✍️ 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 1, 2020
Differential equations are at the heart of mathematical modeling for all the sciences: physical, social, financial, et cetera.
Mathematical modeling is the use of math to describe physical situations. Essentially, math functions as a language. While some sciences prefer to put emphasis on traditional language to describe phenomena (we use words to describe explain biological happenings, for example) other sciences, like physics, rely on mathematical modeling to express ideas and concepts.
Mathematical modeling is useful because it often yields logical derivations that could not have been predicted in the first place. For example, the mathematical modeling of the behavior of particles directly led to the conclusion that subatomic particles can travel through a barrier of infinite energy, a deduction that would seem impossible through a different lens*.
Mathematical modeling also helps scientists confirm logical conclusions. Predictable actions like the behavior of a swinging pendulum with significant air friction** or the diffusion of diseases*** are supported and confirmed by mathematical modeling.
Differential equations serve as scaffolding for mathematical modeling because they relate something’s rate of change to itself. For example, the most popular model for long-term population-change is a logistic differential equation****:
In this case, P is the population (a function of time, t), L is the carrying capacity, and k is a constant.
Expressed literally, Eq. 1 says that any function satisfying the conditions that the function is proportional to its first derivative and that (L-P) is proportional to its first derivative. This specific model will be discussed in more detail in section 6.
Differential equations are also very handy in finance. Specifically, differential equations are used examining stock investments. The value of a stock will change over time relative to how many people are willing to invest in the stock, which is also a function of the price of the stock. The changing relationship of a stock’s rate of change to itself screams “differential equation,” as the stock’s value changes over time in relation to the original price of the stock.
*This is true because the differential equation that describes the probability density of a particle’s position under a delta function potential gives non-zero probability solutions for a particle tunneling through an infinite-potential barrier (if confined to a single axis):
**We logically expect the position of a swinging pendulum to look like a cosine graph that decreases in height with time. This matches the solution under a certain set of parameters:
***The diffusion of diseases in humans is complicated because medical research and other social factors are variables that are computationally difficult to account for. However, for more vulnerable species, such as bananas, one would expect the area of effect of the disease to exponentially increase with time, which approximately matches statistical data.
****This differential equation is often more useful than the exponential differential equation because it accounts for population capacity. Notice that for P << L, the differential equation has exponential solutions.
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