oral quiz

1. How do you multiply fractions, divide fractions, add fractions, subtract fraction, and solve fraction equations?

You multiply fractions straight across, divide fractions by flipping the second fraction and multiplying

straight across, and add fractions by using the butterfly method. You subtract fractions with the butterfly

method, but you subtract the antennae instead. You solve fractions by cross multiplying across the equal

signs and then solving with factoring or SADMEP.

2. How do you find a fraction power (such as 8^2/3 )? How do you find a negative power like 2^−3?

You do a fraction power by taking the root of the bottom number and then the power of the top number

(therefore 8^2/3 = 3√8^2 = 2^2 = 4). You do a negative power by putting it on the bottom of the fraction and

then powering it (therefore 2^−3 = 1/2^3=1/8)

3. What can 𝑥^𝑎+𝑏 also be written as? What can 𝑥^𝑎−𝑏 also be written as? What can 𝑥^𝑎∙𝑏 also be written as?

𝑥^𝑎+𝑏 = 𝑥𝑎 ∙ 𝑥𝑏 𝑥^𝑎−𝑏 = 𝑥𝑎/𝑥𝑏 𝑥^𝑎∙𝑏 = (𝑥^𝑎)^𝑏

4. What is the formula for finding the area of a circle, a semicircle, a quarter-circle, a rectangle, a triangle, and a square? What is formula for the perimeter of a square and the circumference of a circle?

Areas:

Circle = 𝜋𝑟^2

Semicircle =1/2 𝜋𝑟^2

Quarter-Circle=1/4 𝜋𝑟^2

Rectangle = length times width

Triangle = one half times base times height

Square = side squared

Perimeter of a square = 4 times the sides

Circumference of a circle = 2𝜋𝑟

5. What is the formula for the surface area of a cube and a sphere? What is the formula for the volume of a cube,

sphere, cylinder and cone?

Surface area of cube = 6𝑠^2

Surface area of a sphere = 4𝜋𝑟^2

Volume of a cube = 𝑠^3

Volume of a sphere = 4/3 𝜋𝑟^3

Volume of a sphere = 𝜋(𝑟^2)ℎ

Volume of a sphere = 1/3 𝜋(𝑟^2)ℎ

6. What is e the zero? What is the number e equivalent to (rounded to one decimal place)? Is e to the anything

ever equal to zero? What is e to the anything always equal to?

e to the zero is 1. 𝑒1 ≈ 2.7. e to the anything is never equal to zero.

e to the anything is always a positive number

7. What is ln 0? What is ln 1? What is ln(-#)? What can ln (ab) (read as natural log of a times b) also be written

as? What can ln (𝑎𝑏) (read as natural log of a divided by b) also be written as? What can ln 𝑎𝑟 also be written

as?

ln 0 = DNE. ln 1 = 0. ln(−#) = nonexistent. ln 𝑎 + ln 𝑏 = ln(𝑎𝑏). ln 𝑎 − ln 𝑏 = ln 𝑎𝑏 𝑟 ln 𝑎 = ln 𝑎𝑟

8. What is the formula for finding the slope between two points? What are four formulas for the equation of a line?

Formula for the slope between two points =

𝑦2 − 𝑦1

𝑥2 − 𝑥1

The four different equations for a line are:

Standard form = 𝐴𝑥 + 𝐵𝑦 = 𝐶, where A, B, and C are whole numbers

Point Slope Form = 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Free Response Form = 𝑦 = 𝑚(𝑥 − 𝑥1) + 𝑦1

Slope Intercept Form = 𝑦 = 𝑚𝑥 + b

9. What can sec 𝑥 be algebraically simplified to? What about csc 𝑥 , tan 𝑥 , and cot 𝑥?

sec 𝑥 =

1

cos 𝑥

csc 𝑥 =

1

sin 𝑥

tan 𝑥 =

sin 𝑥

cos 𝑥

cot 𝑥 =

cos 𝑥

sin 𝑥

10. What is the difference between sec x and arccos x? What is the difference between csc x and arcsin x?

sec x is 1 over cosine and arccos is the opposite of cosine (it undoes a cosine in solving a problem).

Csc x is 1 over sine and arcsine is the opposite of sine (it undoes a sine in solving problems). They

are not related to each other at all.

11. What is SOHCAHTOA? What is the Pythagorean theorem?

For SOHCAHTOA, first the opposite side is the one across from the angle 𝜃 and adjacent is the side

touching the angle 𝜃 and hypotenuse is always across the right angle. Then,

sin 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

cos 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

tan 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑒𝑐𝑒𝑛𝑡

csc 𝜃 =

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 sec 𝜃 =

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 cot 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

Pythagorean Theorem is (𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡)2 + (𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒)2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒)2

12. When solving an equation how do you know if you have to use SADMEP or factoring? Explain what SADMEP means?

When solving equations, you will use SADMEP when the variable in the problem is only written once.

And, you will use factoring when the variable is written more than one time.

13. What is the opposite of adding, the opposite of multiplying, the opposite of squared, the opposite of cubed, the opposite of sin, the opposite of cos, the opposite of tan, the opposite of ln?

The opposite of adding is subtracting. The opposite of multiplying is dividing. The opposite of

squared is plus and minus square root. The opposite of cubed is cube root. The opposite of sine is

arcsine (sin−1 𝑥 in your calculator). The opposite of cosine is arccosine (cos−1 𝑥 in your calculator).

The opposite of tangent is arctangent (tan−1 𝑥 in your calculator). The opposite of ln is e to the

power

14. There are five things that you should be able to do in a calculator: find 𝑓(#), find where an equation equals 0,

find the intersection of two graphs, find the derivative, find the integral. What do you do first for all of them?

Then I will ask how you do each one.

The first step is to push the button and type the equation into the calculator using the

button.

- To find 𝑓(#) you will either type alpha trace. and choose the right equation then push 2nd

quit

and type the x-value you want inside of parenthesis or you can type graph and then trace and

then the x-value.

- To find the zero of an equation type the equation into 𝑦1 and 0 into 𝑦2. Then type 2nd

trace and

then 5 . Push enter and enter and then guess the x-value where the graph crosses the x-axis.

- To find the intersections of two graphs put one into 𝑦1 and the other into 𝑦2. Then type 2nd

trace

and then 5 . Push enter and enter and then guess the x-value it looks like the two graphs

cross at.

- To find a derivative, you push math 8 on the main screen.

- To find an integral, you push math 9 on the main screen.

15. How do you know if numbers with parenthesis like (2, 7) are an interval or a point? What is the difference

between a point and an interval?

15. If the word before the parenthesis and numbers is “on” or “in” it is an interval. But, if the word before is

“at” then it is a point. An interval is a distance from one x to another (like walking along the boardwalk).

A point is one spot on the graph/rollercoaster (gives the x and height of the track).

16. How do you know if a graph is continuous? How do you know if a graph is differentiable? If something is

continuous does it have to be differentiable, and if something is differentiable is it continuous (I will only ask

one so you need to know the difference between the two)?

A graph is continuous if you can graph the graph without lifting your pencil. A graph is differentiable if it

has no gaps in the graph, sharp turns, or vertical points. If a graph is continuous it does not have to be

differentiable, but if a graph is differentiable then it must be continuous.

17. What are the four reasons that a function would be discontinuous?

A function would be discontinuous if you divide by zero, square root a negative, ln a 0 or a negative, or if

there is a piecewise function whose heights don’t match at the boundary.

18. How do you find the intervals for which a function is continuous if there is a ln or square root involved?

To find continuity algebraically for ln or square root make a discontinuity sign chart:

1. Find the discontinuous point (the inside equation equaling zero)

2. Start at negative infinity in top row of chart and end at infinity, stopping at each discontinuity point

3. Pick any # in these intervals and put in row 2

4. plug in the # from row 2 into the inside equation to see if it is positive or negative and write this

down in row 3

5. Put DNE or exists in row 4

6. answer in interval form

19. How do you know if piecewise function is continuous or not?

To find continuity of a piecewise function check to see if the heights match at the boundary. If they do not,

then the boundary is a discontinuity point.

20. Of the four reasons a function is discontinuous (divide by zero, piecewise functions, ln of 0 or negative, square

root of a negative) all of them give non-removable discontinuities except one. What is the one that can give a

removable discontinuity? How do you if this discontinuity is removable (hole) or non-removable (vertical

asymptote)?

Divide by zero is the only function that can give removable discontinuities. If you plug the zero from the

bottom into the top and get zero then you have a hole/removable discontinuity. If you plug the zero on the

bottom into the top and get anything other than zero then you have a vertical asymptote/non-removable

discontinuity.

21. Jump discontinuities, infinite discontinuities, and essential discontinuities are all non-removable

discontinuities. Graphically, what does a jump discontinuity look like? What does an infinite discontinuity

look like? What does an essential discontinuity look like? Mathematically, how do you know if a graph has a

jump discontinuity? What about an infinite discontinuity? What about an essential discontinuity?

22. What is the blanket statement for the theorems? What gives you a hint that a problem is probably a theorem

problem?

Due to the name of the theorem, since constraints necessary, there is a c in (a, b) such that what the theorem

says. If the problem asks if there “must be” or if it is a yes or no question it is a good chance it’s a theorem

problem

23. How do you find the height of a hole? How do you find the one sided limits?

You find the height of a hole by taking the limit there. You find a one sided limit with the same steps as

limits. However, if a piecewise function is involved you must only plug the x value into the correct

boundary side.

24. What does a limit tell you about a graph (use your hands to explain)? What are the steps to finding limits?

How do you know if a limit is undefined

A limit tells you what height (y-value) that a graph is approaching as the graph approaches the x value you

are taking the limit at, or you can talk about it is the height that two roller coasters collide at or not. Steps

to limits: 1-plug in the number, 2 – l’hopitals rule 3-table (using your calculator). A limit is undefined if

the y-values approach different numbers in your table.

25. How do you find horizontal asymptotes? How do you find limits as x approaches infinity? What is the order of

functions from slowest to fastest?

You find a horizontal asymptote by finding the limit as x approaches infinity and negative infinity. You

find the limit as x approaches infinity by comparing the top and bottom of an equation (if the top is faster

then you get infinity or undefined, if the bottom is faster then you get zero, and if they match then you get

their coefficients). The order of functions from slowest to fastest → trig, constant, ln x,

square root, linear, polynomial, exponential

When taking a derivative or integral of an equation like 1/𝑥² what would you need to do first? What

about for ³√𝑥^5

?

To take the derivative you will need to change 1/𝑥² into 𝑥^-2 and change ³√𝑥^5 into 𝑥^5/4

How do you take the derivative of two equations being multiplied together? How do you take the derivative of

two equations being divided? How do you take the derivative of an equation that is inside of another

equation?

to take the derivative of two equations being multiplied use the Product Rule:

𝑑

𝑑𝑥 [𝑓(𝑥)𝑔(𝑥)] = 𝑓’(𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔′(𝑥)

To take the derivative of two equations being divided use the Quotient Rule:

𝑑

𝑑𝑥 = 𝑓’(𝑥)𝑔(𝑥) − 𝑓(𝑥)𝑔’(𝑥)/g’(x)'²

z𝑔(𝑥){!

To take the derivative of an equation that is inside of another equation use the Chain Rule:

𝑑

𝑑𝑥 [𝑓(𝑢)] = 𝑢′ ∙ 𝑓′(𝑢

8. What are the two limit definitions of the derivative? What is each one used for?

The limit of the difference quotient:

lim

h→0

𝑓(𝑥 + ℎ) − 𝑓(𝑥)

ℎ

and the alternative form:

lim*→#

𝑓(𝑥) − 𝑓(#)

𝑥 − #

The limit of the difference quotient will be on multiple choice and will need to be recognized. The alternative form is used to define differentiability.

. How do you find the derivative of an equation that has x and y on the same side?

29. To find the derivative of an equation that has x and y on the same side use implicit differentiation. To implicit differentiate1.

Take the derivative of each piece

2. Put dy/dx when take derivative of a y.

3. Get dy/dx on one side and the other stuff on the other side

4. Pull out dydx (if more than one)

5. Divide the dydx stuff to the other side

What are the different words for saying the derivative equals zero?

The other words for derivative equaling zero are horizontal tangent line or critical number.

31. How do you find if a graph is increasing or decreasing?

To find intervals of increasing/decreasing do a sign chart for 𝑓′(𝑥)…

1.find critical numbers

2.set up intervals from −∞ to ∞ in a table stopping at each critical number.

3.pick x value in intervals

4.plug those x into 𝑓′(𝑥) and write the sign

5.if positive then write increasing; if negative then write decreasing

6.write intervals in interval form

7.justify your answer with 𝑓’ < 0 or 𝑓’ > 0

What is the average rate of change and the instantaneous rate of change on a graph

The average rate of change is the same as the slope of the secant line which gives the slope from one point to another on the graph. The instantaneous rate of change is the same as the slope of the tangent line which gives the slope of one point on the graph.

3. What is the other word for an instantaneous rate of change? What is the blanket statement for rates of change?

The other word for an instantaneous rate of change is the derivative. The blanket statement for rates of change (the derivative) is: The words of the equation is increasing/decreasing at a rate of |answer| units

What does the word extrema stand for? How do you know if you are finding relative extrema or absolute

extrema?

The word extrema is plural for maximums and minimums. You know you are finding a relative extrema if it says to find the local or relative extrema. You know you are finding an absolute extrema if it says to find the absolute extrema, global extrema, or just the word maximum or minimum by itself.

How do you find relative extrema?

To find relative extrema do a sign chart for 𝑓′:

1. find critical numbers

2. set up intervals from −∞ to ∞ in a table stopping at each critical number.

3. pick x value in intervals

4. plug those x into f ‘ (x) and write the sign

5. if positive then write increasing; if negative then write decreasing

6. if 𝑓2changes from positive to negative then it is a max; if 𝑓2 changes from negative to positive then it is

a min

7. see if asking for when or what

8. answer

9. justify by saying a relative max b/c 𝑓’ changes from positive to negative there; or a relative min b/c

𝑓’ changes from negative to positive there

What is the second derivative test? How does it work?

36. The second derivative test is a shortcut to find relative extrema. If 𝑓′′(𝑥) is positive at a critical number

than f has a relative minimum there. If 𝑓′′(𝑥) is negative at a critical number than f has a relative

maximum there (understand how this works, don’t just memorize it).

How do you find intervals of concavity for a function?

38. To find intervals of concavity do a sign chart for 𝑓′′(𝑥)…

1. find test numbers (when 𝑓′′(𝑥) = 0 or DNE)

2. set up intervals from −∞ to ∞ in a table stopping at each test number.

3. pick x value in intervals

4. plug x value into f " (x)

5. if f " (x) is positive write conc. up; if f " (x) is negative then write conc. down

6. Answer

7. Justify concave up with 𝑓’’(𝑥) > 0, justify concave down with 𝑓’’(𝑥) < 0

39. How do you find points of inflection?

39. To find points of inflectoin do a sign chart for 𝑓′′(𝑥)…

1. find test numbers(when 𝑓′′(𝑥) = 0 or DNE)

2. set up intervals from −∞ to ∞ in a table stopping at each test number.

3. pick x value in intervals

4. plug x value into f " (x)

5. if f " (x) is positive write conc. up; if f " (x) is negative then write conc. down

6. if f " (x) changes from positive to negative or from negative to positive then there is a point of inflection

there.

7. Answer

8. justify inflection points with 𝑓′′(𝑥) changes signs.