuity: Occurs whenever there is a sudden difference in infinite y-values from the left and right approaches at the same x-value. Unlike infinite discontinuity, the y-values of asymptotic discontinuity go in opposite directions toward positive and negative infinity.

How are jump and asymptotic discontinuities similar and different from one another?

Both occur when both the left and right approaches of a given x-value give different y-values, but they differ with regard to jump discontinuity being finite y-values and asymptotic discontinuity being infinite y-values.

What are the two branches for which calculus can be divided and what are the two problem types for which calculus is concerned with solving?

Calculus can be divided into two branches: Differential and Integral. 1. Finding tangent lines to curves. 2. Computing areas under curves.

What is the definition of continuity and how to solve a problem with it?

Definition of continuity: 1. f(a) is defined. 2. lim x -> a f(x) = L 3. f(a) = lim x -> a f(x) Solving using the continuity formula simply requires that you solve 1, 2, and 3 in that order and if all three are solvable then the function is indeed continuous at that interval. Given that the function that is being solved for is usually a piecewise function, there is usually a discontinuity at a point but the rest of the function most likely is continuous.

How to determine the limit of a rational function that approaches infinity and what does each type of answer mean?

Divide the highest degree numerator by the highest degree denominator to find the answer type. If N > D then there is no limit or it increases without bound. If N < D then the limit is zero. If N = D then the limit is whatever the rational coefficient of the two highest degree terms are.

Explain how to shift a function across the x-axis, y-axis, and origin and why it is the case for each reflection.

In the equation y = (x + h)^2 + k: - (x + h)^2 reflects the graph across the y-axis because the y-value will always be negative (unless the variable k has something to say about it) which in effect makes the y-value always negative, hence it being flipped across the y-axis. (- x + h)^2 reflects the graph across the x-axis because the x will always be negative. This makes sense as the calculated (x - h) value will always yield the same y-value as (x + h) even though the y-value is the same. To reflect across the origin, simply combine the two methods for flipping across the x-axis and y-axis.

What are the basic transformations for function with regard to up, down, left, right, and explain why the transformations hold true for most functions.

In the equation y = (x - h)^2 + k: x - h = right shift (not - x because that means x is always negative) x + h = left shift (not + x because that means x is always positive) + y = up shift - y = down shift Most of these transformations remain the same with regard to other functions besides simple quadratics because the functions both have roughly the same format, but mileage varies.

What is indeterminant form and how is this answer usually corrected?

Indeterminant Form: 0/0 This type of answer isn't valid and the use of conjugates is usually required to stop the answer from becoming 0/0.

Simply explain what infinite discontinuity is and when it occurs.

Infinite Discontinuity: Essentially vertical asymptotes and they occur whenever the limit of x -> # gives an answer of either positive or negative infinity.

Simply explain what instantaneous and average velocity is and what it is equivalent to.

Instantaneous and average velocity is the change in position over a given period of time and it is equivalent to the slope of a tangent line or the average of a secant line.

How to determine instantaneous velocity?

Instantaneous velocity is determined by the slope of a tangent line (the difference formula). So just take the equation that is given for instantaneous velocity and find the slope by plugging the equation into the difference formula.

Simply explain what jump discontinuity is and when it occurs.

Jump Discontinuity: Occurs whenever there is a sudden difference or "jump" in finite y-values from the left and right approaches at the same x-value. Put in other words, it occurs when the the two approaches of a given x-value are not the same.

Name all the limit laws. (Repeat 10x)

Limit Laws: Sum Rule, Difference Rule, Product Rule, Quotient Rule, Power Rule, Root Rule, Constant Rule, and the x = c Rule.

List every type of function, its equation in either vertex form or standard form, and the shape, domain, and range of each graph.

Linear Function: y = mx + b - Straight line that has both an infinite domain and range. Quadratic Function: y = (x + h)^2 +k - U-shaped and has an infinite domain but a limited range depending on which way the tongs point. Power Function: y = (x + h)^3 + k - Has a squiggly line that goes up, down, up (or visa versa) around the origin and has an infinite domain and range. Square Root Function: y = sqrt(x), y = cbrt(x), etc - Inverse exponential curve in which the y-value becomes smaller as the x becomes larger. The domains and range of the function are determined by whether the root is odd or even. Exponential Function: y = x^2, y = x^3, etc - Exponential curve is either only positive or negative or both if the function has an odd power. The domain and range are also determined by whether the exponent is either even or odd. Logarithmic Function: y = log(x) - The y-value basically flatlines as the x-value gets bigger. There aren't really any set guidelines for finding the domain and range, so each logarithmic function must be analyzed on its own. Trigonometric Functions: y = sin(x), y = cos(x), etc - Shape varies wildly based on the specific trigonometric function, but sin and cos generally have squiggly intervals, tan and cot have lines that run up and down to infinite, and csc and sec are the opposite of sin and cos. The domain and range is determined by the type of trig function in use. Absolute Value Function: y = absvlu(x +h) + k - V-shaped with the domain being infinite and the range being restricted to whichever direction the V is facing.

Make sure to do what with every answer on the test?

Make sure to circle every answer on the test.

What to do when drawing lines on a graph?

Make sure to put an arrow at the end as that signifies the function continues forever.

How are exponential and sqrt functions stretched and shrunk? Simply explain why this is the case.

Multiplying by a number > 1: Exponential Function: Increases the take-off of the exponential trend. Square Root Function: Increases the take-off of the inverse exponential trend. Multiplying by a number < 1: Exponential Function: Decreases the take-off of the exponential trend. Square Root Function: Decreases the take-off of the inverse exponential trend. This is because the number > 1 is still being multiplied by a given factor, so both function types should increase as a result and visa-versa.

Does a limit actually need to be reached when approaching from the left or the right side of a graph? Give an example.

No, the actual limit value does not need to be reached in order for the limit to exist. For example, when x -> 4 the limit can exist as say 8 even if the value of 8 is a circle point.

How do you determine the average rate of change on a graph? Give an example.

Plot the two points the mark the beginning and the end of the interval on the line or function. Then create a triangle that measures the distance between the two points in terms of up/down for one side and left/right for the other. Then divide the two sides to find the average rate of change. Don't solve using the Pythagorean Theorem for some reason.

How to draw f'(x) from only the graph of f(x)? (Repeat 40x)

Points that produce a horizontal tangent line on f(x) are the zeros of f'(x). If f(x) decreasing -> f'(x) is negative or all the values are above the x-axis. If f(x) increasing -> f'(x) is positive or all the values are below the x-axis.

Where are polynomial and absolute value functions differentiable?

Polynomials are always differentiable, but absolute value functions are only differentiable at the center and not the sides.

When to put limit notation? (Repeat 40x)

Put limit notation (lim x -> #) if there is an x in play in the equation!!!!!!!!!!!!!!!!

Question 10:

Remember that it's f(x + h) not minus and double-check differentiating using the power method. a. Use the difference formula to differentiate the function. b. Find the equation of the tangent line by using the slope of the now differentiated equation and plug in the values of the points into the point-slope formula (y - y) = m(x - x).

Question 1: (Repeat the remember portion and use your mind's eye to visualize the graph 20x)

Remember that the graph is not linear as it ends on the left at (-4, -3), use the graphing calculator to graph the function, and draw an arrow on the right side.

Question 11: (Repeat what to remember 10x)

Remember that the question is asking you where the function is discontinuous and not if the entire function is discontinuous and a sharp point is not differentiable.

Question 9:

Remember that the zeros of f'(x) are the same as the horizontal tangent lines of f(x). If f(x) decreasing -> f'(x) is negative or all the values are above the x-axis. If f(x) increasing -> f'(x) is positive or all the values are below the x-axis.

Question 4:

Remember to always put the limit notation in every step while solving the problem in order until substitution is possible and make sure to enter a value that is close to the limit to double-check the answer. a. Factor and cancel. b. Factor and multiply the factor by itself if it's a negative to get a positive number. c. Conjugate the numerator and cancel. d. Replicate factors on both sides, subtract, factor, and cancel. e. Determine if the two approaches of the piecewise function exist and state whether the function is continuous.

Question 5:

Remember to always put the limit notation in every step while solving the problem in order until substitution is possible. Use the squeeze theorem and write: "By the squeeze theorem: g(x) ≤ f(x) ≤ h(x)" "For 2x−1: lim⁡ x→1 (2x−1)=2(1)−1=1" "For 2x^2: lim⁡ x→1 (x^2)=1" "Therefore if 1 ≤ f(x) ≤ 1 then lim x -> 1 f(x) = 1"

Question 3:

Remember to put in negative or positive infinity instead of DNE.

Simply explain what removable discontinuity is and when it occurs.

Removable Discontinuity: Occurs whenever there is a hole in an otherwise straight line on a graph. Put in other words, this type of discontinuity occurs whenever there are two lines that arrive at the same hollow point, so there is technically a limit from both left, right, and combined, but the actual function value is not present.

What is the best way to solve a limit problem that involves a trig function?

Set up and use a table because trig functions are weird.

In the delta-epsilon proof, what must be done at the end of the proof?

Show both proofs for the epsilon equation with one equation where the delta is solved for and one where the delta is switched out mid-equation so the left side of the equation remains unchanged.

What is the easiest way to solve a trig value for a given angle on the unit circle?

Simply draw a triangle and solve using the Pythagorean Theorem and trig identities.

How to find the slope of a tangent line at a given point?

Since the point is given, all you need to do is use the slope point form and plug in the points for x and y which are given, and the slope by using the difference formula.

Question 6:

Start by determining if the function is continuous or not, and since the function is a polynomial it will always be continuous. Write: "Since f(x) = x^3 - x^2 + x - 2 = 0 is continuous on interval (0, 3) and" Next, drop the "= 0" portion and plug in the x-values (0, 3) to determine if f(a) > 0 and f(b) < 0 or visa-versa and write: "(0)^3 - (0)^2 + (0) - 2 < 0" "(3)^3 - (3)^2 + (3) - 2 > 0" Write: "Then by the intermediate value theorem, there is a root on (0, 3)."

With regard to the graph of two piecewise functions when does and doesn't a value like f(2) exist (not a limit)?

The answer for f(2) exists when there is a point on the graph that is filled in which signifies that the point contains a value. The answer does not exist if there is no line crossing the point or all points the are over the x-value of 2 are not filled in.

The average rate of change and instantaneous rate of change can be solved with which two equations and lines and why?

The average rate of change can be solved with the (f(b) - f(a)) / (b - a) it uses the slope of the secant line because there are two points. The instantaneous rate of change can be solved with the difference formula and it uses the tangent line because there is only one point.

What is the derivative of a function also the same as and what are the other notations for a derivative?

The derivative of a function is also the same as the slope of a tangent line, instantaneous velocity, and rate of change.

How is the derivative of f(x) denoted?

The derivative of f(x) is denoted as f'(x).

What is the equation that is used to find the slope of a tangent line and why?

The difference formula is used to find the slope of a tangent line because it measures the slop of or instantaneous rate of change at a given point on a line.

How to determine the domain of a rational function?

The domain of a rational function is any x-value that doesn't leave the denominator as zero. So to solve for the domain simply set the denominator equal to zero and solve.

Is x^70 or e^x greater at infinity and why?

The e^x is greater at infinity because, unlike x^70 which has a constant power, the x the exponent which means that it will yield a much larger number at infinity than will an exponent of 70.

What are epsilon and delta defined as in a limit proof and why are they defined as this?

The epsilon and delta are defined as the epsilon > 0 and delta > 0. This is because epsilon and delta are the distances away from the x-value and y-value of the limit. Since they are distances this means that both epsilon and delta must be positive, hence they are greater than zero.

In the limit for the equation x -> -2 f(x) = 1.4, what is the limit for the left, right, combined, and f(-2). Why is this the case?

The equation states that as x approaches the limit of -2 the equation becomes 1.4. This means that both the left and right must both be yield the same answer (which it does) for the limit to exist.

If point x = 2 is on the line of a piecewise function and isn't an endpoint, is the function continuous at this point?

The function is continuous at this point, as it is not subject to any of the various forms of discontinuity like being at an end-point or infinite slide would be.

How to determine if a function is discontinuous at a given point algebraically?

The function is discontinuous if both equations arrive at different answers with the same x-value input.

How is a hole marked on a graph that represents a rational function?

The function is otherwise continuous with the exception of a hollow point where the hole's x-value is.

In the function y = 2^x in what way would the graph be reflected in the function became - f(x) and why is this the case?

The graph would be flipped across the y-axis because the function y = 2^x would become y = 2^-x. This is because as the x-value gets bigger the y-value becomes smaller due to the negative exponent.

What does the intermediate value theorem essentially state and why?

The intermediate value theorem essentially states in a given closed interval, if the function is continuous along the given interval and the y-value is f(a) < N < f(b) then a real number N exists.

Very simply explain what the intermediate value theorem is.

The intermediate value theorem states that in the interval [a, b],

How are the left and right approaches represented in algebraic form?

The left approach is represented with an x < # and the right approach is represented with an x > #.

What is the limit for the entire function if both sides of the function are both equivalent to either positive or negative infinity and why is this the case?

The limit answer will be DNE because even if both sides of the limit are either positive or negative infinity the answer will not yield the same for both sides at their respective theoretical limit.

What is the difference equation and what is the limit?

The limit is lim h -> 0 because the h represents distance and the distance is being zeroed out.

When does the limit of a rational function not exist?

The limit of a rational function DNE when the number chosen to be the limit, such as x -> 2, is where the rational function's asymptote is also located.

What is the limit proof trying to explain?

The limit proof essentially states that there exists a distance delta for the x-axis and distance epsilon which if corrected for on both sides should give the limit of the function.

What is the limit of a function that oscillates rapidly as the limit is approached?

The limit will be DNE.

As a limit approaches infinity, when does it either negative, positive infinity, or DNE?

The limit will be either negative or positive infinity is consecutive points have a trend of becoming bigger and bigger. However, if the line trends by being greater and then smaller, then the limit DNE.

What is the equation for determining the rate of change algebraically and how is it used? Explain why this equation makes sense.

The rate of change equation is (f(b) - f(a)) / (b - a) about the interval [a, b], where the a and b variables are the interval that is being solved. This makes sense as the rate of change essentially measures the difference in "elevation" between two different points or intervals on the graph. It can also be described as a quasi-slope formula where the f(b) - f(a) solves for the y-value and the b - a for the x-value to get a y/x for slope.

How to find the rate of change of a function from a graph?

The rate of change is essentially the difference in elevation or slope of two points or intervals. Start by drawing a right triangle between the two points with the vertical line representing the change in y and the horizontal line representing the change in x. Next, divide the change in y by the change in x or y/x to get the rate of change within that interval.

What formula is used to measure tangent and what does it represent?

The tangent line is determined with the distance formula and it represents the slope of a given function at a certain point.

How to determine the zeros of f'(x) from the graph of f(x) and when a side is positive or negative?

The zeros of f'(x) are the same as the horizontal tangent lines of f(x). If f(x) decreasing -> f'(x) is negative or all the values are above the x-axis. If f(x) increasing -> f'(x) is positive or all the values are below the x-axis.

Use calculator for what on the test?

Use the calculator to graph polynomials, piecewise functions, and to check the answers to limits.

What is the equation that is used to find the slope of a secant line and why?

Use the rate of change formula to find the slope of a secant line. This is because a secant line can used to model the rate of change over a given interval.

If the answer is in indeterminant form after direct substitution, what should be done?

Using a conjugate will solve the vast majority of the problems, but canceling out is a secondary option.

When is direct substitution appropriate for solving limits?

Whenever c is in the domain of a polynomial or rational function, then substitute c for x.

When to drop the limit notation in a difference formula?

Whenever the 0 in h -> 0 is sent in.

When must both equations of a piecewise function be solved when solving continuity?

Whenever the point overlaps with the two functions or lies on an end cap like with x > 2 and x < 2, then both equations must be solved to determine if there is any overlap. If so, then the function is considered continuous.