AP Calculus BC - Review Notes

Flashcards for AP Calculus BC review, focusing on Riemann sums, trapezoidal approximations, and integration techniques.

A tank contains 50 liters of oil at time t=0 hours. How can you approximate the number of liters at t=3 using the provided rate data?

The number of liters of oil in the tank at time t=3 hours is approximately 118.2 liters, calculated using a right Riemann sum.

How to approximate the average level of cholesterol over a period of time using the trapezoidal rule?

The average cholesterol level over the 10-week period is approximately 195, using a trapezoidal approximation.

When does a right Riemann sum provide an underestimate of the integral?

A right Riemann sum underestimates the value of the integral when the function is decreasing.

How to approximate the value of a definite integral using Riemann sum?

The approximate value of the definite integral is 20, using a right Riemann sum with four subintervals.

How to calculate the approximate definite integral using left Riemann sum?

The approximation of the definite integral obtained from a left Riemann sum is 14.

How to calculate the approximate area using trapezoidal rule?

The approximate area of the shaded region is calculated using the trapezoidal rule.

How do you calculate the trapezoidal approximation of a definite integral given subintervals?

The trapezoidal approximation of is 130, found using subintervals [2,5], [5,7], and [7,8].

Estimate a definite integral using a right Riemann sum with given subintervals?

The approximation of is 312 using a right Riemann sum with subintervals [2,5], [5,10], and [10,14].

How do you compute the approximate value of velocity at a specific time using left riemann sum?

The approximate value of the velocity at t=6 is 37 ft/sec, computed using a left-hand Riemann sum with three subintervals.

How do you approximate the value of an integral using the trapezoidal rule?

The trapezoidal approximation of the integral is 12.

How do you approximate the distance traveled using trapezoidal sum?

The approximate distance the truck traveled is 125 miles, calculated using a trapezoidal sum.

How can you approximate the amount of liquid in a tank using trapezoidal sum?

The approximation of the number of gallons of water in the tank at time t = 9 is 67 gallons, using a trapezoidal sum.

How do you approximate the value of an integral using left Riemann sum?

The value of the left Riemann sum approximation is 380.

How to estimate the value of a definite integral using the midpoint Riemann sum?

The approximation of the definite integral obtained from a midpoint Riemann sum is 3.

How approximate the total amount of material deposited using a trapezoidal sum?

The approximate number of tons of material deposited in the first 9 hours is 68 tons, calculated using a trapezoidal sum.

How to approximate the total people in an evacuation drill using Riemann sum?

The approximation of the number of people who leave the building during the first 15 minutes is 2075, using a right Riemann sum.

Approximate the value of the definite integral using a left Riemann sum?

The approximation is using a left Riemann sum with three subintervals of equal length.

How to approximate the value of the definite integral using the right Riemann sum?

The approximation is obtained by using a right Riemann sum with the subintervals.

How do you approximate in Riemann Sums if the function is increasing?

Left Riemann sum underestimates and a right Riemann sum overestimates when the function is increasing.

When is right riemann sum an underestimate?

A right Riemann sum approximation would be an underestimate for since f is decreasing.

Define midpoint sum

The midpoint sum is formed using the midpoint of each of the four intervals.

Definition of tangent line

Here, for the expression An equation of the tangent line is therefore .

Fundamental Theorem of Calculus, Intermediate Value Theorem

This option is correct. By the Fundamental Theorem of Calculus,,Since is continuous on the closed interval and then by the Intermediate Value Theorem there must be a value in the open interval such that

Midpoint Riemann sum concavity condition

If changes from increasing to decreasing, and the graph of changes from concave up to concave down, the midpoint Riemann sum approximation is The midpoint sum is formed using the midpoint of each of the four intervals.

Trapezoidal sum approximates what graph shape

The differentiable function is increasing, and the graph of is concave down. Selected values of at selected values of are given in the table above. The trapezoidal sum approximates . Which of the following statements is true. is increasing

Correct. The definite integral can be used to define an accumulation function according to the Fundamental Theorem of Calculus

Correct. The definite integral can be used to define an accumulation function according to the Fundamental Theorem of Calculus

tangent line and slope for the graph

the line tangent to the graph of will have slope If the line tangent to the graph of at parallel to the line tangent to the graph of what is the value

Because is the chain rule gives .

Because is the chain rule gives .

By the Fundamental Theorem of Calculus and using area and geometry what is given

Correct. By the Fundamental Theorem of Calculus and using area and geometry to calculate the definite integral,

By the Fundamental Theorem of Calculus how is determined

By the Fundamental Theorem of Calculus if then f(4) =

Right Riemann sum explained

Correct. The sum can be interpreted as a right Riemann sum i. The value of corresponds to an interval of length . The sum starts with the right endpoint and ends with the right endpoint so the Riemann sum is over the interval

Condition of a slope field

A The line segments in the slope field have slopes given by = x2 + y at the point ( x, y ). B In Quadrants I and II, all slopes must be positive or zero since y > 0 in those quadrants and x2 ≥ 0. C This is the only option in which that condition is true.

What differential equation will relate to above graph of a slope field

10. Shown above is a slope field for which of the following differential equations.

By the second derivative test, the graph of a solution to the differential equation has a local minimum at .

For a solution to this differential equation, and By the second derivative test, the graph of a solution to the differential equation has a local minimum at. Therefore, this graph could be the graph of a solution.

Derivatives of Equation

A 3-B. 6. C A solution to a differential equation can be verified by substituting the function and its derivative(s) into the equation. For the derivatives are and for

slope field for a differential equation of the form

In the slope field for a differential equation of the form, the slope at a point depends only on the value of The line segments in the slope field at each point on a vertical line perpendicular to the axis should therefore all have the same slope. The line segments in this slope field show that behavior and therefore this could be a slope field for a differential equation of the form

differentiable equations using slop field

The slope field indicates that the slopes are undefined at. A solution and as

differential equation and it characteristics

For a solution to this differential equation the second derivative test graph has local min

the slope field indicate what condition on graph

If the solution and linear it fall on a graph starting

The exponential growth model has a solution

The exponential growth model has a solution of the form where and since the amount of the drug in the bloodstream is decreasing.

particle moves linear to location

a particle moves along the -axis. , At t = 0, the particle is at position x = 1 then distance to location

what is the distance in what condition

if velocity given what is the distance

The definite integral is what on calculator

The definite integral is evaluated on the calculator.

how to find an exponential growth model

where and since the amount of the drug in the bloodstream is decreasing.

ⅆWhat graph to be aware for a solution

the particle is growing ⅆ. At at ⅆ

Note about evaluating a graph

Note: Absolute value was not used in the antiderivative of since the initial value of is positive.

Using substitution what is the final result

how solve integral 3. where integral 2

Using what to solve the problem

how will 3integral be 2integral with u

what conditions for integral result

Which of the following must be true I1 3 and -x

2x32 what are the solutions

2x32. which are solutions to where is a constant 123123123

Fundamental Theorem for with initial

B By the Fundamental Theorem of (5) with initial

what about Volume when revolving

What volume to solid when revolving about

How many vertical asymptotes does the graph of have?

How many vertical asymptotes does the graph of have?