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

Concepts of Calculus

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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



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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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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September 11, 2019



2.1 Tangent

tangent 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

y = x2

secant line

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



2.1 Tangent

tangent 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 = 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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September 11, 2019



2.1 Tangent

tangent 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 = 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


msec > mtan

msec > 2




2.1 Tangent

tangent 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


msec > 2

(1,1) ?


(1.0001,1.00020001) 2.0001?




September 11, 2019



2.1 Tangent

tangent 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


msec > mtan but cannot use slope formula to find

msec > 2

(1,1) ?

But 11 DNE DIV BY 0 11


(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



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




September 11, 2019



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



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




September 11, 2019



2.1 Tangent

tangent line

skip

Homework on the WebCalculus Home PageClass Notes: Prof. G. Battaly

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




September 11, 2019



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



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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September 11, 2019



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?



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




September 11, 2019



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.



2.1 Velocity

tangent line




September 11, 2019

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




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.





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.



September 11, 2019



2.1 Velocity

tangent line



2.1 Velocity

tangent line

vavg = 5.2




September 11, 2019

How do we find the area of:

2.1 Area



How do we find the area of:

2.1 Area



A = s2 A = πr2

A = 1 bh 2

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September 11, 2019

What about the shaded area below?

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

2.1 Area



How can we estimate the area?2.1 Area






September 11, 2019

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



Using 8 rectangles:

y = x(x4)

2.1 Area






September 11, 2019

16

Using 16 rectangles:

y = x(x4)

2.1 Area



2.1 Area






September 11, 2019

2.1 Area



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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