Section 2.1 – The Derivative and the Tangent Line Problem

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

The blue line intersects the pink curve twice. Yet it is a tangent. 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.

Local Linearity If a function has a tangent line at a point, it is at least locally linear. Tangent Line Exists.

Local Linearity If a function has a tangent line at a point, it is at least locally linear. Tangent Line Does Not Exist. The function is NOT smooth at this point.

A Tangent Line Exists at Every Point. Local Linearity If a function has a tangent line at a point, it is at least locally linear. A Tangent Line Exists at Every Point. The function is Always smooth.

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 (x1,y1) and (x2,y2): (x2,y2) Δx (x1,y1) 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 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) The blue line is a Tangent Line with Slope m that also contains the point ( c, f(c) ) m (c, f(c) )

Slope can only be positive, negative, zero, or undefined. Classifying Slope Determine the best way to describe the slope of the tangent line at each point. 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. D A C. Since the curve is increasing, the slope will also be increasing. Thus the slope is Positive. C D. Since the curve has a sharp turn, the tangent line will be vertical. Thus the slope is Undefined. B Slope can only be positive, negative, zero, or undefined.

c is the x-coordinate of the point on the curve Example 1 Find the instantaneous rate of change to at (3,-6). Substitute into the function c is the x-coordinate of the point on the curve Simplify in order to cancel the denominator Direct substitution

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

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. A variable. A constant. A function whose output is the slope of a tangent line at any x. The slope of a tangent line at the point x = c. The slope of a tangent line or the instantaneous rate of change, will be referred to as a Derivative of a function at a value of x. The slope function or an instantaneous rate of change function, will be referred to as the Derivative of a Function.

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:

Vocab Differentiation: The process of finding the derivative of a function. Differentiable: The derivative exists. Able to be differentiated. Possess a derivative. Etc.

Example 1 Derive a formula for the slope of the tangent line to the graph of . Multiply by a common denominator Simplify in order to cancel the denominator Substitute into the function Find the Derivative A formula to find the slope of any tangent line at x. Direct substitution

Substitute into the function Example 2 Differentiate . Simplify in order to cancel the denominator Substitute into the function Make the problem easier by factoring out common constants Direct substitution

Example 3 Find the tangent line equation(s) for such that the tangent line has a slope of 12. Find when the derivative equals 12 Find the derivative first since the derivative finds the slope for an x value Find the output of the function for every input Use the point-slope formula to find the equations

Alternate Definition of a Derivative The slope of a secant line between x and x0 is: The derivative of a function provides us with a measure of the instantaneous rate of change. Thus, we get the derivative at x0 or f'(x0) if we take the limit as the denominator goes to 0: Known as The Difference Quotient at a particular point

Substitute into the function Example Find f '(2) if : Since the derivative needs to be evaluated at a point, the alternate definition can be used. Simplify in order to cancel the denominator Factor Substitute into the function Direct substitution

This is a difficult limit to evaluate. AP Type Question Evaluate the limit: Notice that it is just applying the definition of a derivative at a point This is a difficult limit to evaluate. The limit is equal to the derivative of sine at Pi/2. We have not calculated this derivative YET, so we are currently not able to answer this question.

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

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, 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

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: 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 1 Determine whether the following derivatives exist for the graph of the function.

Example 2 Show that does exist if Prove that it is Continuous The limit exists f(1) exists Since the function is continuous at x=1. Prove the Right Hand Derivative is the same as the Left Hand Derivative (and non-infinite) The one-sided derivatives are equal and non-infinite. Thus the derivative exists, at x=1.

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

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

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

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 2 to ∞ is 1 The slope from -7 to 2 is 0

Example 2 Sketch a graph of the derivative of the function graphed below at left. Negative Negative Increasing Increasing Increasing Decreasing Decreasing Positive Positive Positive 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

Example 3 Sketch a graph of the derivative of the function graphed below at left. Positive Negative Decreasing Increasing Increasing Positive 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