Section 2.1: The Derivative and the Tangent Line Problem

Section 2.1 – Classwork 1 TANGENT LINE Slope = -48 Slope = -32 Slope = -24 Slope = Slope ≈ -16 Secant Lines These can be considered average slopes or average rates of change.

Secant Line A line that passes through two points on a curve.

Tangent Line Most people believe that a tangent line only intersects a curve once. For instance, the first time most students see a tangent line is with a circle: Although this is true for circles, it is not true for every curve: Every blue line intersects the pink curve only once. Yet none are tangents. The blue line intersects the pink curve twice. Yet it is a tangent.

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Tangent Line As two points of a secant line are brought together, a tangent line is formed. The slope of which is the instantaneous rate of change

Slope of a Tangent Line In order to find a formula for the slope of a tangent line, first look at the slope of a secant line that contains (x 1,y 1 ) and (x 2,y 2 ) : (x2,y2)(x2,y2) (x1,y1)(x1,y1) ΔxΔx In order to find the slope of the tangent line, the change in x needs to be as small as possible.

Instantaneous Rate of Change Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). f(x)f(x) (c, f(c)) m

Example 1 Determine the best way to describe the slope of the tangent line at each point. A B C A. Since the curve is decreasing, the slope will also be decreasing. Thus, the slope is negative. B. The vertex is where the curve goes from increasing to decreasing. Thus, the slope must be zero. C. Since the curve is Increasing, the slope will also be increasing. Thus the slope is Positive.

Example 2 Find the instantaneous rate of change to at (3,-6). c is the x-coordinate of the point on the curve Direct substitution Substitute into the function Simplify in order to cancel the denominator

Example 3 Find the equation of the tangent line to at (2,10). c is the x-coordinate of the point on the curve Direct substitution Substitute into the function Simplify in order to cancel the denominator Just the slope. Now use the point-slope formula to find the equation

A Function to Describe Slope In the preceding notes, we considered the slope of a tangent line of a function f at a number c. Now, we change our point of view and let the number c vary by replacing it with x. The slope of a tangent line at the point x = c. A constant. A function whose output is the slope of a tangent line at any x. A variable.

Example Derive a formula for the slope of the tangent line to the graph of. Substitute into the function Direct substitution Multiply by a common denominator Simplify in order to cancel the denominator A formula to find the slope of any tangent line at x.

The Derivative of a Function The limit used to define the slope of a tangent line is also used to define one of the two fundamental operations of calculus: The derivative of f at x is given by Provided the limit exists. For all x for which this limit exists, f’ is a function of x. READ: “f prime of x.” Other Notations for a Derivative:

Example 1 Differentiate. Substitute into the function Direct substitution Simplify in order to cancel the denominator Make the problem easier by factoring out common constants

Example 2 Find the tangent line equation(s) for such that the tangent line has a slope of 12. Find the derivative first since the derivative finds the slope for an x value Find when the derivative equals 12 Find the output of the function for every input Use the point-slope formula to find the equations

How Do the Function and Derivative Function compare? Domain: f is not differentiable at x = -½

Differentiability Justification 1 In order to prove that a function is differentiable at x = c, you must show the following: In other words, the derivative from the left side MUST EQUAL the derivative from the right side. Common Example of a way for a derivative to fail: Other common examples: Corners or Cusps Not differentiable at x = -4

Differentiability Justification 2 In order to prove that a function is differentiable at x = c, you must show the following: In other words, the function must be continuous. Common Example of a way for a derivative to fail: Other common examples: Gaps, Jumps, Asymptotes Not differentiable at x = 0

Differentiability Justification 3 In order to prove that a function is differentiable at x = c, you must show the following: In other words, the tangent line can not be a vertical line. An Example of a Vertical tangent where the derivative to Fails to exist: Not differentiable at x = 0

Example Determine whether the following derivatives exist for the graph of the function.

Example 2 Sketch a graph of the function with the following characteristics: The derivative does not exist at x = -2. The function is continuous on (-6,3) The Range is (-7,-1]

Example 3 Show that does not exist if. First rewrite the absolute value function as a piecewise function Since the one-sided limits are not equal, the derivative does not exist Find the Left Hand Derivative Find the Right Hand Derivative

Function v Derivative Compare and contrast the function and its derivative. FUNCTIONDERIVATIVE -5 Vertex x-intercept Decreasing Negative Slopes Increasing Positive Slopes

Function v Derivative Compare and contrast the function and its derivative. FUNCTIONDERIVATIVE -5 Local Max x-intercept Decreasing Negative Derivatives Increasing Positive Derivatives 55 Local Min x-intercept Positive Derivatives Increasing

Example 1 Accurately graph the derivative of the function graphed below at left. The Derivative does not exist at a corner. Make sure the x-value does not have a derivative The slope from -∞ to -7 is -2 The slope from -7 to 2 is 0 The slope from 2 to ∞ is 1

Example 2 Sketch a graph of the derivative of the function graphed below at left. 1. Find the x values where the slope of the tangent line is zero (max, mins, twists) 2. Determine whether the function is increasing or decreasing on each interval Increasing Positive Decreasing Negative Increasing Positive Decreasing Negative Increasing Positive

Example 3 Sketch a graph of the derivative of the function graphed below at left. 1. Find the x values where the slope of the tangent line is zero (max, mins, twists) 2. Determine whether the function is increasing or decreasing on each interval Decreasing Negative Increasing Positive Increasing Positive

Example 4 Sketch a graph of the derivative of the function graphed below. (-3,0)(3,0) (-2.2,-4)(2.2,-4) (0,0)

Example 5 Sketch a graph of the function with the following characteristics: The derivative is only positive for -6<x<-3 and 5<x<10. The function is differentiable on (-6,10)

Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c. The contrapositive of this statement is true: If f is NOT continuous at x = c, then f is NOT differentiable at x = c. The converse of this statement is not always true (be careful): If f is continuous at x = c, then f is differentiable at x = c. The inverse of this statement is not always true: If f is NOT differentiable at x = c, then f is NOT continuous at x = c.

Example Sketch a graph of the function with the following characteristics: The derivative does NOT exist at x = -2. The derivative equals 0 at x = 1. The derivative does NOT exist at x = 4. The function is continuous on [-4,-2)U(-2,5]