AP Calculus BC 2.2: Derivative Definition

Practice AP Calculus BC questions on recognizing derivative definitions in limits and representing instantaneous rates of change.

[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

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CHA-2.B—Represent the derivative of a function as the limit of a difference quotient—Topic 2.2 question 1