✍️ 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)
June 11, 2020
🎥Watch: AP Calculus AB/BC - Related Rates
Related Rates are exactly what they sound like. It is using formulas and expressions that we know involve the same variables to find missing rates. These problems can look very different but are really following a similar process. 💻
Draw a picture! Assign variables to anything you have or need to find, and label the picture as so.
Find out which rates you know, and which rates you are looking for. Label them using the variables you used in part 1 as derivatives. Make sure it is with respect to time!
Ex. If you labeled a ladder length c, and you know the rate at which the length of the ladder is changing, then you should label that rate dc/dt.
Find an equation that relates to what you are looking for and what you already know. (This will not have derivatives in it yet!)
If there is a variable that you do not know the derivative to and you do not need to find the derivative to, plug that known value in right away!
Take the derivative of both sides of the equation.
Plug in what you know and solve for what you do not.
** Don’t forget units! 📌
A 10-ft ladder is leaning against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house. When the base has slid to 8 ft from the house, it is moving horizontally at the rate of 2 ft/sec. How fast is the ladder’s top sliding down the wall when the base is 8 ft from the house?
We can model this as a right triangle with a hypotenuse 10 and legs x and y:
Note that dy/dt and dx/dt are the rates of change of the lengths of the triangle with respect of time. They are the rate at which the ladder is sliding in either direction.
With rights triangles, we typically like to use the Pythagorean Theorem. Trying this we find that x^2 + y^2 = 100. Deriving with respect to time we find that 2x dx/dt + 2y dy/dt = 0.
Plugging in from the problem we find:
8^2 + y^2 = 100 ==> y = 6
(2(8) * 2) + (2(6) dy/dt) = 0
12 dy/dt = -32
dy/dt = -32/12 = -8/3 ft/s
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