FRQ TIPS TO REMEMBER-AP CALC AB J

How would you find if the amount at a processing plant would be increasing or decreasing at t=x (calc)

Calculate the A’(t)(total amount) =G(t)= at t=x

in this ex: A’(5)=G(5)=90+45cos((5²)/18) =98.141 tons

Since 98.141<100 and the gravel is being processed at a rate of 100 tons per hour, the amount of unprocessed gravel is decreasing at t=5

How do you calculate the maximum amount of unprocessed gravel at the plant during hours of operation (calc)

Plug in the equation for the rate at which it arrives - the rate at which it processes to find the x value for the time. Then calculate the initial amount at t=0 + the integral from 0 to t for the amount at which gravel arrives - the rate at which gravel gets processed, plug in the t value you got before to find the max amount. Then check both interval values to see if they are the max either.

How do you find the speed of the particle at a certain time? (calc)

It is the absolute value of velocity, so plug it in, then in the second y equation make it equal to the time asked and see what values they intersect at between the intervals given

How do you write an expression involving an integral that gives the position s(t). and find the position of the particle at a time?(calc)

The equation of s(t)= initial amount at s(0)=initial amount + the intergral from o to t=x of (v(t))dx

When will the particle change direction?(calc)

Whenever v(t)=0. and also changes signs, so look at the graph of v(t) to see the x values and justify by saying that since the graph shows at t=x the sign changes from - to + or the other way around.

How do you know whether the speed of a particle increases or decreases at t=x (calc)

If a(x)= v’(x) has the same sign as v(x) when t=x the function is increasing and decreasing if func doesn’t have the same signs for velocity and acceleration

What do you write when something asks for a approximation

~

When asked, is there a time t, 2<t<4 where c’(t)=2, and justify answer, how do you figure that out (no calc)

Find the AROC from 4 to 2 and if it equals 2 you can justify by saying since the function is differentiable meaning its continuous then by Mean Value Theorem there must be a value in the int 2<x<4 where c’(t)=2

How do you use a midpoint sum and what does the equation of 1/6(integralfrom 0 to 6 for C(t)dt mean?

ex: wid=2

Midpoint sum is calculated by 1/b-a so in this case 1/6-0 or just, 1/6 then mult that by the int from 0 to 6 for each mid point value times the width like 2(c(X))+2(c(x))+ect.. then get that value. This equation tells us the average amount of coffee in the cup over the interval 1 to 6

What is the average value integral?

1/b-a times the int from a to b times f(x)dx

what does it mean the func on a OPEN interval?

DO NOT include the endpoints

The graph is a f’(x) graph, how can we tell when f(x) has a local minimun on a open interval of 0<x<8

Look for the x value where the graph f’(x) changes from a neg to positive meaning that it is concave up and a local min, but dont look at the end point values since its open int

How do you find the open int at which f(X) is both concave down and increasing when given the graph of f’(x)?

The graph of f(x) will be both conc down and increasing when f’(x) is decreasing but positive so find those values. ex: 0<x<1, and 3<x<4.

When given f(x)= func and g(x)= func then what is the area R in between

first: plug in 0 to find wich func is greater than zero and in this case g(x) is so to find the area it would be a(x)= the int from 0 to 2 for g(x)-f(x)

How do you write an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y=4

since y=4 is above the f(x) and g(x) funcs then the volume equation is modeled by piin time integral of (4-(f(x))²-(4-(g(x))²dx

If region R is the base of a solid. Each cross sectipn perpendicular to the x-axis is a square, how do you write the integral expression that gives the volume of the solid?

Since the cross sectipns perpendicular to the x-aixs are squares, the area of each square is given by: the integral from 0 to 2 (g(x)-f(x)²dx

find the y=f(x), the particular solution to the diff equation that passes through (1,0) how do you do this?

for the particular solution often you must use the separation of variables, you find c, then you plug in c and make it equal to y for an equation