16.3 Tangent to a Curve

(Don’t write this! )

What if you were asked to find the slope of a curve?

Could you do this? Does it make sense?

(No, not really, slopes are for lines, they are straight, curves

might not be straight)

So, what if I told you this is exactly what we are going to do!

We will be utilizing limits!!

Let’s think about geometry for a second.

A secant was a line that intersected a circle at two points.

A tangent was a line that intersected a circle at just one point.

Let’s extend this to a curve – any curve!

Secant Line to a Curve on Desmos

(Slide the dot on the right slowly towards the dot on the left)

The secant line becomes a tangent line!

( ) ( )f x f cPQ

x c

(Start writing)So, we want to make the secant become the tangent.

What is the slope of

We want Q to get closer to P

So x needs to get closer to c

this is the difference quotient!!

( ) ( )limx c

f x f c

x c

c x

Q

P(c, f (c))

(x, f (x))

5 5

22

5 5

5 5

4 3( ) (5) 4 3lim lim

5 5 4 3

4 3 4 9lim lim

5 4 3 5 4 3

5 1 1 1 1lim lim

3 3 64 3 9 35 4 3

x x

x x

x x

xf x f xm

x x x

x x

x x x x

x

xx x

( ) 4f x x Ex 1) Find the slope of the line tangent to the curveat P(5, 3)

We can’t substitute 5 in, so algebra to work!

*now we plug in 5*

slope

Draw your own sketch(5, 3)any other point(x, f (x))

If we know the slope of the tangent line, we can write the equation of the tangent line.

Ex 2) Find equation of tangent line to f (x) = x3 – 2x2 + 3 at P(1, 2).*to find equation of a line, we need two things: (1) slope (2) point

3 2 3 2

1 1 1

22

1 1

( ) (1) 2 3 2 2 1lim lim lim

1 1 1

( 1)( 1)lim lim( 1) 1 1 1 1

1

x x x

x x

f x f x x x xm

x x x

x x xx x

x

m = –1 P(1, 2) y – 2 = –1(x – 1) *you can leave like this – that is what calculus does

So back to an original question – how to find slope of a curve…the slope of a curve at point P the slope of the tangent at point P

The slope of a curve might vary from point to point, so it is helpful to be able to represent it in generic form using an arbitrary point. Then we can use it with specific slope values.

Ex 3) Find equation of line with slope 5 tangent to the graph of f (x) = x2 + 3x – 1

This time we have slope, but not the pointgeneral terms: f (x) and f (c)

2 2

2 2 2 2

( ) ( ) 3 1 3 1lim lim

3 3 3 3lim lim

( )( ) 3( ) ( ) 3lim lim

lim( 3) 3 2 3

x c x c

x c x c

x c x c

x c

f x f c x x c cm

x c x c

x x c c x c x c

x c x cx c x c x c x c x c

x c x cx c c c c m

want 2c + 3 = 5 2c = 2 c = 1

point (1, ? )f (1) = 1 + 3 – 1 = 3 (1, 3) y – 3 = 5(x – 1)

Physical quantities can also be found using the idea of a secant becoming a tangent.

Average rates are similar to secants (slope of line)Instantaneous rates are similar to tangents (limit of slope of line)Let’s look at velocity (a rate!)A position function f (t) describes the path something takes.

slope of secant

(4) (2) (16 12 5) (4 6 5) 63 ft/s

4 2 2 2

f f

slope of tangent

f (t) & f (2)

2 2

2 2 2

2 2

( ) (2) 3 5 3 3 2lim lim lim

2 2 2( 2)( 1)

lim lim( 1) 2 1 1 ft/s2

t t t

t t

f t f t t t tm

t t tt t

tt

b) What is the instantaneous velocity of the object at time t = 2 s?

Ex 4) The motion of an object is given by the function f (t) = t2 – 3t + 5 where f (t) is height of object in feet at time t seconds.a) What is the average velocity of the object between t = 2 s and t = 4 s?

Homework

#1603 Pg 869 #1, 5, 9, 12, 13, 15, 16, 18, 20, 21, 23, 25, 30, 32, 34, 35, 37, 40, 42