Chapter 4 Test

Question 1

Review homework questions

Question 2

How do you find the slope of a tangent line to a graph using implicit differentiation?

Answer 2

Find the derivative using implicit differentiation, then plug the point into the derivative to find the slope

Question 3

What is the antiderivative of velocity?

Answer 3

Position

Question 4

Initial Condition

Answer 4

A point that is on the original curve

Question 5

How do you find the area of a semicircle?

Answer 5

1/2 times pi radius^2

Question 6

How do you set up a Reimann Sum?

Answer 6

Do the capital sigma symbol, the number on top is where we stop adding and the number that i equals is where we start. Then you multiply the height by the width for every subinterval.

Question 7

Area of a Trapezoid Formula

Answer 7

( length of bases added together/2)( height of trapezoid)

Question 8

What’s the equation of a semicircle?

Answer 8

y= sq rt(radius^2 - x^2)

Question 9

Derivative of cot x

Answer 9

-csc^2(x)

Question 10

When speed is decreasing, velocity and acceleration have ________ signs

Answer 10

opposite

Question 11

When speed is increasing, velocity and acceleration have ________ signs

Answer 11

same

Question 12

How do you deal with definite integrals with a constant attached?

Answer 12

Find the definite integral of the basic function and multiply the constant out at the end

Question 13

Derivative of sec x

Answer 13

sec x tan x

Question 14

Product rule

Answer 14

fg’ + gf’

Question 15

How do you find when something hits the ground?

Answer 15

Make the position function equal to 0 then solve the quadratic function for t

Question 16

How do you find velocity?

Answer 16

It is the first derivative of the position function

Question 17

How do you write the equation of a line tangent to a graph at a particular point?

Answer 17

First find the derivative. Then plug the x value into the derivative to get the slope. Then, use y=mx+b and plug the point into the equation to get the x intercept.

Question 18

How are continuity and integrability related?

Answer 18

If a function is continuous it has an integral that exists. Meaning it has an antiderivative and the area under its curve is defined

Question 19

How do you find the area of regions without using Reimann Sums?

Answer 19

If they are easy shapes you know how to find the area of then just find the area of them that way

Question 20

How do you use implicit differentiation?

Answer 20

Whenever you find the DERIVATIVE of something with a y, you put a dy/dx after it. Then solve for dy/dx.

Question 21

Derivative of cos x

Answer 21

• sin x

Question 22

How do you get a more accurate approximation using Reimann Sums and how do you get an exact answer?

Answer 22

Increase the number of subintervals; And use limits

Question 23

Derivative of csc x

Answer 23

• csc x cot x

Question 24

How do you find the antiderivative of a fraction like: (x+1) over the sq rt of x?

Answer 24

You have to make sq rt x a negative exponent, then make them “multiplied” and multiply them out, then do the antiderivatives separately, and then add them together.

Question 25

How do you find acceleration?

Answer 25

It is the second derivative of the position function

Question 26

How do you set up a definite integral?

Answer 26

Do the S symbol, the number on top is where we stop adding and the number on the bottom is where we start. Then you multiply the height by the width for every subinterval, but you write f(x)dx

Question 27

Indefinite integral vs definite integral

Answer 27

Indefinite- Antiderivative Definite- Area between the function and the x axis of a graph

Question 28

How do you deal with added/subtracted definite integrals?

Answer 28

Find the definite integrals of the two separately then add or subtract

Question 29

When should implicit differentiation be used?

Answer 29

If you cannot easily, if at all, solve for y.

Question 30

How do you approximate area using a trapezoid?

Answer 30

Create the trapezoids by making lines all the way to the function then connecting the dots. Then, the widths of the subintervals are the heights of the trapezoid. And the outputs of the function at the endpoints of each subinterval are the needed base lengths. You find area of a trapezoid by doing (base1 + base2 all over 2)(height).

Question 31

What is implicit differentiation?

Answer 31

When you solve for dy/ dx

Question 32

How do you utilize initial conditions with the antiderivative?

Answer 32

You plug the initial condition into the antiderivative and solve for c, then rewrite the antiderivative

Question 33

Additive Identity Property

Answer 33

If f is integrable on three intervals [a,c], [a,b] and [a,c] where a

Question 34

Chain Rule

Answer 34

f'(g(x)) times g'(x)

Question 35

Quotient Rule

Answer 35

BT'- TB' all over B^2 B= Bottom T=Top

Question 36

How do you find the approximation of the area of a region with rectangles using Left Riemann Sum?

Answer 36

First see how many subintervals you have and try to make the widths even. To do this find the [a,b], subtract those values and divide it over the subintervals. Then, use the farthest left interval to draw the rectangles. Then, multiply width times the farthest left interval into the function to get height. Then, you will get your approximation. It is the same for the right riemann sum and the midpoint reimann sum.

Question 37

Derivative of sin x

Answer 37

cos x

Question 38

How do you set up an antiderivative?

Answer 38

Do the "S" symbol then f(x)dx, the answer will be the antiderivative plus c

Question 39

How do you find speed?

Answer 39

It is the absolute value of velocity

Question 40

Derivative of tan x

Answer 40

sec^2(x)

Question 41

What is the antiderivative of acceleration?

Answer 41

Velocity

Question 42

How do you find an antiderivative?

Answer 42

S(x^n) dx= (1 over n+1) x^(n+1) + c

Question 43

How could you find a position function given an initial height and velocity?

Answer 43

Find the antiderivative of acceleration then plug in in the initial velocity for the c value. Then find the antiderivative of your velocity equation, then plug in the initial height to find the c value for the position function.

Question 44

What should you remember for every time you do an antiderivative?

Answer 44

Add a c value

Question 45

Review ap review questions

Question 46

If a definite integral seems like it is "backwards"(goes from a larger number to a smaller number(6 to 3)), what should you do?

Answer 46

You should reverse the integral and what it equals(change the sign)

Question 47

How do you solve an added definite integral?

Answer 47

Find the area for each part of it then add it all together

Question 48

How do you do the definite integral of a piecewise function?

Answer 48

Graph it if you can but break it up into two shapes you know then find the area for both

Question 49

How do you know if the position of something is moving left or right if given the position function?

Answer 49

Find the velocity function and plug in your time and if it is positive it is moving to the right. If negative it is moving to the right.

Question 50

How do you know if the position of something is speeding up or slowing down if given the position function?

Answer 50

Find the acceleration function and plug in your time and if it has an opposite sign from the velocity then it is slowing down(speed is decreasing). If they are the same speed is increasing(the function is speeding up).

Question 51

If you are asked to find the slope of a tangent line to the graph of the function at a given point how do you solve this?

Answer 51

First, find the derivative of the function. Then, plug the x value into your derivative.