 # Slope and Curve Analysis

## Challenge Options

Many of the handouts have embedded challenge options

## Outline

— We will determine slope functions for most basic functions both graphically and analytically. We will discover the Power Rule of differentiation by finding patterns among the slopes of tangent lines. We will then use limits, secant lines, and tangent lines to formalize a definition of instantaneous rates of change (IROC)

— We will compare three methods to approximate the slope of a curve at a point and apply limits to each method to arrive at a formal definition of the derivative. We will use the definition of the derivative to formalize the Power Rule.

— We will sketch functions and their first and second derivatives. We will describe the shapes of these graphs as increasing, decreasing, concave up, and concave down and we will justify our description by using the graphs of the first and second derivatives.

— We will investigate conditions under which a function is differentiable at a point. You will also examine differentiability of a function geometrically by examining tangent lines and local linearity. ## Goals

• Write slope functions for graphs of basic functions.
• Derive and use the formal definition of a derivative.
• Write derivatives of sine, cosine, and power functions.
• Examine the role of the second derivative to describe a
function’s shape.
• Understand the relationship between derivatives and velocity
and acceleration.
• Sketch first and second derivative curves.
• Determine antiderivatives.
• Investigate points of non-differentiability