Graphs of Exponentials - Lecture Notes | MATH 250A, Study notes of Differential Equations

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MATH 250aFall Semester 2007Section 2 (J. M. Cushing) Thursday, August 30 http://math.arizona.edu/~cushing/250a.html

GRAPHS OF EXPONENTIALS

Notes (3)^

for all^ ( graph lies above the horizontal

-axis )

( )^0

x

P x^

x > 0

(4)^

( graph passes through point (0,1) )

Sketch the graph of (0) 1 P e = =

x P ( x ) = e

(1)^

x^ implies graph is^0

increasing

dP^ e =^ > dx^^2 (2) 2

x^ implies graph is^0

concave up

d P^ e =^ > dx^ (5) lim^ ( )^

,^ lim^ ( )

x^

x

P x^

P x

→+∞^

= +∞^ =→ −∞

GRAPHS OF EXPONENTIALS^ Sketch the graph of

x − (^) P ( x ) = e

Reflect graph of

x^ e through the vertical axis

GRAPHS OF EXPONENTIALS^ Sketch the graph of

x − (^) P ( x ) = e^ x -3^ -^

P 311 3 -

GRAPHS OF EXPONENTIALS

x

-2^ -1^0

(^2) P 1.0 0.8 0.6 0.4 0.2 1 2 Sketch the graph of

x − ( ) P x e =

2 (1)^

for^ implies graph is

for

2 0

0

0

x^

decreasing

dP^ xe^

x^

x

−= − (^) dx <^ >^

⇒^

(^ )^

2 2 2 2

concave down for

2 / 2

(2)

concave up for

(^0) 2 / 2 2 2^1

x

x x

d P^ x^

e dx

−

< ⎧⎪^

<^ < ⎪ =^ −^

⇒ ⎨⎪⎪⎩ (3)^ ( )^ 0,

(0)^ 1,^ lim

( )^0 x P x^ P^

P x → +∞

^ =^

=^ means graph is symmetricwith respect to the vertical axis.Such a function is called

(^ )^ ( ) P x^ P x −^ =^ even.

GRAPHS OF EXPONENTIALS

x

-2^ -1^0

(^2) P 1.0 0.8 0.6 0.4 0.2 1 2 Sketch the graph of

x − ( ) P x e =

2 (1)^

for^ implies graph is

for

2 0

0

0

x^

decreasing

dP^ xe^

x^

x

−= − (^) dx <^ >^

⇒^

(^ )^

2 2 2 2

concave down for

2 / 2

(2)

concave up for

(^0) 2 / 2 2 2^1

x

x x

d P^ x^

e dx

−

< ⎧⎪^

<^ < ⎪ =^ −^

⇒ ⎨⎪⎪⎩ (3)^ ( )^ 0,

(0)^ 1,^ lim

( )^0 x P x^ P^

P x → +∞

^ =^

=^ means graph is symmetricwith respect to the vertical axis.Such a function is called

(^ )^ ( ) P x^ P x −^ =^ even.

GRAPHS OF LOGARITHMIC FUNCTIONS^ Sketch the graph of

P ( x )^ = ln ( x

) Notes (4)^

( graph passes through point (1, 0) )(1) 0 P =

0 (5)^

( why? )

lim^ ( )^

,^ lim^ ( )

x^

t

P x^

P x

→+∞^

for^ implies graph is = +∞^ = −∞→ +

increasing

dP^

x

=^ >^ dx x

2 2 2

1 implies graph is^0

concave down

d P^ = −^ dx^ x

APPLICATIONS OF EXPONENTIAL FUNCTIONS How long does it take a decreasing exponential function

P ( t )

that starts at^

P at time^ t^ =^0

0 to decrease by 50%? How does this halving time

t depend on h^

P?^0

How does^ th

(1) Exponential Decaydepend on the exponential rate of decay?

APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay How long does it take a decreasing exponential function

P ( t )

that starts at^

P at time^ t^ =^0

0 to decrease by 50%? ANSWER:where is the exponential decay rate.

1 ln 2 t hr r

How does this halving time

t depend on h^

P?^0

ANSWER: it doesn’t depend on

P at all!^0

How does^ th

depend on the exponential rate of decay? (^) ANSWER: it is^ inversely proportional

to^ r^.

APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay How long does it take a decreasing exponential function

P ( t )

that starts at^

P at time^ t^ =^0

0 to decrease by 50%? ANSWER:where is the exponential decay rate.

1 ln 2 t hr r

t is called the h^

half life^ of^ P

APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay^ EXAMPLE Suppose that the amount

P^ of a drug of a medicinal drug present in a patient’s bloodstream decreases by

p % per hour.

(a)^ Show that

P^ is a decreasing exponential functionand find its half-life. (1)^ (0) 1^

p 100 ⎛^ P P ⎞⎟⎜ =^ −^

⎟⎜ ⎟⎜⎝ ⎠

2 (2)^ (1) 1^

p^ p (0) 1 100 100 P^ P^ ⎛^ ⎞^ ⎛^ ⎞⎟^ ⎟⎜^ ⎜ P ⇒^ =^

−^ =^

−⎟ ⎟ ⎜^

⎜⎟ ⎟ ⎜^

⎜ ⎝^ ⎠^

⎝^ ⎠^3 (3)^ (2) 1^

p^ p (0) 1 100 100 P^ P^ ⎛^ ⎞^ ⎛^ ⎞⎟^ ⎟⎜^ ⎜ P ⇒^ =^

−^ =^

−⎟ ⎟ ⎜^

⎜⎟ ⎟ ⎜^

⎜ ⎝^ ⎠^

⎝^ ⎠

APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay^ EXAMPLE Suppose that the amount

P^ of a drug of a medicinal drug present in a patient’s bloodstream decreases by

p % per hour.

(a)^ Show that

P^ is a decreasing exponential functionand find its half-life. (1)^ (0) 1^

p 100 ⎛^ P P ⎞⎟⎜ =^ −^

⎟⎜ ⎟⎜⎝ ⎠^2 (2)^ (0) 1^

p 100 ⎛^ P P ⎞⎟⎜ =^ −^

⎟⎜ ⎟⎜⎝ ⎠^3 (3)^ (0) 1^

p 100 ⎛^ P P ⎞⎟⎜ =^ −^

Observe the pattern … and induct :⎟⎜ ⎟⎜⎝ ⎠ ( )^ (0) 1^

t p 100 ⎛^ P t P ⎞⎟⎜ =^ −^

⎟⎜ ⎟⎜⎝ ⎠ This is an exponential function.

APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay^ EXAMPLE Suppose that the amount

P^ of a drug of a medicinal drug present in a patient’s bloodstream decreases by

p % per hour.

(a)^ Show that

P^ is a decreasing exponential functionand find its half-life.^ (ln

)

100 where^

1 where ( )^ (0)^

1 ( )^ (0) ( )^ (0)^

ln^0 t a t rt

p r P t^ P^ a

a P t^ P^ e P t^ P^ e

a =^ −

=^ −^ < = =^

= −^ >^ (^

)

Half life:^

ln 2^2

ln^ ln 1

t^ h r^

a^

p =^ =^

=− −^ −

Suppose that the amount

P^ of a drug of a medicinal drug present in a patient’s bloodstream decreases by

p % per hour.

(b) For the antibiotic ampicillis p = 40%. What is the half-life?Draw a graph of

P^ as a function of time. APPLICATIONS OF EXPONENTIAL FUNCTIONS

(1) Exponential Decay^ EXAMPLE