AP Calculus AB/BC FRQ Practice

♾️AP Calculus AB/BC

Practice FRQ 1 of 81/8

1. A ramp is being designed so that its height above the ground is modeled for $x ≥ 0$ by the function $H$ , where $H(x)=\frac{x^2-4}{x-2}$ for $x ≠ 2$ . The value of $H(2)$ is not defined by this expression. A table of selected values of $H(x)$ near $x=2$ is shown. Let $m(x)=\frac{H(x)-H(2.5)}{x-2.5}$ for $x ≠ 2.5$ .

Selected values of $$H(x)$$ near $$x=2$$

1.9	3.900
1.99	3.990
1.999	3.999
2.001	4.001
2.01	4.010
2.1	4.100

Use the table to estimate $\lim_{x\to 2} H(x)$ . Write your answer using correct limit notation.

Find $\lim_{x\to 2} H(x)$ by rewriting $H(x)$ as an equivalent expression for $x≠ 2$ . Show your work.

Write a limit expression that represents the instantaneous rate of change of $H$ at $x=2.5$ in terms of average rates of change over intervals containing $x=2.5$ . Then evaluate the limit.

A new function $K$ is defined by $K(x)=\frac{x^2-4}{x-2}$ for $x≠ 2$ and $K(2)=c$ , where $c$ is a constant. Find the value of $c$ such that $K$ is continuous at $x=2$ . Justify your answer using the definition of continuity at a point. The function $K$ matches the ramp model $H$ for all $x≠ 2$ , but the height at $x=2$ is defined separately as $K(2)=c$ (in feet).