AP®︎ Calculus BC: Unit 2 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 2 of AP Calculus BC – Differentiation: Definition and Fundamental Properties.
Questions List
Unit 2 (All Topics)
Question 1 Topic 2.1Easy
Let .
What is the average rate of change of over ?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus BC Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.
Unit 2: Differentiation: Definition and Fundamental Properties
Define the derivative using limits and learn fundamental differentiation rules.
Topic 2.1: Defining Average and Instantaneous Rates of Change at a Point
Learning Objective: CHA-2.A
Determine average rates of change using difference quotients.
Essential Knowledge: CHA-2.A.1
The difference quotients (f(a+h) - f(a))/h and (f(x) - f(a))/(x - a) express the average rate of change of a function over an interval.
Learning Objective: CHA-2.B
Represent the derivative of a function as the limit of a difference quotient.
Essential Knowledge: CHA-2.B.1
The instantaneous rate of change of a function at x = a can be expressed by lim_{h→0} (f(a+h) - f(a))/h or lim_{x→a} (f(x) - f(a))/(x - a) provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f'(a).
Topic 2.2: Defining the Derivative of a Function and Using Derivative Notation
Learning Objective: CHA-2.B
Represent the derivative of a function as the limit of a difference quotient.
Essential Knowledge: CHA-2.B.2
The derivative of f is the function whose value at x is f'(x) = lim (h→0) (f(x+h) − f(x))/h, provided this limit exists.
Essential Knowledge: CHA-2.B.3
For y = f(x), notations for the derivative include dy/dx, f'(x), and y'.
Essential Knowledge: CHA-2.B.4
The derivative can be represented graphically, numerically, analytically, and verbally.
Learning Objective: CHA-2.C
Determine the equation of a line tangent to a curve at a given point.
Essential Knowledge: CHA-2.C.1
The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.
Topic 2.3: Estimating Derivatives of a Function at a Point
Learning Objective: CHA-2.D
Estimate derivatives.
Essential Knowledge: CHA-2.D.1
The derivative at a point can be estimated from information given in tables or graphs.
Essential Knowledge: CHA-2.D.2
Technology can be used to calculate or estimate the value of a derivative of a function at a point.
Topic 2.4: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Learning Objective: FUN-2.A
Explain the relationship between differentiability and continuity.
Essential Knowledge: FUN-2.A.1
If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of f, then it is not in the domain of f'.
Essential Knowledge: FUN-2.A.2
A continuous function may fail to be differentiable at a point in its domain. Illustrative examples: The left hand and right hand limits of the difference quotient are not equal, as in f(x) = |x| at x = 0; The tangent line is vertical and has no slope, as in f(x) = x^(1/3) at x = 0.
Topic 2.5: Applying the Power Rule
Learning Objective: FUN-3.A
Calculate derivatives of familiar functions.
Essential Knowledge: FUN-3.A.1
Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x) = x^r.
Topic 2.6: Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Learning Objective: FUN-3.A
Calculate derivatives of familiar functions.
Essential Knowledge: FUN-3.A.2
Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
Essential Knowledge: FUN-3.A.3
The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.
Topic 2.7: Derivatives of Cos x, Sin x, e^x, and ln x
Learning Objective: FUN-3.A
Calculate derivatives of familiar functions.
Essential Knowledge: FUN-3.A.4
Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.
Learning Objective: LIM-3.A
Interpret a limit as a definition of a derivative.
Essential Knowledge: LIM-3.A.1
In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.
Topic 2.8: The Product Rule
Learning Objective: FUN-3.B
Calculate derivatives of products and quotients of differentiable functions.
Essential Knowledge: FUN-3.B.1
Derivatives of products of differentiable functions can be found using the product rule.
Topic 2.9: The Quotient Rule
Learning Objective: FUN-3.B
Calculate derivatives of products and quotients of differentiable functions.
Essential Knowledge: FUN-3.B.2
Derivatives of quotients of differentiable functions can be found using the quotient rule.
Topic 2.10: Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Learning Objective: FUN-3.B
Calculate derivatives of products and quotients of differentiable functions.
Essential Knowledge: FUN-3.B.3
Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.
