AP Calculus BC Unit 2: Differentiation

Practice AP Calculus BC Unit 2 questions on defining, estimating, and calculating derivatives across numerical, graphical, and analytical forms.

Covered syllabus points

2.1 Defining Average and Instantaneous Rates of Change at a Point 2.2 Defining the Derivative of a Function and Using Derivative Notation 2.3 Estimating Derivatives of a Function at a Point 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist 2.5 Applying the Power Rule 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple 2.7 Derivatives of cos x, sin x, eˣ, and ln x 2.8 The Product Rule 2.9 The Quotient Rule 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

[Maximum number: 1]

Grass clippings are placed in a bin, where they decompose. For $0 \leq t \leq 30$ , the amount of grass clippings remaining in the bin is modeled by $A(t)=6.687(0.931)^{t}$ , where A(t) is measured in pounds and t is measured in days.

Find the average rate of change of A(t) over the interval $0 \leq t \leq 30$ . Indicate units of measure.

$\frac{A(30)-A(0)}{30-0}=-0.197$ (or -0.196 ) lbs/day

[Maximum number: 2]

Consider the differential equation $\frac{d y}{d x}=y^{2}(2 x+2)$ . Let y=f(x) be the particular solution to the differential equation with initial condition f(0)=-1.

Find $\lim _{x \rightarrow 0} \frac{f(x)+1}{\sin x}$ . Show the work that leads to your answer.

$\lim _{x \rightarrow 0}(f(x)+1)=-1+1=0$ and $\lim _{x \rightarrow 0} \sin x=0$
Using L'Hospital's Rule,

\lim _{x \rightarrow 0} \frac{f(x)+1}{\sin x}=\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{\cos x}=\frac{f^{\prime}(0)}{\cos 0}=\frac{(-1)^{2} \cdot 2}{1}=2

[Maximum number: 2]

A student starts reading a book at time t=0 minutes and continues reading for the next 10

minutes. The rate at which the student reads is modeled by the differentiable function R, where

R(t) is measured in words per minute. Selected values of R(t) are given in the table shown.

Table for Question Unit 2: Differentiation: Definition and Fundamental Properties question 3 — AP Calculus BC

Approximate $R^{\prime}(1)$ using the average rate of change of R over the interval $0 \leq t \leq 2$ . Show

the work that leads to your answer. Indicate units of measure.

A student starts reading a book at time t=0 minutes and continues reading for the next 10 minutes. The rate

at which the student reads is modeled by the differentiable function R, where R(t) is measured in words per

minute. Selected values of R(t) are given in the table shown.

t (minutes)	0	2	8	10
R(t) (words per minute)	90	100	150	162

Model Solution

Scoring

A Approximate $R^{\prime}(1)$ using the average rate of change of R over the interval $0 \leq t \leq 2$ . Show the work

that leads to your answer. Indicate units of measure.

$\begin{aligned} & R^{\prime}(1) \approx \frac{R(2)-R(0)}{2-0} \\ & =\frac{100-90}{2}=\frac{10}{2}=5 \text { words per minute per minute } \end{aligned}$

Answer with setup	Point 1 (P1)
Units	Point 2 (P2)

Scoring Notes for Part A

- To earn P1, a response must present the answer along with the supporting work of a difference and a

quotient using values from the table.

○ $\frac{100-90}{2-0}, \frac{10}{2-0}, \frac{100-90}{2}$ , or $\frac{R(2)-R(0)}{2-0}=5$ is sufficient to earn P1.

○ $\frac{R(2)-R(0)}{2-0}$ by itself is not sufficient to earn P1.

- P2 is earned for correct units, whether or not they are attached to a numerical value for the average

rate of change.

- P2 is also earned for the units "words/minute ${ }^{2}$ ."

[Maximum number: 1]

36. If g is a differentiable function with g(1)=4 and $g^{\prime}(1)=3$ , which of the following statements could be false?

$\lim _{x \rightarrow 1^{+}} g(x)=\lim _{x \rightarrow 1^{-}} g(x)$

$\lim _{x \rightarrow 1} g^{\prime}(x)=3$

$\lim _{x \rightarrow 1} g(x)=4$

$\lim _{h \rightarrow 0} \frac{g(1+h)-g(1)}{h}=3$

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Covered syllabus points

Unit 2: Differentiation: Definition and Fundamental Properties question 1

Unit 2: Differentiation: Definition and Fundamental Properties question 2

Unit 2: Differentiation: Definition and Fundamental Properties question 3

Unit 2: Differentiation: Definition and Fundamental Properties question 4