AP Calculus BC 2.2 Defining the Derivative of a Function and Using Derivative Notation Questions

Review derivative definition questions by using difference quotients, derivative limits and tangent-line interpretations.

Covered syllabus points

CHA-2.B—Represent the derivative of a function as the limit of a difference quotient—Topic 2.2 CHA-2.C—Determine the equation of a line tangent to a curve at a given point

[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]

Let f be the function defined by $f(x)=\frac{3}{2 x^{2}-7 x+5}$ .

Find the slope of the line tangent to the graph of f at x=3.

$f^{\prime}(x)=\frac{-3(4 x-7)}{\left(2 x^{2}-7 x+5\right)^{2}}$

f^{\prime}(3)=\frac{(-3)(5)}{(18-21+5)^{2}}=-\frac{15}{4}

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Covered syllabus points

Question 1

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