$\newenvironment {prompt}{}{} \newcommand {\ungraded }{} \newcommand {\todo }{} \newcommand {\mooculus }{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }{\nprounddigits {-1}} \newcommand {\npnoroundexp }{\nproundexpdigits {-1}} \newcommand {\npunitcommand }{\ensuremath {\mathrm {#1}}} \newcommand {\tkzTabSlope }{\foreach \x /\y /\z in {#1}{\node [left = 3pt] at (Z\x 1) {\scriptsize \y }; \node [right = 3pt] at (Z\x 1) {\scriptsize \z }; }} \newcommand {\RR }{\mathbb R} \newcommand {\R }{\mathbb R} \newcommand {\N }{\mathbb N} \newcommand {\Z }{\mathbb Z} \newcommand {\sagemath }{\textsf {SageMath}} \newcommand {\d }{\mathop {}\!d} \newcommand {\l }{\ell } \newcommand {\ddx }{\frac {d}{\d x}} \newcommand {\ddt }{\frac {d}{\d t}} \newcommand {\zeroOverZero }{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }{\textbf } \newcommand {\unit }{\mathop {}\!\mathrm } \newcommand {\eval }{\bigg [ #1 \bigg ]} \newcommand {\seq }{\left ( #1 \right )} \newcommand {\epsilon }{\varepsilon } \newcommand {\phi }{\varphi } \newcommand {\iff }{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{\vec {proj}} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }{\overrightarrow } \newcommand {\vec }{\mathbf } \newcommand {\veci }{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }{\boldsymbol {\l }} \newcommand {\uvec }{\mathbf {\hat {#1}}} \newcommand {\utan }{\mathbf {\hat {t}}} \newcommand {\unormal }{\mathbf {\hat {n}}} \newcommand {\ubinormal }{\mathbf {\hat {b}}} \newcommand {\dotp }{\bullet } \newcommand {\cross }{\boldsymbol \times } \newcommand {\grad }{\boldsymbol \nabla } \newcommand {\divergence }{\grad \dotp } \newcommand {\curl }{\grad \cross } \newcommand {\lto }{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }{\overline } \newcommand {\surfaceColor }{violet} \newcommand {\surfaceColorTwo }{redyellow} \newcommand {\sliceColor }{greenyellow} \newcommand {\vector }{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }{} \newcommand {\HyperFirstAtBeginDocument }{\AtBeginDocument } \newcommand {\descriptionlabel }{\hspace {\labelsep }\textbf {#1:}}$

Two young mathematicians discuss the novel idea of the “slope of a curve.”

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley, do you remember “slope?’
Riley
Most definitely. “Rise over run.”
Devyn
You know it.
Riley
“Change in $y$ over change in $x$.”
Devny
That’s right.
Riley
Brought to you by the letter “$m$.”
Devny
Enough! My important question is: could we define “slope” for a curve that’s not a straight line?
Riley
Well, maybe if we “zoom in” on a curve, it would look like a line, and then we could call it “the slope at that point.”
Devyn
Ah! And this “zoom in” idea sounds like a limit!
Riley
This is so awesome. We just made math!

The concept introduced above, of the “slope of a curve at a point,” is in fact one of the central concepts of calculus. It will, of course, be completely explained. Let’s explore Devyn and Riley’s ideas a little more, first.

To find the “slope of a curve at a point,” Devyn and Riley spoke of “zooming in” on a curve until it looks like a line. When you zoom in on a smooth curve, it will eventually look like a line. This line is called the tangent line.

Which of the following approximate the slope of the “zoomed line”?
$\frac {(f(a)+h) - f(a)}{(a+h)-a}$ $\frac {f(a+h) - f(a)}{(a+h)-a}$ $\frac {(f(a)-h) - f(a)}{(a-h)-a}$ $\frac {f(a-h) - f(a)}{(a-h)-a}$ $\frac {f(a) - (f(a)+h)}{a-(a+h)}$ $\frac {f(a) - f(a+h)}{a-(a+h)}$ $\frac {f(a) - (f(a)-h)}{a-(a-h)}$ $\frac {f(a) - f(a-h)}{a-(a-h)}$
Let $f(x) = 3x-1$. Zoom in on the curve around $a = -2$ so that $h = 0.1$. Use one of the formulations in the problem above to approximate the slope of the curve. The slope of the curve at $a = -2$ is approximately…$\answer {3}$
Repeat the previous problem for $f(x) = x^2 - 1$, $a = 0$, and $h = 0.2$. Choose a formulation that will give you a positive answer for the slope. The (positive) slope of the curve at $a = 0$ is approximately… $\answer {0.2}$
Zoom in on the curve $f(x) = x^2 - 1$ near $x=0$ again. By looking at the graph, what is your best guess for the actual slope of the curve at zero?
impossible to say zero one infinity