3.6 finding the equation of the tangent line to a curve
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3.6 Finding the Equation of the Tangent Line to a Curve. Remember: Derivative=Slope of the Tangent Line. What is another way to find the slope of this line?. The DERIVATIVE!!!!. What is another way to find the slope of this line?. - PowerPoint PPT PresentationTRANSCRIPT
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Remember: Derivative=Slope of the Tangent Line
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What is another way to find the slope of this line?The
DERIVATIVE!!!!)(' af
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What is another way to find the slope of this line? xxf 2)('
2)1(2)1(')(' faf
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Both ways give you the slope of the tangent to the curve at point A.
That means you can _____________________________.set them equal to each other
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That means you can set them equal to each other:
axafy
xy
af
)(
12
2)('
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That means you can set them equal to each other:
12
2
xy
)1(22 xy
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Therefore,
)1(22 xyIs the slope of the tangent
line for f(x)=x2+1
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y-f(a)=f’(a)(x-a)
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Step 1: Find the point of contact by plugging in the x-
value in f(x). This is f(a).
39)3(4)3(3)3()( 2 faf
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Step 2: Find f’(x). Plug in x-value for f’(a)
46)(' xxf 224)3(6)3(')(' faf
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Step 3: Plug all known values into formula
y-f(a)=f’(a)(x-a)
))3((2239 xy
)3(2239 xy
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• Find the equation of the tangent to y=x3+2x at:– x=2
– x=-1
– x=-2
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f’(x)=0
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Step 1: Find the derivative, f’(x)
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Step 2: Set derivative equal to zero and solve, f’(x)=0
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Step 3: Plug solutions into original formula to find y-value, (solution, y-
value) is the coordinates.
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Note: If it asks for the equation then you will write y=y value found when
you plugged in the solutions for f’(x)=0
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What do you notice about the labeled
minimum and maximum?
They are the coordinates where the tangent is horizontal
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Where is the graph increasing?
{x| x<-3, x>1}
What is the ‘sign’ of the derivative for these
intervals?
-3 1
+ +
This is called a sign diagram
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Where is the graph decreasing?
{x| -3<x<1}
What is the ‘sign’ of the derivative for this interval?
-3 1
+ + –
What can we hypothesize about how the sign of the derivative relates to the
graph?f’(x)=+, then graph
increasesf’(x)= – , then graph
decreases
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We can see this:
When the graph is increasing then the gradient
of the tangent line is positive (derivative is +)
When the graph is decreasing then the
gradient of the tangent line is negative (derivative is - )
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So back to the question…Why does the fact that the
relative max/min of a graph have horizontal tangents make sense?
A relative max or min is where the graph goes
from increasing to decreasing (max) or from decreasing to increasing (min). This means that
your derivative needs to change signs.
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Okay…So what?
To go from being positive to negative, the derivative like any function must go through zero. Where the
derivative is zero is where the graph changes direction, aka the relative
max/min
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Take a look at f(x)=x3. What is the coordinates of the point on the function where the derivative is equal to 0? Find
the graph in your calculator, is this coordinate a relative maximum or a
relative minimum?NO – the graph only flattened out then
continued in the same direction
This is called a HORIZONTAL INFLECTION
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It is necessary to make a sign diagram to determine whether the coordinate where f’(x)=0 is a relative maximum, minimum, or a horizontal inflection.
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Anywhere that f’(x)=0 is called a stationary point; a stationary point could be a relative
minimum, a relative maximum, or a horizontal inflection
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• What do you know about the graph of f(x) when f’(x) is a) Positive b) Negative c) Zero
• What do you know about the slope of the tangent line at a relative extrema? Why is this so?
• Sketch a graph of f(x) when the sign diagram of f’(x) looks like
• What are the types of stationary points? What do they all have in common? What do the sign diagrams for each type look like?
-5 1
– – +
Stationary Point
? ?
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3.7 – Critical Points & Extrema
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Vocabulary• Critical Points – points on a graph in which
a line drawn tangent to the curve is horizontal or vertical– Maximum– Minimum– Point of Inflection
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Maximum
• When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c.
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Minimum
• When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c
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Point of Inflection
• Not a maximum or minimum
• “Leveling-off Point”
• When a tangent line is drawn here, it is vertical