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Section 2.1 – The Derivative and the Tangent Line Problem

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

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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.

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

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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.

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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.

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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

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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

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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

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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

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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

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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

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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.

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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)f(x) (c, f(c) ) m The blue line is a Tangent Line with Slope m that also contains the point ( c, f(c) )

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Classifying Slope 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. Slope can only be positive, negative, zero, or undefined. D. Since the curve has a sharp turn, the tangent line will be vertical. Thus the slope is Undefined. D

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Example 1 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

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Example 2 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

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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. The slope function or an instantaneous rate of change function, will be referred to as the Derivative of a Function. 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.

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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:

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

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Example 1 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. Find the Derivative

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

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Example 3 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

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

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

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AP Type Question Evaluate the limit: This is a difficult limit to evaluate. Notice that it is just applying the definition of a derivative at a point 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.

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

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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

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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

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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

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Differentiability Implies Continuity 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. If f is differentiable at x = c, then f is continuous at x = c.

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

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

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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

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

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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

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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

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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

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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

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