# ap calculus ab free response fundamental theorem of calculus

Slope fields Slope fields are coordinate planes of 1 by 1 sections of slopes at each x and y value. If given a differential equation and asked to find a slope field, you plug in an x and y pair into the differential equation, the value it outputs is the slope at the point, and is the steepness you use to draw a line at that particular pair. Logistic Models with Differential Equations This is a joint proportional differential equation.

You can interpret the initial condition and the differential equation without solving the differential equation. The limiting value, also known as the carrying capacity can be determined using a logistic growth model. The logistic growth model can also be used to determine the value of the dependent variable in the logistic differential equation at the point when it is changing the fastest.

Big takeaways: Unit 8 teaches us more useful applications of the integral as we enter the 3D graph world! Content to Focus On: Average Value of a function The average value of a function is denoted by 1 b-a a b f x dx. We can only find the average value of a function that we have a definite integral for. Position, Velocity, Acceleration Position is the integral of velocity, and velocity is the integral of acceleration!

If you take the absolute value of velocity, you are able to find speed. Distance, the integral of the absolute value of velocity Displacement, the integral of velocity To remember this, I think about a track! After one lap around, your distance is m, but your displacement is 0.

Make sure if you are asked for distance, you remember to use absolute value! Finding area between curves If given two curves on a coordinate plane and endpoints, students should be able to integrate those functions, either with respect to the x or y axis, in order to determine the area between them.

When talking about the x axis, all functions should be in terms of x, the bounds should be in terms of x, and it will be the integral of the upper functions minus the lower. When talking about with respect to the y axis, all functions should be in terms of y, the bounds should be in terms of y, and it will be an integral of an outer function minus the inner.

Finding the area between curves that intersect at more than two points. Sometimes, your function will have different parts where you may need to split the area and add your answers together, because the upper functions changes, or another function changes.

When in doubt, break it up into pieces you know how to work with, and add those together. Volumes of cross sections When using cross sections, you integrate the area of whatever shape is given to you this could be a square, rectangle, triangle, or semicircle.

You will integrate the area on a given interval, but it is important that you interpret the function for the variable in the shape that it represents. If it says perpendicular to the x-axis, all should be in terms of x. Perpendicular to the y-axis, all should be in terms of y. Volumes by disks and washers In this case we are integrating an area again, but this time the shape will always be a circle, so we will always be using the area formula for a circle.

The variable that changes here is r, the radius, which you can interpret using the function. If the rotation is around just the x axis, all should be in terms of x bounds and functions and the radius will be your function. If the rotation is around just the y axis, all should be in terms of y bounds and functions and the radius will be your function. If it is rotated around any other vertical or horizontal line, you may need to add or subtract value from your function to find the radius.

A washer is when two functions are used, and you must subtract out the volume the inner function would have added into your total volume to account for the missing piece. A definite integral can be used to calculate this. Big takeaways: Unit 10 is only BC and covers many topics under the umbrella of sequences and series.

Content to Focus On: The harmonic series is a divergent infinite series. Notes and Examples University of Texas at Austin. Example Problems University of Colorado. Self-Score - fix mistakes, understand correct work. Chapter 7 — Applications of Integration. W asher Method Animation. Skills Check: Volumes of Rotated Solids.

Chapter 7 Multiple Choice Practice. Guided Notes - KEY. Chapter 6 — Differential Equations. Accumulation Functions. Differential Equations and Exponential Growth. Area Between Curves. What is the value of? The given information tells us that , which means that. Thus, we will substitute 3 into the equation. The given information tells us that. The equation then becomes. We can now solve for.

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The way a Riemann sum works is that it approximates the area by summing up the area of rectangles and then finding the What is a limit? In mathematical All Rights Reserved. The Princeton Review is not affiliated with Princeton University. Recently viewed. Find Your Dream School.

By submitting my email address. Any time a definite integral needs to be evaluated, the Fundamental Theorem of Calculus can come to the rescue. If a function f x is nonnegative on an interval [ a , b ], then the area of the region under the curve can be computed by a definite integral.

The limits of integration, a and b , specify the left and right boundaries of the region. The bottom boundary is the x -axis, and the top boundary is the graph of f x itself. The graph is a diagonal line through the origin. First set up the definite integral that computes the area. By working the problem out using a different method, I hope to show you the remarkable connection between areas and antiderivatives.