6.2 Related Ratesap Calculus

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Free practice questions for AP Calculus AB - Modeling rates of change, including related rates problems­. Includes full solutions and score reporting. AP Calculus Units. There are too many Related Rate Problems and videos to put here if you want more help go to youtube and search Related Rates. 6.2 Related Rates Suppose we have two variables and (in most problems the letters will be different, but for now let's use and) which are both changing with time. A 'related rates' problem is a problem in which we know one of the rates of change at a given instant—say, —and we want to find the other rate at that instant.

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Section 3-11 : Related Rates

  1. In the following assume that (x) and (y) are both functions of (t). Given (x = - 2), (y = 1) and (x' = - 4) determine (y') for the following equation. [6{y^2} + {x^2} = 2 - {x^3}{{bf{e}}^{4 - 4y}}] Solution
  2. In the following assume that (x), (y) and (z) are all functions of (t). Given (x = 4), (y = - 2), (z = 1), (x' = 9) and (y' = - 3) determine (z') for the following equation. [xleft( {1 - y} right) + 5{z^3} = {y^2}{z^2} + {x^2} - 3] Solution
  3. For a certain rectangle the length of one side is always three times the length of the other side.
    1. If the shorter side is decreasing at a rate of 2 inches/minute at what rate is the longer side decreasing?
    2. At what rate is the enclosed area decreasing when the shorter side is 6 inches long and is decreasing at a rate of 2 inches/minute?
    Solution
  4. A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec at what rate is the radius decreasing when the area of the sheet is 12 m2? Solution
  5. A person is standing 350 feet away from a model rocket that is fired straight up into the air at a rate of 15 ft/sec. At what rate is the distance between the person and the rocket increasing (a) 20 seconds after liftoff? (b) 1 minute after liftoff? Solution
  6. A plane is 750 meters in the air flying parallel to the ground at a speed of 100 m/s and is initially 2.5 kilometers away from a radar station. At what rate is the distance between the plane and the radar station changing (a) initially and (b) 30 seconds after it passes over the radar station? Solution
  7. Two people are at an elevator. At the same time one person starts to walk away from the elevator at a rate of 2 ft/sec and the other person starts going up in the elevator at a rate of 7 ft/sec. What rate is the distance between the two people changing 15 seconds later? Solution
  8. Two people on bikes are at the same place. One of the bikers starts riding directly north at a rate of 8 m/sec. Five seconds after the first biker started riding north the second starts to ride directly east at a rate of 5 m/sec. At what rate is the distance between the two riders increasing 20 seconds after the second person started riding? Solution
  9. A light is mounted on a wall 5 meters above the ground. A 2 meter tall person is initially 10 meters from the wall and is moving towards the wall at a rate of 0.5 m/sec. After 4 seconds of moving is the tip of the shadow moving (a) towards or away from the person and (b) towards or away from the wall? Solution
  10. A tank of water in the shape of a cone is being filled with water at a rate of 12 m3/sec. The base radius of the tank is 26 meters and the height of the tank is 8 meters. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 10 meters? Note the image below is not completely to scale…. Solution
  11. 11. The angle of elevation is the angle formed by a horizontal line and a line joining the observer’s eye to an object above the horizontal line. A person is 500 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, (theta ), changing when the hot air balloon is 200 feet above the ground. See the (probably bad) sketch below to help visualize the angle of elevation if you are having trouble seeing it. Solution

By its nature, Calculus can be intimidating. But you can take some of the fear of studying Calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Also discover a few basic rules applied to Calculus like Cramer’s Rule, and the Constant Multiple Rule, and a few others, and you’ll be on your way to acing the course.

The Most Important Derivatives and Antiderivatives to Know

The table below shows you how to differentiate and integrate 18 of the most common functions. As you can see, integration reverses differentiation, returning the function to its original state, up to a constant C.

3) Related Rates- Water in a Tank 4) Complete 'Related Rates (Volume & Surface Area)' pre-quiz Watch Spencer Munroe (APC '14) explain TW question #2 here. 6.2.3 Find the volume of a solid of revolution with a cavity using the washer method. In the preceding section, we used definite integrals to find the area between two curves. In this section, we use definite integrals to find volumes of three-dimensional solids.

The Riemann Sum Formula For the Definite Integral

The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. The Riemann Sum formula is as follows:

Below are the steps for approximating an integral using six rectangles:

  1. Increase the number of rectangles (n) to create a better approximation:

  2. Simplify this formula by factoring out w from each term:

  3. Use the summation symbol to make this formula even more compact:

    The value w is the width of each rectangle:

    Each h valueis the height of a different rectangle:

    So here is the Riemann Sum formula for approximating an integral using n rectangles:

  4. For a better approximation, use the limit

  5. to allow the number of rectangles to approach infinity:

Integration by Parts with the DI-agonal Method

6.2 related ratesap calculus worksheet

The DI-agonal method is basically integration by parts with a chart that helps you organize information. This method is especially useful when you need to integrate by parts more than once to solve a problem. Use the following table for integration by parts using the DI-agonal method:

The Sum Rule, the Constant Multiple Rule, and the Power Rule for Integration

When you perform integration, there are three important rules that you need to know: the Sum Rule, the Constant Multiple Rule, and the Power Rule.

The Sum Rule for Integration tells you that it’s okay to integrate long expressions term by term. Here it is formally:

The Constant Multiple Rule for Integration tells you that it’s okay to move a constant outside of an integral before you integrate. Here it is expressed in symbols:

The Power Rule for Integration allows you to integrate any real power of x (except –1). Here’s the Power Rule expressed formally:

where n ≠ –1

How to Solve Integrals with Variable Substitution

In Calculus, you can use variable substitution to evaluate a complex integral. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work.

Related
  1. Declare a variable u,set it equal to an algebraic expression that appears in the integral, and then substitute u for this expression in the integral.

  2. Differentiate u to find

    and then isolate all x variables on one side of the equal sign.

  3. Make another substitution to change dx and all other occurrences of x in the integral to an expression that includes du.

  4. Integrate by using u as your new variable of integration.

  5. Express this answer in terms of x.

How to Use Integration by Parts

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When doing Calculus, the formula for integration by parts gives you the option to break down the product of two functions to its factors and integrate it in an altered form. To use integration by parts in Calculus, follow these steps:

  1. Decompose the entire integral (including dx) into two factors.

  2. Let the factor without dx equal u and the factor with dx equal dv.

  3. Differentiate u to find du, and integrate dv to find v.

  4. Use the formula:

  5. Evaluate the right side of this equation to solve the integral.

How to Solve Compound Functions Where the Inner Function Is ax + b

Some integrals of compound functions f(g(x)) are easy to do quickly in Calculus. These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax + b — that is, it differentiates to a constant.

Here are some examples:

Solve Compound Functions Where the Inner Function Is ax

When figuring Calculus problems, some integrals of compound functions f(g(x)) are easy to do quickly. These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax — that is, it differentiates to a constant.

Here are some examples: