AP Calc

Question 1

Integrals and anti-derivatives topics

Answer 1

1. Riemann Sums
2. Definite integrals
3. Fundamental Theorem of Calculus

Question 2

A Riemann Sum equivalent to the definite integral from a to b of f(x)dx

Answer 2

limit as delta x -> 0 of the sum of f(xi)deltax for a

a Riemann sum is an approximation:

Question 3

Given the anti derivative of any given function, what will you almost always need to add at the end?

Answer 3

C, the constant that will go to 0 once differentiated

Question 4

integral from a to b f(x)dx

Answer 4

f(b) - f(a) where F’ = f

Question 5

Derivative with respect to x of an integral from a to x? ( f(t) and dt )

Answer 5

f(x)

Question 6

d/dx of a to g(x) f(t)dt?

Answer 6

f(g(x))*g’(x)

Question 7

Given the fact that an antiderivative is the opposite direction of the norm, what must we consider?

Answer 7

We do not have the same level of certainty: the constant, C, accounts for this.

Question 8

SS signifys antiderivative: SS sec u * tan u du

Answer 8

sec u + C

Question 9

SS csc u * cot u du

Answer 9

-csc u + C

Question 10

SS tan u du

Answer 10

ln |sec u| +C

Question 11

What is the antiderivative of cot(x)?

Answer 11

1. cot(x)=cos(x)/sin(x)
2. Let t=sin(x);

=> dt=cos(x)dx.

3. Integral 1/t == ln |t|

=> Integral cot(x) = ln | sin(x) | + C

ln(|x|)+C

Question 12

Integral(square(csc(x)))

Answer 12

-cot(x) + C

Question 13

SS secu du

Answer 13

ln| secu + tanu | + C

Question 14

SS cscu du

Answer 14

-ln| csc u + cot u | +c

Question 15

SS du(sqrt (1-u²⁾ )

Answer 15

arcsin u + C

Question 16

SS du/(1+u²)

Answer 16

arctanu + C

Question 17

SS du/(u*sqrt(u² -1))

Answer 17

arcsec u u du

Question 18

SS e^u du

Answer 18

e^u + C

Question 19

SS a^u du

Answer 19

a^u /ln a

Question 20

SS du/u

Answer 20

ln |u| + C

Question 21

Displacement

Answer 21

Integral of velocity with respect to time

Question 22

Distance

Answer 22

Absolute value of the integral of velocity with respect to time:

distance is different from displacement. Cannot have negative distance.

Question 23

Area between f and g, where f(x) >= g(x) from a to b

Answer 23

SS from a to b of ( f(x) - g(x) ) dx

Question 24

For any problem dealing with change:

If R(t) represents a rate of change, then the SS fomr a to b R(t)dt represents…

Answer 24

the accumulated amount of change during the timer interval from a to b.

MPH ==> total miles driven (minus negative displacement)

10 gals/sec ==> how many gallons are in the pool

etc etc

Question 25

Volume by disks

Answer 25

1. Rotate function about x-axis- dx rectangles become dx-height cylinders e.g “disks”
2. Same approximating model- limit as dx goes to 0 of sum of disks
3. Volume of each disk ==> pi*r²*h = pi * ( f(x) )² * dx
4. SS pi*f(x)² dx

Question 26

Volume by washers

Answer 26

dV = pi(R² - r²) dx

Question 27

How do you compute the Left Riemann Sum of a function?

Answer 27

For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 0, 1, …, n − 1, and adding up the resulting areas gives:

Question 28

The left Riemann sum amounts to an overestimation if f is ?

Answer 28

The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

Question 29

What is the Left Riemann Sum of a function?

Answer 29

• Let
• If for all i, then S is called a left Riemann sum.

Question 30

Draw a diagram of the Left Riemann sum of x³over [0,2] using 4 subdivisions

Answer 30

Left Riemann sum of x³over [0,2] using 4 subdivisions

Question 31

Draw a diagram of the right Riemann sum methods of x³over [0,2] using 4 subdivisions

Answer 31

Riemann sum methods of x³over [0,2] using 4 subdivisions

Right

Question 32

Riemann sum methods of x³over [0,2] using middle subdivisions

Answer 32

USE MRAM

Question 33

Draw a diagram of Riemann sum methods of x³over [0,2] using trapezoidal subdivisions.

Answer 33

Riemann sum methods of x3over [0,2] using trapezoidal subdivisions

Question 34

Compute the area under the curve of y = x² between 0 and 2 using Riemann’s method.

Answer 34

1. The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of ; these are the widths of the Riemann rectangles (hereafter “boxes”).
2. Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be .
3. Therefore, the sequence of the heights of the boxes will be .
4. It is an important fact that , and .
5. The area of each box will be and therefore the nth right Riemann sum will be:

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1. Take the limit as n → ∞

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This method agrees with the definite integral as calculated in more mechanical ways:

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Question 35

Draw the diagrams for the 4 Riemann Sum methods for an arbitrary function f.

Answer 35

Four of the Riemann summation methods for approximating the area under curves.

1. Right and leftmethods make the approximation using the right and left endpoints of each subinterval, respectively.
2. Maximum and minimum methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively.
3. The values of the sums converge as the subintervals halve from top-left to bottom-right.