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Concepts of Calculus © G. Battaly 2019 1 September 11, 2019 Homework on the Web Calculus Home Page Class Notes: Prof. G. Battaly GOALS: Understand 1. What are tangent lines? 2. What are secant lines? 3. How do slopes of tangent lines relate to slopes of secant lines? 4. How does the slope of a tangent line relate to the slope of the function at the point in common to both. 5. What is average velocity? 6. What is instantaneous velocity? 7. How can we find areas beneath curves? 2.1 Tangent, Velocity, Area Study 2.1 # 1, 2, 3, 7, 9, 16, 17, 24, 25 classnotes Homework on the Web Calculus Home Page Class Notes: Prof. G. Battaly What is a secant line ? Secant Line: A line that intersects a curve in two or more points. 2.1 Tangent tangent line y=x 2 secant line

# 2.1 Tangent, Velocity, Area - Battaly2.1 Tangent tangent line Calculus Home Page Homework on the Web Class Notes: Prof. G. Battaly How do slopes of tangent lines relate to slopes of

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• Concepts of Calculus

September 11, 2019

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

GOALS:   Understand 1.  What are tangent lines?2.  What are secant lines?3.  How do slopes of tangent lines relate to slopes of secant lines?4.  How does the slope of a tangent line relate to the slope of the function at the point in common to both.5.  What is average velocity?6.  What is instantaneous velocity?7.  How can we find areas beneath curves?

2.1  Tangent, Velocity, Area

Study 2.1  # 1, 2, 3, 7, 9, 16, 17, 24, 25

classnotes

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

What is a secant line ?

Secant Line:

∎A line that intersects a curve in two or more points.

2.1  Tangent

tangent line

y = x2

secant line

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• Concepts of Calculus

September 11, 2019

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

y = x2

secant line

Slope is the rate of change of y with respect to x.For a straight line, this is constant.ie:  y changes at the same rate no matter what values of x we select.For a curve, this is variable.ie:  we expect y to change at a different rate for different values of x

Calculus:  How do we find the rate of change of y with respect to x for curves?

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• Concepts of Calculus

September 11, 2019

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

What is the slope of a secant line?

∎ msec = y2  y1   =   4  1   =  3 = 3               x2  x1        2  1       1

y = x2

secant line

Calculus:  Examine the slope of the secant line as point A approaches point C.

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

∎ msec = y2  y1   =   4  1   =  3 = 3               x2  x1        2  1       1

What is the slope of a secant line?

∎ msec = y2  y1  =   2.25  1  =  1.25 = 2.5               x2  x1        1.5  1        0.5

y = x2secant line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)(1.01,1.0201)

slope of a secant line from C(1,1) to:

432.52.1

2.01

What value does msec approach as A(x,y) gets closer to C(1,1)?

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Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

∎ msec = y2  y1   =   4  1   =  3 = 3               x2  x1        2  1       1

What is the slope of a secant line?

∎ msec = y2  y1  =   2.25  1  =  1.25 = 2.5               x2  x1        1.5  1        0.5

y = x2secant line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)

432.52.1

(1.01,1.02) 2.01

slope of a secant line from C(1,1) to:

msec >  mtan

msec >  2

What value does msec approach as A(x,y) gets closer to C(1,1)?

Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

∎ msec = y2  y1   =   4  1   =  3 = 3               x2  x1        2  1       1

∎ msec = y2  y1  =   2.25  1  =  1.25 = 2.5               x2  x1        1.5  1        0.5

y = x2

tangent line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)

432.52.1

(1.01,1.02) 2.01

slope of a secant line from C(1,1) to:

msec >  2

(1,1) ?

What value does msec approach as A(x,y) gets closer to C(1,1)?

(1.0001,1.00020001) 2.0001?

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Homework on the WebCalculus Home Page

Class Notes:  Prof. G. Battaly

2.1  Tangent

tangent line

What is the slope of a secant line?

∎ msec = y2  y1   =   9  1   =  8 = 4               x2  x1        3  1       2

∎ msec = y2  y1   =   4  1   =  3 = 3               x2  x1        2  1       1

∎ msec = y2  y1  =   2.25  1  =  1.25 = 2.5               x2  x1        1.5  1        0.5

y = x2

tangent line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)

432.52.1

(1.01,1.02) 2.01

slope of a secant line from C(1,1) to:

msec >  mtan  but cannot use slope formula to find

msec >  2

(1,1) ?

But  11   DNE     DIV BY 0        11

What value does msec approach as A(x,y) gets closer to C(1,1)?

(1.0001,1.00020001)      2.0001

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What is a tangent line ?

Tangent Line:

∎A line that touches a curve at a point without crossing over. ∎A line which intersects a (differentiable) curve at a point where the slope of the curve equals the slope of the line.

2.1  Tangent

tangent line

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How do slopes of tangent lines relate to slopes of secant lines?

∎They are only related if one of the points defining the secant line is also a point on the curve and on the tangent line.

2.1  Tangent

∎As the point not in common approaches the common point, the slope of the secant approaches the slope of the tangent. tangent line

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2.1  Tangent

tangent line

y = x2

tangent line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)

432.52.1

(1.01,1.02) 2.01

(1,1) ?(1.0001,1.00020001)      2.0001

msec >  2

Find:  equation of tangent line to           y = x2 at C(1,1)?

conclude mtan= 2

DNE

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2.1  Tangent

tangent line

y = x2

tangent line

A(x,y)  msec(3,9)(2,4)(1.5,2.25)(1.1,1.21)

432.52.1

(1.01,1.02) 2.01

(1,1) ?(1.0001,1.00020001)      2.0001

msec >  2

Find:  equation of tangent line to           y = x2 at C(1,1)?

conclude mtan= 2

Pointslope form of straight line:           y  y1 = mtan(x  x1)

y  1 = 2(x  1)

y  1 = 2x  2

y = 2x  1 equation of tangent line to           y = x2 at C(1,1)

DNE

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2.1  Tangent

tangent line

skip

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2.1  Tangent

tangent line

x        y        msec

mtan = 3

As Q(x,y) gets closer and closer to P(1,1),  msec approaches mtan = 3

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Example:   2.1 # 2 (stewart)2.1  Tangent

tangent line

Cardiac monitor: number of heart eats after t minutes

t(min) 36  38 40   42 44

beats     2530   2661    2806   2948    3080

Estimate heart rate after 42 minutes using secant line between

a)  t=36  and t=42            b)  t=38   and t=42

c)  t=40 and t=42             d)  t=42 and t=44.

What are your conclusions?

skip

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2.1  Tangent

but need to avoid division by zero, so change 42 in L1 to 42.01Not really conclusive, but can average all, w/o 0;  orcould average the 2 closestGet 68.5 for 2 closest.Get 69.6 for allLinear regression gets slope of 69.2

Example:   2.1 # 2 (stewart)

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2.1  Velocity

tangent line

Montauk Point is 150 miles away.  I am planning to go birding there this weekend.  If it takes me 3 hours to get there, how fast will I travel?

What is this speed of travel called?

Do I travel at that speed when I leave my driveway?  When I am on the LI expressway?

How fast could I be travelling at any time during the trip?  What is this called?

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2.1  Velocity

tangent line

Montauk Point is 150 miles away.  I am planning to go birding there this weekend.  If it takes me 3 hours to get there, how fast will I travel?

What is this speed of travel called?

Do I travel at that speed when I leave my driveway?  When I am on the LI expressway?

How fast could I be travelling at any time during the trip?  What is this called?

150  = 50 mph   3

average velocity, or speed  (no direction)

No.  probably drive at 5 mph backing upNo.  probably drive at 55 mph or 60 mph

instantaneous velocity:  rate of change at a specific time, t

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2.1  Velocity

tangent line

Average Velocitys(t) is the position of an object moving along a coordinate axis at time t.  The average velocity of the object over a time interval [a, t] is

vave =  s(t)s(a)  =  Δs              t  a          Δt

Instantaneous VelocityThe instantaneous velocity at the time t=a is the value that the average velocities approach as t approaches a, assuming that the value exists.

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2.1  Velocity

tangent line

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average speed between t=1 and t=1.05 sec = 4.96

average speed between t=1 and t=1.005 sec = 5.2

average speed between t=1 and t=1.001 sec = 5.2

average speed between t=1 and t=1.01 sec = 5.2

Doing this by hand, we used too few decimal places to actually get a meaningful difference among the results  eg: three of four above are 5.2.  Need to use an extra decimal place.  See next page.

average speed between t=1 and t=1.05 sec = 4.956

average speed between t=1 and t=1.005 sec = 5.18

average speed between t=1 and t=1.001 sec = 5.20

average speed between t=1 and t=1.01 sec = 5.12

Adding one more decimal place to the h values, we get results that discriminate between outcomes.

We see that, as t approaches the value of 1, the average speed approaching 5.2.

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2.1  Velocity

tangent line

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2.1  Velocity

tangent line

vavg = 5.2

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How do we find the area of:

2.1  Area

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How do we find the area of:

2.1  Area

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A = s2 A = πr2

A = 1 bh       2

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Can we find the area with a formulaor other tools from algebra or geometry?

2.1  Area

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How can we estimate the area?2.1  Area

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Using 4 rectangles:

Area of rectange 1 = 1 x 3 = 3Area of rectange 2 = 1 x 4 =4Area of rectange 3 = 1 x 3 = 3Area of rectange 4 = 1 x 0 = 0

Ʃ = 10ΔX=1

2.1  Area

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Using 8 rectangles:

y = x(x4)

2.1  Area

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16

Using 16 rectangles:

y = x(x4)

2.1  Area

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2.1  Area

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2.1  Area

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0.4*0.8 + 0.4*0.980+0.4*0.980+0.4*0.8+0.4*0= 0.4(0.8 + 0.980+0.980+0.8+0) = 0.4*3.560 = 1.424

A = (1/2) π r2  = (1/2) π (1)   = π/2 = 1.5708 actualFor 10 rectangles of width 0.2 would get:0.2*f(0.8)+0.2*f(0.6)+0.2*f(0.4)+0.2*f(0.2)+0.2*f(0)+0.2*f(0.2)+0.2*f(0.4)+0.2*f(0.6) +0.2*f(0.8)+0.2*f(1) = 0.2[0.6+0.8+0.9165+0.9798+1+0.9798+0.9165+0.8+0.6+0]=0.2*7.5926= 1.51852    getting closer to actual

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