of 28 /28
Calculus AB Review Slopes of non-linear secant/tangent lines, and point-slope formula Kenyon Hundley Tuesday, September 25, 12

# Calculus AB - Slope of secant and tangent lines

Embed Size (px)

DESCRIPTION

### Text of Calculus AB - Slope of secant and tangent lines Calculus AB ReviewSlopes of non-linear secant/tangent

lines, and point-slope formula

Kenyon Hundley

Tuesday, September 25, 12 Important Formulas

Slope of a secant line at two points:

Slope of a tangent line at one point:

Point-Slope Equation:

m=--------f(x₀)-f(x₁)x₀-x₁

m=--------f(h+x₀)-f(x₀)h

y-y₀=m(x-x₀)

Tuesday, September 25, 12 1. f(x)=---

a) find the slope of the secant line between x = 3 and x = 4

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

x-2

x-5

Tuesday, September 25, 12 1.a) find the slope of the secant line between x = 3 and x = 4

f(x)=---x-2x-5

Our Line’s Equation

m=--------f(x₀)-f(x₁)x₀-x₁

Secant Line Slope Formula

Tuesday, September 25, 12 1.a) Steps1.Using the secant line slope formula we can

plug in the two values of ‘x’ that are given to us. We’ll just say x₀=3 and x₁=4. This looks like:

2.So on the top we see we have f(3) and f(4), which we interpret as “plug in 3 and 4 as x-values into the function and the solve for the y-value (or f(x) in other words)”. When we plug 3 and 4 in as x-values to our given equation, it looks like this:

m=--------f(3)-f(4)3-4

f(3)=--- = -1/23-23-5

f(4)=--- = -24-24-5

Tuesday, September 25, 12 1.a) Steps3.So now that we know f(3)=-1/2 and f(4)=-2,

we can plug these values back into our secant slope equation which will look like:

4.So from here we’ll do basic simplification, which results in our answer:

m=--------(-1/2)-(-2)3-4

m = -1.5

Tuesday, September 25, 12 1.b) find the slope of the tangent line at x = 2

f(x)=---x-2x-5

Our Line’s Equation

Tangent Line Slope Formula

m=--------f(h+x₀)-f(x₀)h

Tuesday, September 25, 12 1.b) Steps1.Using the tangent line slope formula we’ll

plug in the value of ‘x’ that is given to us. Since x=2, this looks like:

2.So in order to find the slope, we want to simplify the equation to a point where, if we plugged 0 in for ‘h’, we would not result in an undefined equation (0 as the denominator). We’ll start by solving the functions that are defined in the numerator, f(2+h) and f(2):

m=--------f(2+h)-f(2)h

f(2+h)= --- = h/(h-3)2+h-22+h-5 f(2)=--- = 02-2

2-5

Tuesday, September 25, 12 1.b) Steps3.So, now we’ll look at the equation with the

function values plugged in (f(h+2)=h/(h-3), f(2)=0).

4.Now with our equation m=1/(h-3), we can see that plugging in 0 for ‘h’ won’t result in an undefined value so we can use this to solve for the slope.

m=--------h/(h-3)h = --------h

h(h-3) = --------1h-3

m = -- = -1/3 as h approaches 010-3

We write the answer like “m = -1/3 as h approaches 0” because we cannot find the exact slope of the tangent line at a given point, but we can get really

close to it by plugging in 0 for ‘h’, or a decimal value extremely close to 0.

h

Tuesday, September 25, 12 1.c) find the equation of the tangent line at x = 2

-1/3

Tangent Line’s Slope

Point-Slope Formula

y-y₁ = m(x-x₁)

f(x)=---x-2x-5

Our Line’s Equation

Tuesday, September 25, 12 1.c) Steps1.To begin solving for the tangent line’s

equation, we substitute x=2 in to our line’s equation:

2.So now we know that 0 is x’s corresponding y-value. We’ll use (2,0) as our x₁ & y₁ in the point-slope equation, and also -1/3 as the slope (we found the slope of the same tangent line in 1.b):

f(x)= --- = 02-22-5

y-0 = -1/3(x-2)

Tuesday, September 25, 12 1.c) Steps3.Now we basically just simplify the equation

to slope-intercept form, then we have our answer:

y = -1/3x+2/3

Tuesday, September 25, 12 2. f(x)=2x²-3x+1

a) find the slope of the secant line between x = -1 and x = 2

b) find the slope of the tangent line at x = 2

c) find the equation of the tangent line at x = 2

Tuesday, September 25, 12 2.a) Steps1.Using the secant line slope formula we can

plug in the two values of ‘x’ that are given to us. We’ll just say x₀=-1 and x₁=2. This looks like:

2.So on the top we see we have the functions f(-1) and f(2). When we plug -1 and 2 in as x-values to our given equation, it looks like this:

m=--------f(-1)-f(2)3-4

f(-1)=2(-1)²-3(-1)+1= 6f(2)=2(2)²-3(2)+1= 3

Tuesday, September 25, 12 2.a) Steps3.So now that we know f(-1)=6 and f(2)=3, we

can plug these values back into our secant slope equation which will look like:

4.So from here we’ll do basic simplification, which results in our answer:

m=--------(6)-(3)-1-2

m = -1

Tuesday, September 25, 12 2.b) find the slope of the tangent line at x = 2

Our Line’s Equation

Tangent Line Slope Formula

m=--------f(h+x₀)-f(x₀)h

f(x)=2x²-3x+1

Tuesday, September 25, 12 2.b) Steps1.Using the tangent line slope formula we’ll

plug in the value of ‘x’ that is given to us. Since x=2, this looks like:

2.So in order to find the slope, we want to simplify the equation to a point where, if we plugged 0 in for ‘h’, we would not result in an undefined equation (0 as the denominator). We’ll start by solving the functions that are defined in the numerator, f(2+h) and f(2):

m=--------f(2+h)-f(2)h

f(2+h)=2(2+h)²-3(2+h)+1=2h²+3+5hf(2)=2(2)²-3(2)+1=3

Tuesday, September 25, 12 2.b) Steps3.So, now we’ll look at the equation with the

function values plugged in.

4.Now with our equation m=2h+5, we can see that plugging in 0 for ‘h’ won’t result in an undefined value so we can use this to solve for the slope.

m=--------2h²+3+5h-3h = --------2h²+5h

h = --------2h+51

m = 0+5 = 5 as h approaches 0

h

Tuesday, September 25, 12 2.c) find the equation of the tangent line at x = 2

5

Tangent Line’s Slope

Point-Slope Formula

y-y₁ = m(x-x₁)

Our Line’s Equation

f(x)=2x²-3x+1

Tuesday, September 25, 12 2.c) Steps1.To begin solving for the tangent line’s

equation, we substitute x=2 in to our line’s equation:

2.So now we know that 3 is x’s corresponding y-value. We’ll use (2,3) as our x₁ & y₁ in the point-slope equation, and also 5 as the slope (we found the slope of the same tangent line in 2.b):

y-3 = 5(x-2)

y=2(2)²-3(2)+1 = 3

Tuesday, September 25, 12 2.c) Steps3.Now we basically just simplify the equation

to slope-intercept form, then we have our answer:

y = 5x-7

Tuesday, September 25, 12 3. If an object travels a distance of s = 2t²-5t+1,

where s is in feet and t is in seconds,find:

a) the average velocity of object within the first 10 seconds.

b) the instantaneous velocity of the object at t = 3 seconds.

Tuesday, September 25, 12 3.a) the average velocity of object within the first 10 feet

Our Line’s Equation

Vavg=--------s(t₀)-s(t₁)

t₀-t₁

Velocity Average (Secant Line Slope) Formula

s(t)= 2t²-5t+1

Tuesday, September 25, 12 3.a) Steps1. In this problem, we basically use the same rules and

formula as the secant line slope formula, except we use different notation. We can plug 10(seconds) and 0(seconds) for t₀ and t₁ in order to find our average velocity during the first ten seconds. We’ll say t₀=0 and t₁ =10. This looks like:

2. So on the top we, once again, see that we have functions; s(0) and s(10) are our functions. After plugging them into our line’s equation, we get this:

Vavg=--------s(0)-s(10)0-10

s(0)= 2(0)²-5(0)+1 = 1s(10)= 2(10)²-5(10)+1= 151

Tuesday, September 25, 12 3.a) Steps3.So now that we know s(0)=1 and f(10)=151,

we can plug these values back into our average velocity equation which will look like:

4.So from here we’ll do basic simplification, which results in our answer:

Vavg=--------(1)-(151)0-10

Vavg = 15 feet per second

Tuesday, September 25, 12 3.b) the instantaneous velocity of the object at t = 3 seconds

Our Line’s Equation

Instantaneous Velocity (Tangent Line Slope) Formula

Vinst= ---------s(h+t₀)-s(t₀)h

s(t)= 2t²-5t+1

Tuesday, September 25, 12 3.b) Steps1. To discover the instantaneous velocity at t=3 seconds, we

basically use the tangent line slope formula but with different notation. We plug in the given value of 3 in for “t” in the Vinst equation. This looks like:

2. So, (like the tangent line slope equations) in order to find the instantaneous velocity, we want to simplify the equation to a point where, if we plugged 0 in for ‘h’, we would not result in an undefined equation (0 as the denominator). To start we’ll solve the functions that are defined in the numerator, s(3+h) and s(3):

Vinst=--------s(3+h)-s(3)h

s(h+3)= 2(h+3)²-5(h+3)+1= 2h²+7h+4s(3)= 2(3)²-5(3)+1= 4

Tuesday, September 25, 12 3.b) Steps3.So, now we’ll look at the equation with the

function values plugged in.

4.Now with our equation Vinst=2h+7, we can see that plugging in 0 for ‘h’ won’t result in an undefined value so we can use this to solve for the instantaneous velocity at 3 seconds.

Vinst=--------2h²+7h+4-4h = --------2h²+7h

h = --------2h+71

Vinst = 0+7 = 7 feet per second

Tuesday, September 25, 12 ##### Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree
Documents ##### TANGENT – THE SLOPE RATIO (TRIGONOMETRY) 4.1.1 …kenstonlocal.org/dubovec/wp-content/uploads/2015/09/Ch4.pdf · TANGENT – THE SLOPE RATIO (TRIGONOMETRY) ... ratio:the steeper
Documents ##### Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Secant and …elyse/100/2016/Deriv_Conceptual.pdf · Chapter 2: Derivatives 2.1: Revisiting Tangent Lines Slope of Secant and
Documents ##### SM2H 6.2 Inscribed Angles, Chord, Tangent and Secant …...SM2H 6.2 Inscribed Angles, Chord, Tangent and Secant Theorems Complete the statement. 1. A(n) _____angle is an angle whose
Documents ##### 11-Secant-Tangent and Tangent-Tangent Angles · ©1 Z2z0d1 B1z QK3uKtNa J nSNoAfBtyw MalrKeZ 3L 8L uCp.X o 0AWlvl 3 GriYghQtws6 0rneLsneErgvZe ZdO.X r vM 6a Sd Ge7 Owbi rt 2h v sIcn
Documents ##### Soldier Pile Walls Soil Nail Systems Secant Piling Soil ... · PDF fileSoldier Pile Walls Soil Nail Systems Secant Piling Soil Mix Walls Underpinning Slope Stabilization
Documents ##### E-Portfolio - GeoGebra€¦ · E-PORTFOLIO Mauro Rábago. Slope of Tangent Line Using Secant Lines. Limits Graphically. Limits at Infinity: Horizontal Asymptotes. Application Limits
Documents ##### TANGENT: THE SLOPE RATIO (TRIGONOMETRY) 3.2.1 … · TANGENT: THE SLOPE RATIO (TRIGONOMETRY) ... The tangent of an angle in a right triangle is the ratio opposite leg ... 549.9 +
Documents ##### Scanned with CamScanner - MR. FAUCETTE · 2019. 4. 22. · Vocab: secant segment, external secant segment, tangent segment Practice: Practice Packet #14 Do Now: (4 min to do, 2 to
Documents ##### CE 402: Part C - Fayoum · CE 406: Foundation Design 15 Secant Pile Walls. • The important advantage of secant and tangent walls is the increased alignment flexibility. • The
Documents ##### The Slope of a Curve or How Secant Lines become Tangent Lines
Documents ##### 2. Slope of a Tangent Blank.notebook...2. Slope of a Tangent Blank.notebook 3 September 07, 2016 Summary: Algebraic computation of slopes of tangents using limits. (Instantaneous Rates
Documents ##### Tangent Lines Section 2.1. Secant Line A secant line is a line that connects two points on a graph. Notice the slopes of secant lines are different depending
Documents ##### 2012 Graphing & Writing Secant, Cosecant, Tangent and ...thsprecalculus.weebly.com/uploads/7/0/8/1/7081416/precal... · Graphing & Writing Secant, Cosecant, Tangent and Cotangent
Documents ##### Math 408R:UT Worksheet 5: Secant and tangent lines; the … · 2014. 8. 21. · Math 408R:UT Worksheet 5: Secant and tangent lines; the derivative 9/16/2014 9.A car travels in a straight
Documents ##### Section 7.3 The Graphs of the Tangent, Cosecant, Secant, and Cotangent Functions. · 2014. 10. 20. · Section 7.3 The Graphs of the Tangent, Cosecant, Secant, and Cotangent Functions
Documents ##### MathPortafolio - GeoGebra...2 of 22 use the ot part to an of the slope of the tangent me to the curve at (2, 3. The on the curve If is the point . the slope of the secant line PO (round
Documents ##### A REVIEW ON NOSE CONE DESIGNS FOR DIFFERENT FLIGHT … · 2020. 8. 24. · Conic, spherically blunted conic, bi-conic, tangent ogive, spherically blunted tangent ogive, secant ogive,
Documents ##### Chapter 3 Employment of Projections and Thematic Base-Map Compilation Definition: Tangent or secant to equator is termed regular, or normal. Tangent or
Documents ##### Tangent Lines and Rates of Change...2 we get Thus the slope of the tangent line is 12. Using the point-slope formula above, or Next we are interested in finding a formula for the slope
Documents ##### Circles - White Plains Middle School€¦ · SWBAT: Apply the Power Theorems 1. Given 2 ... The models below show segment relationships in circles. Secant-Tangent Secant ÂX and tangent
Documents ##### AnalyC… · Tangent and Cotangent Graphs ... Notice the relationship between your answers to (1) and (2) and to the tangent graph. ... Secant and Cosecant Graphs Name Period
Documents ##### Organize the following into 2 categories: DERIVATIVES & INTEGRALS Slope of a Tangent Line Slope of a…
Documents ##### Specifications for the Construction of Secant and …... · Specifications for the Construction of Secant and Tangent Pile ... pile wall design process to ... Secondary Pile A vertical
Documents Documents