✍️ 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 7, 2020
One way we apply derivatives to real life through rates of change involves position and motion. These problems involve rectilinear motion. 🚗
These problems always start with a particle, and that particle is moving in some way. You can think of position, x(t), as the original function at any given time. When you plug in time to the position function, you get out the location of the particle at that time.
That particle had to get to that position somehow! That is where velocity comes in. Velocity is always a measurement with respect to time, so it this is the derivative of our function with respect to time. You will see it written as v(t), but it may be helpful to picture it as x'(t), so you remember how to relate it to its position!
When velocity is negative, the particle is moving to the left. ➖ = ⬅️
When velocity is positive, the particle is moving to the right. ➕ = ➡️
If you want to talk about how fast the particle is moving but not about the direction, you are talking about speed. Speed is just velocity without direction. In math, the way we make direction not matter is by using absolute value. So we can think of speed as v(t)
What do you get when you find the rate of change of the velocity, you get how fast your velocity is changing, which is acceleration! Acceleration is the derivative of velocity, meaning it is the second derivative of position. a(t) = v'(t) = x''(t).
When v(t), a(t) have the same sign, then the particle is speeding up! ➕➕ = ⬆️
When v(t), a(t) have different signs, then the particle is slowing down! ➕➖ = ⬇️
When solving problems about rectilinear motion, you will often be given the position, velocity, or acceleration in any form (graph, function, table), and asked to find information about another measurement, that you may not have been given, at a specific time. You must be able to understand and move freely between position, velocity, and acceleration.
You may also be asked to analyze the motion of the particle. When this is asked, you need to find a few things. ❗
Find where v(t)= 0 and a(t) = 0
Use those values to figure out where the velocity and acceleration functions are negative and positive.
Use the velocity function to figure out what direction the particle is moving.
Use the acceleration and velocity to figure out where the function is speeding up and slowing down.
Use the values of t from where velocity and acceleration are zero and plug them into the position function to figure out where the particle was when it decided to change position.
This is a lot of information, so sign charts can help!
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.