Fungsi e Dan Laju Berkaitan

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    LN 7-1 Kok o Martono , FMIPA - ITB MBM 070

    Exponential Functions

    Exponential Function The equation f ( x) = b x, b > 0, b 1 defines anexponential function for each different constant b, called the base . Thedomain of f is the set of real numbers, and the range of f is the set of all

    positive real numbers.

    y

    0 x

    Basic Properties of the Exponential Graph All graphs will pass through the point (0,1),b0 = 1 for any permissible base b.All graphs are continuous curves on \ , with

    no hole or jumps.The x-axis is a horizontal asymptote.If b > 1, then y = b x increases as x increase.If 0 < b < 1, then y = b x decreases as x increases.

    Properties of the Exponential Function For a , b > 0, and a , b 1 we have

    Exponent laws:

    ( ), , ( ) , ( ) , x x

    y x

    x x y x y x y x y x y x x xa a a

    b baa a a a a a ab a b

    + - = = = = =

    x ya a x y= = , x xa b a b= = .

    Base e Exponential Function

    e = ( ) ( ) ( )1/1 1 0lim 1 lim 1 lim 1n x x

    n xn x x x

    + = + = + = 2.718 281 828 459

    Exponential function with base e is defined by y = e x. The graph is increases.

    1

    x y b

    b

    =

    >

    0 1

    x y b

    b

    =

    < <

    1

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    LN 7- 2

    MBM071

    Exponential Function and Related Rates

    Simple Interest and Continuous Compound Interest

    Simple Interest If a principal P is borrowed at an annual rate of r , then after t years at the simple interest the borrower will owe the lender an amount A, where

    A = P + Prt = P (1 + rt )Compound Interest If a principal P is borrowed at an interest is compounded n times a year, then the borrower will owe the lender at amount A given by

    ( )1nt r

    n A P= +

    where r /n is the interest rate per compounding period and nt is the number of compounding period.

    Continuous Compound Interest From compound interest formula, keeping P ,r , and t fixed and compute limit of this function for n . The result is

    ( ) ( ) ( )( )1/0lim 1 lim 1 lim 1n rt nt rt sr r rt r

    n nn n sP P P s Pe

    + = + = + =

    Formula for Continuous Compound Interest : rt A Pe= , where P = principal, r = annual nominal interest rate compound continuously,

    t = time in years, and A = amount at time t .

    Example 7.1

    If P = $100, r = 0.06 = 6%, and t = 2 years, then the borrower at the simple interest will owe the lender an amount A = 100(1 + 0.06 2) = 100 1.12 = $112. If P = $100, r = 0.06 = 6%, t = 2 years, and n = 4, then the borrower at the com-

    pound interest will owe the lender an amount

    A = ( )4 20.06 8

    4100 1 100 (1.015)

    + = = $112.65.

    If P = $100, r = 0.06 = 6%, t = 2 years, and n = 4, then the borrower at the conti-nuous compound interest will owe the lender an amount

    A = 100 e (0.06) (2) = 100 e 0.12 = $112.75.

    Logarithmic Function

    The inverse of an exponential function is called a logarithmic function y = b log x x = b y, b > 0 and b 1

    We write y = log x if b = 10; y = log x x = 10 y, x > 0 and y \ .We write y = ln x (natural logarithmic) if b = e; y = ln x x = e y. From y = ln x x = e y we have ln e y = y, y \ and e ln x = x, x > 0; ln e = 1.

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    LN 7- 3

    MBM072

    Exponential Function and Related Rates

    Example 7.2 (a) How long will it take an investment of $ 5,000 to grow to$8,000 if it is invested at 5% compound continuously? (b) How long will it take

    money to double if it is invested at 6.5% compound continuously?Solution (a) Starting with formula A = Pe r t , we must solve 8,000 = 5,000 e 0.05 t for t . From

    this equation we have e 0.05 t = 1.6. Take the natural logarithm of both side and

    solved for t , we have 0.05ln ln1.6t e = , then 0.05 t = ln 1.6. Thus 0.05ln1.6 9.4t =

    years (use calculator!)

    (b) Starting with formula A = Pe r t , we must solve 2 P = Pe 0.065 t for t . From thisequation we have e 0.065 t = 2. Take the natural logarithm of both side and solved

    for t , we have ln e 0.065 t = ln 2, then 0.065 t = ln 2. Thus 0.05ln 2 10.66t = years.

    (use calculator!)

    Derivative Formula for Natural Logarithmic and Exponential

    Theorem 1ln , 0d dx x x x= > and , x xd

    dx e e x= \ .

    y

    y = x

    x

    Proof

    ( ) ( )( ) ( )

    0 0

    /

    0 0

    / 1/

    0 0

    ln ( ) ln 1

    1 1

    1 1

    1 1

    ln lim lim ln

    lim ln 1 lim ln 1

    ln lim 1 ln lim 1

    ln

    h h

    x h

    h h

    x h s

    h s

    d x h x x hdx h h x

    x h h x h x x x

    h x x x

    x x

    x

    s

    e

    + - += =

    = + = +

    = + = +

    = =

    Use the fact0

    1lim 1hh

    eh- = (use calculator or table,

    without proof), then

    0 0

    0

    ( 1)

    1

    lim lim

    lim 1

    x h x x h

    h

    x

    h h

    x x x

    h

    d e e e edx h h

    eh

    e

    e e e

    +

    - -

    -

    = =

    = = =

    x y e=

    ln y x=

    ln y x=

    x y e= 1

    0 1 e

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    LN 7- 4

    MBM073

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems An investor bought stock for $20,000. Five year later, the stock was sold for $30.000, If interest is compound continuously, what annual nominal rate of interest did the original $20,000 investment earn?

    Solution

    Problems Show that the doubling time t in years at an annual rate r compounded continuouslyis given by t = ln2r . If 0.02

    r 0.30, explain why these restrictions on r reasonable.

    Solution

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    LN 7- 5

    MBM074

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems If ( ) ln f x x x= + and 2( ) 1 x f x e= + , find f ( x) and g ( x).

    Solution

    Problems Find the absolute extreme value of f ( x) = x2(2 ln x) and g( x) = e

    x / x

    2.

    Solution

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    LN 7- 6

    MBM075

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems Given the function20.5( ) , x f x e x-= - <

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    MBM076

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems Drug concentration The concentration of a drug in the bloodstream t hours after injection is given approximately by ( ) 4.35 , 0 5t C t e t -= . What is the rate of change of con-centration after 1 hour and after 4 hours?

    Solution

    Problems Blood pressure The mathematical model relating systolic blood pressure and ageis given approximately by ( ) 40 25ln ( 1), 0 65P x x x= + + , where P ( x) is pressure measured in mmHg and x is age in years. What is the rate of change of pressure at the end of 10 years?

    Solution

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    MBM077

    Exponential Function and Related Rates

    Implicit Differentiation

    Implicit Differentiation If F ( x, y) = 0 defines y as differentiable function of x,we can find the derivative of y with respect to x by process implicit differentiation .

    If ( )u u y= and ( ) y y x= , the chain rule gives du du dydx dy dx= . Thus the derivative dydx

    can find by differentiate each side of F ( x, y) = 0 respect to x and simplify.

    Example 7.3 Find y from the equation 2 2 1 xy y x- = + and find the equation

    of tangent line to its graph at the points where x =

    1.

    Solution If 2u y= , then 2 2du du dydx dy dxu y y y y= = = = , so that2 2 .d dx y y y

    =

    Differentiating 2 2 1 xy y x- = + implicitly, we have

    2

    2

    2

    22 1

    2 2

    (2 1) 2 x y xy

    x yy y y x

    y xy x y

    y-

    -

    + - =

    - = -

    =

    Find y when x = 1. From 2 2 1 xy y x- = + with x = 1 we obtain2 2

    2

    1 1 1

    2 0( 1)( 2) 0

    1 or 2

    y y

    y y

    y y

    y y

    - = +

    - - =

    + - =

    = - =

    Therefore, the tangent point at the graph of function is (1, 1) and (1,2).

    Find the slope of tangent line,22 1 ( 1) 1 1

    2 1 ( 1) 1 3 3(1, 1) y - - - - -

    - = = = - and 22 1 (2) 2 22 1 2 1 3 3(1,2) y - - -

    = = = -

    Equation of tangent line at (1, 1) is13

    ( 1) ( 1)

    3 3 13 2 0.

    y x

    y x

    x y

    - - = - -

    + = - +

    + + =

    Equation of tangent line at (1,2) is23

    2 ( 1)

    3 6 2 22 3 8 0.

    y x

    y x

    x y

    - = - -

    - = - +

    + - =

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    LN 7- 9

    MBM078

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems If the function x = x(t ) defined implicitly by equation x2 + t 3 = t 2 x 11, show thatthe graph of function through ( 2,1) and find x ( 2,1).

    Solution

    Problems Biophysics In biophysics, the fundamental equation of muscle contraction is given by ( L + m)

    (V + n) = k where m, n, k are constants, V is the velocity of the shortening of muscle

    fibers for a muscle subjected to a load of L. Find dLdV and dV dL by using implicit differentiation.

    Solution

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    LN 7-10

    MBM079

    Exponential Function and Related Rates

    Related Rates

    In related rates problem , we have two variables, say x and y, are both function of athird variable t though this function are given implicitly. An equation connecting x

    and y can be found. If we know dxdt , then we can find dydt .

    Suggestions for Solving Related Rates ProblemsSketch a figure if helpful.Identity all relevant variables, including those whose rate are given and to befound; express these rates as derivatives. Find an equation connecting these variables and implicitly differentiate .

    Solve for the derivative that will give the unknown rate.

    Example 7.4 Suppose that two motorboats leave from the same point at thesame time. If one travels north at 15 miles per hour and the other travels east at 20miles per hour, how fast will distance between them be changing after 2 hours?

    N

    z

    y

    E

    Solution Draw a picture as shown in left figure. Inthis problem, all variable x, y, and z are changing withtime; x = x(t ), y = y(t ), and z = z(t ) given implicitly. Itnow makes sense to take derivatives of each variablewith respect to time. From the Pythagorean theoremwe have

    z2 = x2 + y2

    We also know that dxdt = 20 miles per hour and dydt

    = 15 miles per hour. We would

    like to find dzdt at the end of 2 hours; that is when x = 40 miles and y = 30 miles.

    When x = 40, y = 30, then z = 2 240 30 2500+ = = 50 miles.

    We differentiate both side equation z2 = x2 + y2 with respect to t and solve for dzdt

    .

    2 z dzdt = 2 x dxdt

    + 2 y dydt

    2(50) dzdt = 2(40)(20) + 2(30)(15)

    dzdt

    = 25

    Thus, the boat will be separating at a rate of 25 miles per hours.

    x

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    LN 7-11

    MBM080

    Exponential Function and Related Rates

    Example Related Rates in Business

    Example 7.5 Suppose that for a company manufacturing transistor radios,equation of the cost is C = 5,000 + 2 x, the revenue is R = 10 x 0.001 x2, and the

    profit is P = R C , where the production output in 1 week is x radios. If productionis increasing at the rate 500 radios per week when production is 2000 radios, find the rate of increase in cost, revenue, and profit.

    Solution If production x is a function of time (it must be since it is changingwith respect to time), then C , R, and P must also be function of time and givenimplicitly. Letting t represent time in weeks, we differentiate both side of each of

    the three equations with respect to t , and substitute x = 2000 and dxdt = 500 to find the desired rates.

    Increase of cost : C = 5,000 + 2 x, C = C (t ), and x = x(t ).

    (5,000) (2 )

    2 2 500 1000

    dC d d dt dt dt

    dC dxdt dt

    x= +

    = = =

    Cost increasing at a rate of $1,000 per week.

    Increase of revenue :

    R = 10 x 0.001 x2, R = R(t ), and x = x(t ).

    ( )

    2(10 ) (0.001 )

    10 0.002 (10 0.002 )

    10 0.002 (2000) 500 3,000

    dR d d dt dt dt

    dR dx dx dxdt dt dt dt

    x x

    x x

    = -

    = - = -

    = - =

    Revenue increasing at a rate of $3,000 per week.

    Increase of profit :

    3, 000 1, 000 2, 000

    dP dR dC dt dt dt

    P R C = -

    = -

    = - =

    Thus, the profit increasing at a rate of $2,000 per week.

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    LN 7-12

    MBM081

    Exponential Function and Related Rates

    Student Work-sheet

    I.N Name Signature

    Problems A point is moving on the graph of xy = 36. When the point is at (4,9), its x coordinateis increasing at 4 units per second. How fast is the y coordinate changing at the moment?

    Solution

    Problems The diameter of a spherical balloon is increasing at the rate of 3 centimeters per minute. How fast is the volume changing when the diameter is 20 centimeters?

    Solution

    Problems Advertising A retail store estimates that weekly sales $s and weekly advertising cost$ x are related by 0.000560,000 40,000 xs e

    -= - . The current weekly advertising cost are $2,000and these cost are increasing at the rate $300 per week. Find the current rate of change of sales.

    Solution

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    LN 7-13

    MBM082

    Exponential Function and Related Rates

    Exercise 7

    1. How long will it take money to triple if it is invested at 8% compounded conti-nuously?

    2. A family paid $40,000 cash for a house. Fifteen years later, that sold the housefor $100,000. If interest is compounded continuously, what annual nominal rateof interest did the original $40,000 investment earn?

    3. A student claims that the tangent line to the graph of f ( x) = e x at x = 3 pass throughthe point (2,0). Are the tangent lines at x = 4 will pass through the point (3,0)?Explain.

    4. Find the absolute extreme value of ( ) 4 ln 7 f x x x x= -

    and 3

    ( ) x

    g x x e-=

    . 5. If f ( x) = x ln x, find the interval where f is increasing, is decreasing, local extreme,

    interval where f is concave upward, is concave downward, and sketch the graphof f .

    6. If the cost of producing x units of a product is given by( ) 600 100 100lnC x x x= + - , x 1

    find the minimum average cost.

    7. Learning psychology A mathematical model for the average of a group of peoplelearning to type is given by ( ) 10 6ln , 1 N t t t = + where N (t ) is the number of words per minute typed after t hour of instruction and practice 2 hours per day, 5days per week. What is the rate of learning after 10 hours?

    8. Find the equations of tangent lines to the graph y2 xy = 6 at the points where x = 1.

    9. Price-demand The price p in dollars and demand x for a product are related by 2 22 5 50 80,000 x xp p+ + =

    (a) If the price increasing at a rate of $ 2 per month when the price is $ 30, find the rate of change of the demand.

    (b) If the demand is decreasing at a rate of 6 units per month when the demand is 150 units, find the rate of change of the price.

    Math Quote The universe cannot be read until we have learnt the languageand become familiar with the characters in which it is written. It was written inmathematical language, and the letters are triangle, circle, and other geometricalfigure, without which means it is humanly impossible to comprehend a singleword. Gottlob Frege (1848 1925)

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