✍️ Free Response Questions (FRQ)
Calculus Free Response Questions
👑 Unit 1: Limits & Continuity
1.5Determining Limits Using Algebraic Properties of Limits
1.6Determining Limits Using Algebraic Manipulation
1.10Exploring Types of Discontinuities
1.11Defining Continuity at a Point
1.12Confirming Continuity over an Interval
🤓 Unit 2: Differentiation: Definition & Fundamental Properties
2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
🤙🏽 Unit 3: Differentiation: Composite, Implicit & Inverse Functions
3.0Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions
3.1The Chain Rule
3.3Differentiating Inverse Functions
3.4Differentiating Inverse Trigonometric Functions
👀 Unit 4: Contextual Applications of the Differentiation
4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.4Intro to Related Rates
4.6Approximating Values of a Function Using Local Linearity and Linearization
✨ Unit 5: Analytical Applications of Differentiation
5.0Unit 5 Overview: Analytical Applications of Differentiation
5.2Extreme Value Theorem, Global vs Local Extrema, and Critical Points
5.3Determining Intervals on Which a Function is Increasing or Decreasing
5.4Using the First Derivative Test to Determine Relative (Local) Extrema
5.7Using the Second Derivative Test to Determine Extrema
🔥 Unit 6: Integration and Accumulation of Change
6.11Integrating Using Integration by Parts
💎 Unit 7: Differential Equations
7.0Unit 7 Overview: Differential Equations
7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
🐶 Unit 8: Applications of Integration
8.1Finding the Average Value of a Function on an Interval
8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4Finding the Area Between Curves Expressed as Functions of x
8.5Finding the Area Between Curves Expressed as Functions of y
8.6Finding the Area Between Curves That Intersect at More Than Two Points
8.7Volumes with Cross Sections: Squares and Rectangles
8.8Volumes with Cross Sections: Triangles and Semicircles
8.9Volume with Disc Method: Revolving Around the x- or y-Axis
8.10Volume with Disc Method: Revolving Around Other Axes
8.11Volume with Washer Method: Revolving Around the x- or y-Axis
🦖 Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
9.0Unit 9 Overview: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.1Defining and Differentiating Parametric Equations
♾ Unit 10: Infinite Sequences and Series (BC Only)
10.0Unit 10 Overview: Infinite Series and Sequences
10.1Defining Convergent and Divergent Infinite Series
10.6Comparison Tests for Convergence
10.7Alternating Series Test for Convergence
10.1110.11 Finding Taylor Polynomial Approximations of Functions
10.14Finding Taylor or Maclaurin Series for a Function
🧐 Multiple Choice Questions (MCQ)
⏱️ 3 min read
June 8, 2020
🎥Watch: AP Calculus AB/BC - Separable Differential Equations
Continued from 7.6 Finding General Solutions Using Separation of Variables.
Let’s look at the differential equation given in Eq. 12:
If we treat the derivative as a fraction, we can do something that would disgust professional mathematicians*:
We can perform step 5 because the two sides of the equation are equivalent, thus we can perform the same operation (ie. integration) on both sides and not lose any information.
Continuing the integration:
Note that we need two constants because we found two different antiderivatives. However, we can combine them into a single constant:
This statement means that, because the set of real numbers is an infinite set** and both of the constants are real numbers, the difference between them is also a real number, ie. another constant we can call C.
Notice that this equation is also equivalent to the solution given in Eq. 12. The solution given in Eq. 27 is a generalization of the solution given in Eq. 12.
Now, let’s say that we have some prior information about the function called an initial condition, eg. y(0) = 1. We can use this information to solve for C:
This implies that C = 0, which means we have found the specific solution to the differential equation in Eq. 12 along with the condition that y(0) = 1.
We can also revisit the differential equation in Eq. 11:
We need to take special precautions regarding the absolute value signs when solving for y:
So, the general solution is y = C2e^x - 5. For an additional exercise, find a specific solution to the differential equation such that y(ln5) = -6.
Another differential equation that is solvable via separation of variables is the equation below:
Notice that if C = 0, the solution to the differential equation is just y = x. If you plug y = x into the differential equation in Eq. 33, you will notice that this returns a valid equality.
Solve the differential equation below given an initial condition and a range restriction. (Hint: rewrite the arcsine expression using the identity that defines a cosine graph as a shifted sine graph and do something similar for the arccosine expression.)
The arc trig expressions are red herrings:
*This mathematical move is actually incorrect because multiplying the dx term over in that fashion implies that dx is a number and therefore dy/dx is a fraction, which is not true. However, this move is completely acceptable for the AP exam.
**“Infinite” in this context is not well-defined. The actual size of the set is called its cardinality, which is not simply infinity as implied in the statement:
Some definitions also include zero in the set of natural numbers (the lastly defined set in the statement).
2550 north lake drive
milwaukee, wi 53211
92% of Fiveable students earned a 3 or higher on their 2020 AP Exams.
*ap® and advanced placement® are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.
© fiveable 2020 | all rights reserved.