Rate of Change Click here to go to the lab titled "Derivatives - - PDF document

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AP Calculus

Derivatives

2015-11-03 www.njctl.org

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Table of Contents Rate of Change Derivative Rules: Power, Constant, Sum/Difference Continuity vs. Differentiability Chain Rule Derivatives of Trig Functions Implicit Differentiation Derivatives of Inverse Functions Derivatives of Logs & e Slope of a Curve (Instantaneous ROC) Higher Order Derivatives Equations of Tangent & Normal Lines Derivative Rules: Product & Quotient Calculating Derivatives Using Tables Derivatives of Piecewise & Abs. Value Functions

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Exploration into the idea of being locally linear... Click here to go to the lab titled "Derivatives Exploration: y = x2"

Derivatives Exploration

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Exploration into the idea of being locally linear... Click here to go to the lab titled "Derivatives Exploration: y = x2"

Derivatives Exploration

Teacher Notes

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Lead students through an exploration by having them graph y=x

2 (or any curve of

their choice) and have them zoom in slowly by changing their window settings little by little. You want the students to see that eventually their curve starts to resemble a line. The realization should be that this foreign concept of Derivatives will allow them to be able to find the slopes of curves in particular places, due to the fact that they are locally linear. URL for Lab: http://njctl.org/courses/math/ ap-calculus-ab/derivatives/x-squared- exploration-lab/

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Rate of Change

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You and your friends take a road trip and leave at 1:00pm, drive 240 miles, and arrive at 5:00pm. How fast were you driving? Consider the following scenario:

Road Trip! Slide 7 (Answer) / 213

You and your friends take a road trip and leave at 1:00pm, drive 240 miles, and arrive at 5:00pm. How fast were you driving? Consider the following scenario:

Road Trip!

Teacher Notes

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When students answer 60 mph, ask the following: · How did you arrive at the answer? · Are you driving 60mph the entire time? · What does 60mph represent? (they should come up with the words average velocity) *they may say speed, which is valid at this point. · How would you calculate how fast you were going at 2:37pm?

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position time

t1 t2 t0 t3 Now, consider the following position vs. time graph:

Position vs. Time Slide 8 (Answer) / 213

position time

t1 t2 t0 t3 Now, consider the following position vs. time graph:

Position vs. Time

Teacher Notes

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Students can often grasp the concept of derivatives when you relate it to something they are familiar with, such as velocity. Discuss with students: · What does the orange line represent? (average velocity over the entire interval) · What does each green segment represent? (instantaneous velocity at t1 and t2)

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We will discuss more about average and instantaneous velocity in the next unit, but hopefully it allowed you to see the difference in calculating slopes at a specific point, rather than over a period of time.

Recap Slide 10 / 213 SECANT vs. TANGENT

a b x1 x2 y1 y2 A secant line connects 2 points on a curve. The slope

• f this line is also known as

the Average Rate of Change. A tangent line touches one point on a curve and is known as the Instantaneous Rate of Change.

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a b

x1 x2 y1 y2

How would you calculate the slope of the secant line?

Slope of a Secant Line Slide 11 (Answer) / 213 Slide 12 / 213

What happens to the slope of the secant line as the point b moves closer to the point a? a b x1 x2 y1 y2 What is the problem with the traditional slope formula when b=a?

Slope of a Secant Line Slide 12 (Answer) / 213

What happens to the slope of the secant line as the point b moves closer to the point a? a b x1 x2 y1 y2 What is the problem with the traditional slope formula when b=a?

Slope of a Secant Line

Teacher Notes

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Allow students to discuss what they think, eventually listening for the conclusion that the secant line resembles the tangent line as those points get closer together. Encourage them to observe the fact that the change in x, #x, gets smaller (approaching 0) as the point b approaches a. In reference to the second question, students should note that when b=a, using the traditional slope formula would result in .

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Slope of a Curve (Instantaneous Rate of Change)

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The derivative of a function is a formula for the slope of the tangent line to that function at any point x. The process of taking derivatives is called differentiation. We now define the derivative of a function f (x) as The derivative gives the instantaneous rate of change. In terms of a graph, the derivative gives the slope of the tangent line.

Derivatives Slide 22 / 213 Slide 23 / 213

"f prime of x" Notation How it's read "y prime" "derivative of y with respect to x" "derivative with respect to x of f(x)" You may see many different notations for the derivative of a

• function. Although they look different and are read differently,

they still refer to the same concept.

Notation

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As you may have noticed, derivatives have an important role in mathematics as they allow us to consider what the slope, or rate of change, is of functions other than lines. In the next unit, you will begin to apply the use of derivatives to real world scenarios, understanding how they are even more useful with things such as velocity, acceleration, and

• ptimization, just to name a few.

Derivatives

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Derivative Rules: Power, Constant & Sum/Difference

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where c is a constant

The Constant Rule

All of these functions have the same

• derivative. Their derivative is 0.

Why do you think this is? Think of the meaning of a derivative, and how it applies to the graph of each of these functions.

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where c is a constant

The Constant Rule

All of these functions have the same

• derivative. Their derivative is 0.

Why do you think this is? Think of the meaning of a derivative, and how it applies to the graph of each of these functions. Teacher Notes

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Lead students in a discussion about what the graphs of each of those functions look like. Hopefully, they will conclude that they are all equations of horizontal lines. Therefore, no matter where you are

• n the graph, the slope of any

tangent line will be zero. Hence, the derivative is zero at any point, regardless of the x-value.

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11 A B C D E

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11 A B C D E

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Answer

C

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12 A x B 1 C 14 D E

• 15

What is the derivative of 15?

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12 A x B 1 C 14 D E

• 15

What is the derivative of 15?

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Answer

D

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14 A B C D Find y' if

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14 A B C D Find y' if

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Answer

C

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19 A B C D Find y'(16) if E

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19 A B C D Find y'(16) if E

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Answer

B

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Higher Order Derivatives

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You may be wondering.... Can you find the derivative of a derivative!!?? The answer is... YES! Finding the derivative of a derivative is called the 2

nd derivative.

Furthermore, taking another derivative would be called the 3rd

• derivative. So on and so forth.

Higher Order Derivatives

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You may be wondering.... Can you find the derivative of a derivative!!?? The answer is... YES! Finding the derivative of a derivative is called the 2

nd derivative.

Furthermore, taking another derivative would be called the 3rd

• derivative. So on and so forth.

Higher Order Derivatives

Teacher Notes

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You may want to mention to students that each derivative they take must be continuous in order to keep taking the next derivative. Continuity and differentiability will be discussed later, but it is helpful to mention here.

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Finding 2nd, 3rd, and higher order derivatives have many practical uses in the real world. In the next unit, you will learn how these derivatives relate to an object's position, velocity, and acceleration. In addition, the 5

th derivative is

helpful in DNA analysis and population modeling.

Applications of Higher Order Derivatives Slide 54 / 213 Slide 54 (Answer) / 213

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22 Find if A B C D E

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22 Find if A B C D E

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Answer A

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Derivatives of Trig Functions

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Derivatives of Trig Functions

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The reason for placing trig derivatives prior to product & quotient rule is to allow for more of a variety of problems during these subsequent sections.

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For example, if asked to take the derivative of , our previous rules would not apply. So far, we have talked about taking derivatives of polynomials, however what about other functions that exist in mathematics? Next, we will explore derivatives of trigonometric functions!

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Let's take a moment to prove one of these derivatives...

Proof

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Let's take a moment to prove one of these derivatives...

Proof

angle sum identity

The proofs of trig derivatives stem from 2 previously learned limits, you can revisit this idea or just mention that they are being used. If students are interested in looking at the proofs of all the trig derivatives, you can direct them to research them on their own time online. There are extensive proofs, simply by searching for trig derivatives.

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Find 34 A B C D E F

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Find 34 A B C D E F

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Answer D

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Find 36 A B C D E F

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Find 36 A B C D E F

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Answer E

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Derivative Rules: Product & Quotient

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44 True False

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44 True False

Teacher Notes

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FALSE Students can share/discuss the functions they use to disprove this statement.

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So far, we have discussed how to take the derivatives of polynomials using the Power Rule, Sum and Difference Rule, and Constant Rule. We have also discussed how to differentiate trigonometric functions, as well as functions which are comprised as the product of two functions using the Product Rule. Next, we will discuss how to approach derivatives of rational functions.

What About Rational Functions? Slide 93 / 213 Slide 93 (Answer) / 213

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f(x), or "top" g(x), or "bottom" Find Given:

Example Slide 94 (Answer) / 213

f(x), or "top" g(x), or "bottom" Find Given:

Example

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Answer

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Given: Find

Example Slide 95 (Answer) / 213 Slide 96 / 213

Now that you have seen the Quotient Rule in action, we can revisit

• ne of the trig derivatives and walk through the proof.

Proof Slide 96 (Answer) / 213

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Calculating Derivatives Using Tables

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On the AP Exam, in addition to calculating derivatives on your own, you must also be able to use tabular data to find derivatives. These problems are not incredibly difficult, but can be distracting due to extraneous information.

Derivatives Using Tables Slide 105 / 213 Slide 106 / 213 Slide 107 / 213 Slide 108 / 213

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Equations of Tangent & Normal Lines

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Recall from Algebra, that in order to write an equation of a line you either need 2 points, or a slope and a point. If we are asked to find the equation of a tangent line to a curve, our line will touch the curve at a particular point, therefore we will need a slope at that specific point. Now that we are familiar with calculating derivatives (slopes) we can use our techniques to write these equations of tangent lines.

Writing Equations of Lines Slide 115 / 213 Slide 115 (Answer) / 213 Slide 116 / 213 Slide 116 (Answer) / 213 Slide 117 / 213

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tangent line at x = 1 normal line at x = 1 y = x2

In addition to finding equations of tangent lines, we also need to find equations of normal lines. Normal lines are defined as the lines which are perpendicular to the tangent line, at the same given point. How do you suppose we would calculate the slope of a normal line?

Normal Lines Slide 118 (Answer) / 213

tangent line at x = 1 normal line at x = 1 y = x2

In addition to finding equations of tangent lines, we also need to find equations of normal lines. Normal lines are defined as the lines which are perpendicular to the tangent line, at the same given point. How do you suppose we would calculate the slope of a normal line?

Normal Lines

Teacher Notes

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This question is meant to be in general, for any normal line, not necessarily this specific curve. Allow students to discuss how to come up with the slope of the normal line, hoping they make the connection that perpendicular lines have opposite reciprocal slopes.

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Derivatives of Logs & e

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The next set of functions we will look at are exponential and logarithmic functions, which have their own set

• f rules for differentiation.

Exponential and Logarithmic Functions Slide 130 / 213

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By considering a particular value of a, , we are able to see the proof for the derivative of exponential functions.

Note: This proof is based on the fact that e, in the realm of calculus, is the unique number for which

Derivatives of Exponential Functions Slide 131 (Answer) / 213 Slide 132 / 213

is the only nontrivial function whose derivative is the same as the function!

cool!

Derivatives of Exponential Functions Slide 132 (Answer) / 213

is the only nontrivial function whose derivative is the same as the function!

cool!

Derivatives of Exponential Functions

Teacher Notes

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Technically, y=0 is also it's

• wn derivative as well, but

does not depend on another variable, so generally people say that is the only one. Consider asking students if they can think of y=0 before telling them.

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62 A B C D E F

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62 A B C D E F

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Answer

C

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Chain Rule

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Derivatives of Inverse Functions

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We have already covered derivatives of inverse trig functions, but it is also necessary to calculate the derivatives of other inverse functions.

Derivatives of Inverse Functions Slide 155 / 213 Slide 155 (Answer) / 213 Slide 156 / 213 Slide 156 (Answer) / 213

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If and find

Example Slide 159 (Answer) / 213 Slide 160 / 213

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Continuity vs. Differentiability

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1) f(a) exists 2) exists 3) Definition of Continuity In the previous Limits unit, we discussed what must be true for a function to be continuous: Differentiability requires the same criterion, as well as a few others.

Definition of Continuity Slide 167 / 213

In order for a function to be considered differentiable, it must contain: · No discontinuities · No vertical tangent lines · No Corners · No Cusps

"sharp points"

Differentiable Functions

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If a function is differentiable, it is also continuous. However, the converse is not true. Just because a function is continuous does not mean it is differentiable. What does this mean??? Consider the function: Notice: If we were asked to find the derivative (slope) at x=0, there is a sharp corner. The slope quickly changes from -1 to 1 as you move closer to x=0. Therefore, this function is not differentiable at x=0.

Differentiability Implies Continuity Slide 168 (Answer) / 213

If a function is differentiable, it is also continuous. However, the converse is not true. Just because a function is continuous does not mean it is differentiable. What does this mean??? Consider the function: Notice: If we were asked to find the derivative (slope) at x=0, there is a sharp corner. The slope quickly changes from -1 to 1 as you move closer to x=0. Therefore, this function is not differentiable at x=0.

Differentiability Implies Continuity

Teacher Notes

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Another way to explain to this to students is to draw several tangent lines at x=0 and show that they all have different slopes. Because there is not one single tangent line that can "balance" at x=0, it is not differentiable at this point. Another explanation: Imagine zooming in on the function, like we have previously done. The function must resemble a line ("locally linear") to be differentiable.

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CORNER CUSP DISCONTINUITY VERTICAL TANGENT

A FUNCTION FAILS TO BE DIFFERENTIABLE IF... Slide 170 / 213 Types of Discontinuities:

removable infinite removable jump essential

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81 True False If f(x) is continuous on a given interval, it is also differentiable.

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81 True False If f(x) is continuous on a given interval, it is also differentiable.

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Answer False

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Derivatives of Piecewise & Abs. Value Functions

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Now that we've discussed the criterion for a function to be differentiable, we can look at how to find the derivatives of piecewise and absolute value functions, which often contain sharp corners, and discontinuities.

Derivatives of Piecewise & Absolute Value Functions Slide 178 / 213

When calculating derivatives of piecewise functions, the same rules apply for each piece; however, you must also consider the point in which the function switches from one portion to another. For a piecewise function to be differentiable EVERYWHERE it must be: · Continuous at all points (equal limits from left and right) · Have equal slopes from left and right

Derivatives of Piecewise & Absolute Value Functions Slide 179 / 213 Slide 179 (Answer) / 213

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Example: Find the derivative of It is apparent that every absolute value function will have a sharp point (thus, not being differentiable at that point). But again, we can still find the derivative, discluding the sharp point. Note: We must first write our function as a piecewise.

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Implicit Differentiation

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Find Find

Find

CHALLENGE!

Find

Find Find

CHALLENGE!

Practice Slide 198 (Answer) / 213 Slide 199 / 213

Why am I being asked to find the derivative with respect to the variable, t, so often? Often in Calculus, we are interested in seeing how things change with respect to TIME, hence taking the derivative (which shows us rate of change) with respect to the variable t. This will become increasingly more apparent in the next unit when we study Related Rates.

Derivatives with Respect to t

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Now that we have practiced using implicit differentiation, we can extend the process to find the derivatives at specific points.

Implicit Differentiation at a Point Slide 206 / 213

Find the slope of the tangent line to the circle given by: at the point

Example

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Find the slope of the tangent line to the circle given by: at the point

Example

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Answer

Find Plug in values for x and y:

Recall:

• 1. Differentiate both sides
• 2. Collect all dy/dx to one side
• 3. Factor out dy/dx
• 4. Solve for dy/dx.

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For this example, note the benefits of implicit differentiation vs. explicit differentiation. As an optional exercise, you may rework the example for the explicit function: which is the upper half of the graph. Then, remember this must be done again if points on the lower half are also desired, given by:

Implicit vs. Explicit Differentiation Slide 208 / 213

As a further step in this example, we can now find the equation

• f the tangent line at the point, (3,4).

Example, Continued Slide 208 (Answer) / 213

As a further step in this example, we can now find the equation

• f the tangent line at the point, (3,4).

Example, Continued

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Answer

Find the equation of the tangent line at (3,4): From before, slope is Recall the equation for a line is:

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Example: Find the slope of the graph of at the point

Example Slide 209 (Answer) / 213

Example: Find the slope of the graph of at the point

Example

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Answer

• 1. Differentiate both sides
• 2. Collect all dy/dx to one side
• 3. Factor out dy/dx
• 4. Solve for dy/dx.

Plug in point values:

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95 Find the slope of the tangent line at x=3 for the equation: A B C D

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95 Find the slope of the tangent line at x=3 for the equation: A B C D

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Answer

note:

Answer: A

Note: Students may need prompting to substitute the x-value into original function to find the corresponding y-value to use in derivative.

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96 Find the slope of the tangent line at the point for the equation: A B C D

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96 Find the slope of the tangent line at the point for the equation: A B C D

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Answer

note:

Answer: D

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97 Find the equation of tangent line through point (1, -1) for the equation: A B C D

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97 Find the equation of tangent line through point (1, -1) for the equation: A B C D

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Answer

note:

slope at (1,-1) = -4

Answer: D