Unit 4 Review Supports - Created by AP Calculus 2020-2021 Students by Nicholas Bianculli

Question 29

koneill52 — 2021-01-28 13:34:30 UTC

Look at 4.7 page 1 to understand the relationship between g(x) and f(x)
Draw a number line of g’(x)
When looking at a number line of the first derivative what does a maximum look like?

Question 30

koneill52 — 2021-01-28 13:35:37 UTC

Remember that g’(x)= f(x)
Remember that the point of inflection of g(x) is when g’’(x)changes signs, so what does that mean for the slopes of g’(x)?

Question 46a

dgreenbaum — 2021-01-28 13:37:09 UTC

Use “CCC” to find absolute minimum

Find Critical points
Compare values of v at critical points and end points
Write a Conclusion

Question 22

amwinton — 2021-01-28 13:37:48 UTC

A). In order to write the acceleration equation, consider the relationship between acceleration and velocity. Then, think about what time the acceleration would be smallest.

B). How do upward and downward movement relate to rightward and leftward movement?

C). Average acceleration is the change in velocity over a time interval. What does this say about average velocity?

D). Recall the relationship between velocity and position to write your position equation?

E). Refer to 4.3 Slide 5 (f) … Remember to consider changes in velocity when writing integrals. Also keep in mind: Can “total distance” include negatives?

Question 31

jseff — 2021-01-28 13:37:49 UTC

Using the given information, come up with a function rule using an integral.

Question 46b

dgreenbaum — 2021-01-28 13:38:17 UTC

Think about which theorem would apply to this question and use TUNE!

Question 42

amwinton — 2021-01-28 13:38:51 UTC

A): Refer to 4.2 Slide 2 #1

B). Consider the difference between “total distance” and the value you compute when using an integral. Can “total distance” include negatives? How can you change the integral to get “total distance” as opposed to “net change?”

C). Recall the factors that relate to increasing/decreasing speed, and how the relationship of those factors determines the effect on speed.

D). In order to determine position remember you have to remember two important factors:

Your starting point
How does position relate to velocity? How can you use that relationship to write an integral that answers the question?

Question 48 Part D

koneill52 — 2021-01-28 13:39:58 UTC

Method 2

Refer to video:

Question 46c

dgreenbaum — 2021-01-28 13:43:13 UTC

Think about the relationship between the sign of the velocity and acceleration on this interval

Question 46 d

dgreenbaum — 2021-01-28 13:43:27 UTC

Think about what does the graph of v(t) would have to look like in order for a(t)<0

Question 23

dgreenbaum — 2021-01-28 22:01:45 UTC

Remember your integral rule:

Question 24

emargolis — 2021-01-28 22:02:05 UTC

Question 25

emargolis — 2021-01-28 22:05:25 UTC

Think about the sign (+ or -) of f' and what it means in relation to the graph of f. Drawing a number line could also help.

Question 26

emwagner2 — 2021-01-29 01:29:04 UTC

Question 27

aleland2 — 2021-01-29 01:36:05 UTC

Think about which trig functions have the derivatives of sin(2x) and cos(2x). You may have to add a coefficient to the antiderivative to ensure its derivative would cancel out to have no coefficient.

Also, remember the following integral rule:

Question 28

emwagner2 — 2021-01-29 01:42:12 UTC

Remember, the antiderivative of a velocity function is position.

Antiderivatives include a “+C.” You are given the value of “C” for this problem in the question - read carefully.

Question 47

aleland2 — 2021-01-29 01:43:49 UTC

a) You can find the antiderivative of this velocity function by finding the total area between the function and the x-axis. Think about what the antiderivative of a velocity function represents.

b)Think about what makes a function non-differentiable. Also, remember that v’(t) is the slope of v(t).

c)To find a(t), apply the same thinking as you did in part b. (See image below if you need help setting up a piecewise function)

d)

Question 47c image

aleland2 — 2021-01-29 01:55:27 UTC

Question 47d image

aleland2 — 2021-01-29 01:55:41 UTC

Question 10

bgerchick — 2021-01-29 15:58:19 UTC

First try to get rid of the denominator. This will make it easier to take the antiderivative.

Question 11

bgerchick — 2021-01-29 15:58:37 UTC

Make sure to know your trig functions! Since each term is separated by an addition/subtraction sign you need to find the antiderivative of each term.

Question 12

bgerchick — 2021-01-29 15:58:59 UTC

Like #11 we must know our trig functions! In order to make this problem simpler try to distribute everything you can. What do we get once you multiply out the sinx? Once you have your two new terms you will be able to take the antiderivative of this easier.

Question 40

bgerchick — 2021-01-29 15:59:26 UTC

a) For g(3) remember that g is equal to the integral of f. You will then be able to figure this out much easier. For g’(3) this is easier than you think! What does g’(x) equal in terms of f (x)? Once you understand what g’(x) equals in terms of f(x) you will be able to take this one step further! What do we do to get from g’(x) to g’’(x)?

b) Remember that we are trying to find the rate of change of g and not f! Use part a and the graph to help you complete this problem. The average rate of change is just looking at the difference between two points.

c) Use the value we found in part b to help you with this question. Remember we want to use that value to see where g’(c)=f(c). Look for points of intersection and make an equation out of your value in part b!

d) Remember that we are trying to find the points of inflection of g and not f. This means that we are trying to find when g”(x) does what? Use part a to guide you in going from the graph of f to g’’(x).

Question 13

jarcohen — 2021-01-29 17:53:45 UTC

Break down the fractions as much as possible using your knowledge of trig and trig reciprocals before finding the respective integrals.

Question 1

climb1 — 2021-01-29 17:55:24 UTC

Use the law of integration (see attached photo or refer to lesson 4.6 slide 1) to separate the definite integral into two separate definite integrals .*Note the subtraction sign*
Note that the graph to the right is f(x). Think about the connection between the integral of f(x) and the graph of f(x)
utilize the antiderivative method from 4.5, OR draw a graph and find the area to find definite integrals other than f(x)

Question 6 Hints

daccurso1 — 2021-01-29 17:55:29 UTC

1. Start by splitting up the integral into 2 pieces that are connected by subtraction
2. For the first integral, find the antiderivative. For the second, graph the function and use the net area method. (Refer to question 15 on 4.5 for reference).
3. Solve.

Question 2

kevfranzblau — 2021-01-29 17:55:29 UTC

Use rules from law of integration (as seen below) to separate the definite integral into definite integrals of single functions and remember their limits of integration are important.

Question 18

jsalim3 — 2021-01-29 17:55:50 UTC

1. Foil completely
2. Use the power rule for antiderivatives

Question 5

gsteinert — 2021-01-29 18:00:38 UTC

1. Think about how you could get rid of the absolute value sign.
2. If you are unable to do that then watch this video to try and get you going in the right direction.
3. Be careful with your signs.

Question 48a

lschwartzman1 — 2021-01-29 18:02:25 UTC

Remember that the acceleration of the particle is the slope of the velocity. How could e^1-t be rewritten to make taking the derivative easier?

Question 48b

lschwartzman1 — 2021-01-29 18:02:43 UTC

Remember that speed and velocity are not the same. Refer to 4.1, slide 7 to see how the values of the velocity and acceleration affect the speed of an object.

Question 48c

lschwartzman1 — 2021-01-29 18:02:49 UTC

What property would change sign when the displacement changes direction? What would the value of this property be when the property changes sign? Set this equation equal to that value to find the time at which the particle changes direction. Make a number line and plug in values to make sure the critical point(s) results in a sign change at that point.

Question 48d

lschwartzman1 — 2021-01-29 18:02:54 UTC

How is distance different from displacement? How should the definite integrals be separated given what we learned in question 48c? Since the derivative of the displacement is the velocity, the antiderivative of the velocity is the displacement. To find the distance, the integrals would have to be separated into separate integrals. Refer to 4.6, slide 6, question 10 to see how to find the distance from a velocity function.

Question 33

climb1 — 2021-01-29 18:07:59 UTC

Note that the graph given is f'
Look at lesson 2.1 for a refresher on critical values of a function
Draw a number line to find intervals where f' is increasing/decreasing

Look at lessons 2.2/2.3 for a refresher on inflection points
Utilize the intervals on your number line from part a) and the graph of f'. Where on the graph of f' is f"<0 and f'<0?

Look at lessons 2.2/2.3 for a refresher on inflection points. Think about how concavity of f is shown on a graph of f' *remember at an inflection point there is a change in something*

How can you find f(x) from a graph of f' using integrals when given a point of f(x). Refer to lesson 4.8 slides 1&2
Once you have an integral of a function, remember that a definite integral of a function is the area under the graph between the interval

to find f(4) and f(-2)

plug in the x=4 and x=2 into the expression of f(x) that you found. Think about the meaning of the integral in our expression. How can we use the given areas of the regions under the graph of the intervals [-2,1] and [1,4]?
*Look carefully at what numbers are in the top and bottom of the integral sign*
- **refer to the law of integration below (also found in lesson 4.6)**

Question 37 Hints

daccurso1 — 2021-01-29 18:09:30 UTC

a. Think about where relative extremum occur (critical points). One of the functions must change signs at the critical points for a min and max to exist. (refer to 4.7 questions 31-33 for reference).

b. Think about what f' and f'' determine in the graph. (refer to 2.1-2.5 Review 1, question 19 for a similar question).

c. Determine what g'(x) is equal to in terms of f(x). Then, find critical points for the function that equals g'(x).

d. Watch video linked below for a refresher on finding points of inflection. Think about what g''(x) is equal to in terms of f(x).

Question 19

otogawa — 2021-01-29 18:13:20 UTC

1. Use the law of integration(posted below) to split the integration into two separate integrals; connect them by subtraction

2. Draw the a graph of y = sqrt(25-x^2) to help find the total area from the integral 0 to 5 of sqrt(25-x^2)dx

Question 43

jsalim3 — 2021-01-29 18:22:33 UTC

A) Make a velocity number line
B) Use your velocity number line that you drew in part A and test all critical points (including endpoints because it is on a closed interval).
C) Remember that you are taking into account TOTAL distance so you're going to have to use absolute value. Also, use your values in part B to help answer part C so you do not have to redo any of your previous work.
D) To find a(t), use the formula for rate of change of v(t) from the interval (7,10). If still stuck, refer to 4.3, slide number 3, question 5a to help find the acceleration equation. To find v(t), remember that velocity is the antiderivative of acceleration. Use the power rule for antiderivatives to get v(t) and solve for C with the points given by the velocity graph.
To find x(t), consider that you only have to find the squirrel's distance starting from the t=7 because the interval is from 7 < t < 10. Then, position is the antiderivative of velocity, so use the v(t) equation you had just found earlier to form an integral and add that to the squirrel's position when t=7.

Question 9

lwexler4 — 2021-01-29 18:32:36 UTC

Split up the integral into two parts (see the image below)
Find the critical points of the inside of absolute value, negate the negative parts of the equation to always have positive value
- https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-8c/v/definite-integral-of-absolute-value
Draw a possible graph that would resemble f(-x) = -f(x). How will this impact the area under the curve?

Question 39 hints

zberger4 — 2021-01-29 18:33:06 UTC

A) Remember that in this scenario g’(x) = f(x)
B) Draw a g’ number line to find when g is positive/negative
C) Use the 3 C’s checklist on 4.7 Slide 4 to accurately justify your answer
D) Recall that the point of inflection in a formula is when its second derivative changes signs. f’(x) = g”(x) as well

Question 3

akornblum — 2021-01-31 18:27:19 UTC

Question 45 a

akornblum — 2021-01-31 18:29:44 UTC

- Remember a(t) = v’(t)
- Use the equation v(b)-v(a)/ b-a to get the acceleration of the bike

Question 44

ecohen77 — 2021-01-31 18:38:50 UTC

Question 21

ecohen77 — 2021-01-31 18:40:28 UTC

Question 45 B

mjulian17 — 2021-01-31 18:42:38 UTC

V(t)= derivative of the position function
Use the Khan Academy video for a refresher on finding definite integrals using area formulas (start at 1:06 end at 3:30)
Remember that the absolute value of something is always positive
See the video lesson in the following post for a refresher on distance vs. displacement

Question 45 B continued

mjulian17 — 2021-01-31 18:43:53 UTC

Here is the video lesson on distance vs. displacement:

Question 45 C

akornblum — 2021-01-31 18:44:57 UTC

- When does the velocity change direction? **Think negative or positive**
- Remember when the velocity is 0, that means Caren is not moving
--> This question is asking when Caren turns around not when she stops moving

Question 45 D

mjulian17 — 2021-01-31 18:49:20 UTC

You need to find both Caren and Larry’s displacement in miles from the school
- For Larry:
  - When finding the integral, don’t forget to multiply by the reciprocal of the “inside part” of the trig functions in order to make up for the chain rule
- For Caren:
  - This is similar to b) but remember that you don’t use absolute values when finding displacement

Question 20

chaimes — 2021-01-31 18:54:55 UTC

Question 7

snardi3 — 2021-02-01 21:44:02 UTC

Remember that the x^3 is in the denominator. Keep in mind how this will effect the antiderivative and your answer. Pay attention to signs!

Question 8

snardi3 — 2021-02-01 21:45:11 UTC

Question 38

snardi3 — 2021-02-01 21:45:47 UTC

a) Keep in mind the area is below the x-axis. Remember that the graph of f is equal to the graph of g'. Take into account differentiability.
b) Remember a point of inflection is where the slope of g' changes signs. Keep in mind g'=f.
c) Pay attention to the placement of x in the integral. Look for points where h(x)=0. Make a table with points on the graph and find the integrals where the areas will cancel out.

Question 48 a

jseff — 2021-02-01 22:26:54 UTC

Remember the relationship that exists between velocity and acceleration. When taking the derivative of an expression with "e" in it do not forget to use the chain rule.

Question 48 b

jseff — 2021-02-01 22:27:46 UTC

Find the velocity at t=3 and compare it to acceleration at t=3. Remember what it means when something is speeding up, are velocity and acceleration in the same direction or are they in opposite directions?

Question 48 c

jseff — 2021-02-01 22:32:43 UTC

A critical point of the function v(t) does not always signify a change in direction, use a number line to see when the particle changes direction. Remember to explain your answer using words!

Question 48 d

jseff — 2021-02-01 22:34:47 UTC

Method 1:
To refresh your memory about working from velocity to position, refer to 4.3 Question 6. Draw a wiggle diagram to help find the total distance traveled by the particle over the time interval [0,3].

Question 2

cbedell1 — 2021-02-01 23:14:12 UTC

Question 38

jzimmerman50 — 2021-02-01 23:14:21 UTC

Question 14

jarcohen — 2021-02-01 23:24:58 UTC

Keep in mind the inverse trig functions (ArcSin, ArcCos, ArcTan)
Break down, simplify, and separate any fractions that you can

Question 15

jarcohen — 2021-02-01 23:27:03 UTC

multiply all aspects and simplify it prior to taking the integral

Question 41

jarcohen — 2021-02-01 23:28:32 UTC

a) after finding your velocity equation for particle P, make a number line
b) place the P and Q velocity timelines on top of each other and use your knowledge of the relationship between velocity signs and direction.
c) think about how the relationship between velocity and acceleration affects a particles change in speed
d) There are 2 ways to go about this problem

think about ways to create a position equation and find the constant if one exists
set up your integral with the given constant and plug in what we know

Question 39 Video

lwexler4 — 2021-02-02 01:12:18 UTC

Question 16

mamariam — 2021-02-02 02:00:21 UTC

Simplify the quadratic equation using division

(Use long division or synthetic division)
1. If you need help on how to long divide: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-div/x2ec2f6f830c9fb89:quad-div-by-linear/v/polynomial-division

Question 17

mamariam — 2021-02-02 02:01:18 UTC

With subtraction, we can separate the two components of the integral and find the antiderivative of them separately
If your stuck on how to find the antiderivative of the second component, think about the anti-derivatives of trig functions
1. If you don’t remember, look at 3.4 ⇒ pages 1-4 (specifically question 13)

Question 35

jschuster9 — 2021-02-02 02:11:30 UTC

a) Look for particular sign change in f': Note this is a local minimum and the graph shown is of f'. (Refer to lessons 2.1-2.3 on Behaviors of Functions)

b)
- Think about where critical points and endpoints are (Lesson 2.1)
- For the minimum, think of where f' changes negative to positive,
- For the maximum, think of where f' changes from positive to negative
- Remember laws of integration, and how this may affect the equation when solving (Refer to lesson 4.6)

c) Refer to Lesson 2.2 on concavity and Lesson 4.2 on particle motion to understand rules and behaviors of functions.

d) Remember to use Chain Rule (Refer to Lesson 1.5 - Chain Rule)

Question 4

sbluth — 2021-02-02 02:26:54 UTC

refer to 4.6 Properties of Integrals where the Law of Integrations are listed
find the area of f(x) from (-1,2)
refer to 4.4 Antiderivatives if you need help finding the antiderivative of 3x^2
plug in 2 and -1, then subtract
Solve

Question 36

sbluth — 2021-02-02 02:28:50 UTC

Hint: the graph of f(x) should be a piecewise function divided by f>0 and f<0
Refer to 1.1 for a refresher on the derivative formula
Determine derivative as h approaches 0 from the left side
Determine the derivative as h approaches 0 from the right side
Determine if the derivatives from the left and right side of 0 are equal

Hint: The average rate of change of f is equal to 0 when the change in y is equal to zero
Find where the change in y = 0 on the graph of f’(x)
Determine the amount of points where the change in y = 0 on the graph of f’(x)

Hint: the interval of (-4,6) can be divided into a piecewise functions
Refer to 2.7 to see the Mean Value Theorem
Input intervals where the average rate of change is 0 by using the mean value theorem
Find the interval where f’(c) = ⅓ (the given output of the mean value theorem)

Hint: if g’ = f, then g’’ = f’
Refer to 2.2 Concavity
Recall that a graph is concave up when its second derivative is increasing.
Determine the intervals where the function g(x) is concave up using concepts form 2.2

Question 42

mkerrick1 — 2021-02-02 02:35:15 UTC

Part A.
1. Open the link, Velocity is the speed of something in a given direction. Plug in a positive and negative velocity to see the difference.
  1. https://phet.colorado.edu/sims/cheerpj/moving-man/latest/moving-man.html?simulation=moving-man
2. Recognize the relationship between velocity and direction.
  1. If the particle moves right, velocity is ________. When the particle moves left velocity is ________.
3. Use a number line to find when the particle moves to the left
Part B
1. There is a difference between total distance and what the integral gives you. Can total distance include negatives? How can you get the integral to include total distance rather than displacement?
Part C
1. Think about the relationship between our velocity equation and acceleration equation. We learned these on 4.1. On the bottom of page 4 of 4.1. You should have these relationships.
2. How do we know when a particle is spending up or slowing down? If you're unsure, go back to the moving man link given in part a).
  1. Acceleration: When positive we are pushing the man to the right. When negative we are pushing him to the left.
  2. Speed: absolute value of velocity/ directionless velocity
  3. Speeding up= signs of velocity and acceleration are the same
  4. Slowing down= songs of velocity and acceleration are different
3. It is recommended to make a velocity number line and acceleration number line to compare the signs.
Part D
1. Recognize the relationship between velocity and position (also on bottom of 4.1)
  1. Furthermore, take the antiderivative of your velocity equation. Remember that you need to multiply by the inverse of the constant attached to “t”). Remember to use proper integral notation.
2. A time and position is given to us, using this we can find a numerical value for “c” in our antiderivative equation.

Question 3

jschuster9 — 2021-02-02 02:43:11 UTC

Split the question into g(x) and -2f(x)

Do not forget to add values for the split integrals at the end

**QUESTION 36 IN LESSON 4.5 WILL BE OF USE

Question 34

rbachmann1 — 2021-02-02 03:33:56 UTC

Steps:

Plug 3 into g(x)
Find the area in between the graph and the x-axis from -3 to 3

HINT: f is the derivative of g… In order to find when f is increasing what does the derivative graph have to look like? and in order to find when f is concave down what does the derivative graph have to be doing?

Steps:

find h’(x) by finding the derivative of g(x)/5x (Quotient Rule!)
Plug in 3 for x
Solve

Steps:

Find p’(x) (Chain Rule!)
Plug in -1 for x
Solve