APSU Notes

Section 1.1 - Preview of Calculus

Warm Up Problem

Calculate the slope of the secant line of $f(x)=x^2$ from the point (1,1) to the point (2,4).

Warm up



  • Secant Line - A Line with 2 points on the curve.
  • Tangent Line - A line that intersects $f(x)$ af $f$ and nowhere else hear $P$.

Motivating Example

Can we determine the slope of the graph $f(x)=x^2$ at the point (1,1)?


$(1, 1)$ to $(1.5, 2.25)$

$\frac{2.25-1}{1.1-1}=\frac{.21}{.1}=2.1 slope$

The Slope of the Tangent Line: $m=2$

$(1, 1)$ to $(x, f(x)):\frac{f(x)-1}{x-1}=\frac{x^2-1}{x-1}$

$\displaystyle{\lim_{x \to 1}}=\frac{x^2-1}{x-1}=2$


Preview of Calculus

  • One of our first goals: Finding the slope of a curve $y=f(x)$.
  • Need to find a the slope of a tangent line at a point $P$: a line that intersects $f(x)$ at $P$ and nowhere else near $P$.
  • We do this by approximating the tangent line with secant lines: lines between two points on the curve.
  • Def: Calculus is the mathematical study of change.


Lecture Video

Handwritten Notes