Practice Problems

1. What is the slope of the line through $(3,9)$ and $(x, y)$ for $y=x^{2}$ and $x=2.97$? $x=3.001$? $\mathrm{x}=3+\mathrm{h}$? What happens to this last slope when $\mathrm{h}$ is very small (close to 0)? Sketch the graph of $y=x^{2}$ for $x$ near $3$.

3. What is the slope of the line through $(2,4)$ and $(x, y)$ for $y=x^{2}+x-2$ and $x=1.99$? $\mathrm{x}=2.004$? $\mathrm{x}=2+\mathrm{h}$? What happens to this last slope when $\mathrm{h}$ is very small? Sketch the graph of $y=x^{2}+x-2$ for $x$ near $2$.

5. Fig. 9 shows the temperature during a day in Ames.

(a) What was the average change in temperature from 9 am to 1 pm?

(b) Estimate how fast the temperature was rising at 10 am and at 7 pm?

Problem 9 defines new functions $A(x)$ in terms of AREAS bounded by the functions $y=3$ and $y=x+1$. This may seem a strange way to define a functions $A(x)$, but this idea will become important later in calculus. We are just getting an early start here.

9. Define $A(x)$ to be the area bounded by the $x$ and y axes, the horizontal line $\mathrm{y}=3$, and the vertical line at $\mathrm{x}$ (Fig. 13). For example, $\mathrm{A}(4)=12$ is the area of the 4 by 3 rectangle.