b1213 exponential logs

5/20/2018 B1213 Exponential Logs

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B1.2 & B1.3Derivatives of

Exponential and Logarithmic

Functions

IB Math HL/SL - Santowski


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(A) Derivatives of Exponential

FunctionsGraphic Perspective

Consider the graph of

f(x) = axand then

predict what the

derivative graphshould look like


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Our exponential fcn isconstantly increasing, it isconcave up and has nomax/min points

So our derivative graphshould be positive,increasing and have no x-intercepts

So then our derivativegraph should look verysimilar to anotherexponential fcn!!


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So when we usetechnology to graphan exponentialfunction and itsderivative, we seethat our prediction iscorrect

Now lets verify thisgraphic predicationalgebraically


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(B) Derivatives of Exponential

FunctionsAlgebraic Perspective

Lets go back to the limitcalculations to find the derivativefunction for f(x) = bx

So we see that the derivative is infact another exponential function

(as seen by the bxequation) whichis simply being multiplied by someconstant (which is given by thelimit expression)

But what is the value of the limit??

So then, the derivative of anexponential function isproportional to the function itself

h

bbxf

hbbxf

h

bbxf

h

bbbxf

h

bbxf

h

xfhxfxf

h

h

x

h

h

x

h

hx

h

xhx

h

xhx

h

h

1lim)(

1limlim)(

)1(lim)(

lim)(

lim)(

)()(lim)(

0

00

0

0

0

0


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(C) Investigating the Limits

Investigate lim h0(2h1)/h

numerically with a table ofvalues

x y

-0.00010 0.69312 -0.00007 0.69313

-0.00003 0.69314

0.00000 undefined

0.00003 0.69316

0.00007 0.69316

0.00010 0.69317

And we see the value of 0.693as an approximation of the limit

Investigate lim h0(3h1)/h

numerically with a table ofvalues

x y

-0.00010 1.09855 -0.00007 1.09857

-0.00003 1.09859

0.00000 undefined

0.00003 1.09863

0.00007 1.09865

0.00010 1.09867

And we see the value of1.0986 as an approximation ofthe limit


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(C) Investigating the Limits

Investigate lim h0 (4h1)/hnumerically with a table ofvalues

x y

-0.00010 1.38620 -0.00007 1.38623

-0.00003 1.38626

0.00000 undefined

0.00003 1.38633

0.00007 1.38636

0.00010 1.38639


Investigate lim h0 (eh1)/hnumerically with a table ofvalues

x y

-0.00010 0.99995 -0.00007 0.99997

-0.00003 0.99998

0.00000 undefined

0.00003 1.00002

0.00007 1.00003

0.00010 1.00005



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(D) Special Limits - Summary

The number 0.693 (coming from our

exponential base 2), 1.0896 (coming from

base = 3), 1.386 (base 4) are, as it turns

out, special numberseach is thenatural logarithm of its base

i.e. ln(2) = 0.693

i.e. ln(3) = 1.0896

i.e. ln(4) = 1.386


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(E) Derivatives of Exponential

Functions - Summary

The derivative of an exponential function was

Which we will now rewrite as

And we will see one special derivativewhen theexponential base is e, then the derivative becomes:

h

bbxf

h

h

x 1lim)(0

)ln()( bbxf x

xx eeexf )ln()(


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(F) Examples

Find the equation of the line normal to f(x)

= x2exat x = 1

Find the absolute maximum value of f(x) =

xe-x

Where is f(x) = ex^2 increasing?


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(G) Derivatives of Logarithmic


Consider the graph of

f(x) = logax and then

predict what the

derivative graphshould look like


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Our log fcn is

constantly increasing,

it is concave down

and has no max/minpoints

So our derivative

graph should be

positive, decreasingand have no x-

intercepts


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So when we usetechnology to graph alogarithmic functionand its derivative, wesee that ourprediction is correct

Now lets verify thisgraphic predicationalgebraically


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(H) Derivatives of Logarithmic

FunctionsAlgebraic Perspective Let logbx = y so then b

y= x

So now we have an exponential equation (forwhich we know the logarithm), so we simply useimplicit differentiation to find dy/dx

d/dx (by) = d/dx (x) [ln(b)] x byx dy/dx = 1

dy/dx = 1/[byln(b)] but recall that by= x

Dy/dx = 1/[x ln(b)]

And in the special case where b = e (i.e. wehave ln(x)), the derivative is 1/[x ln(e)] = 1/x


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(I) Derivatives of Logarithmic

Functions - Summary

The derivative of a logarithmic function is

And we will see one special derivativewhen

the exponential base is e, then the derivative of

f(x) = ln(x) becomes

)ln(

1)(

bx

xf

xexxf

1

)ln(

1)(


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(J) Examples

Find the maximum value of f(x) = [ln(x)] x

Find f `(x) if f(x) = log10(3x + 1)10

Find where the function y = ln(x2

1) isincreasing and decreasing

Find the equation of the tangent line to y =

ln(2x1) at x = 1


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(K) Internet Links

Calculus I (Math 2413) - Derivatives -

Derivatives of Exponential and Logarithm

Functions from Paul Dawkins

Visual Calculus - Derivative of Exponential

Function

From pkving
http://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://www.geocities.com/pkving4math2tor4/4_the_elem_transc_func/4_02_03_gen_exp_and_log_func.htmhttp://www.geocities.com/pkving4math2tor4/4_the_elem_transc_func/4_02_03_gen_exp_and_log_func.htmhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://archives.math.utk.edu/visual.calculus/2/exp.2/index.htmlhttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asphttp://tutorial.math.lamar.edu/AllBrowsers/2413/DiffExpLogFcns.asp


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(L) Homework

Stewart, 1989, Chap 8.2, p366, Q4-10

Stewart, 1989, Chap 8.4, p384, Q1-7

b1213 exponential logs

Documents

b1213 exponential logs

exponential functionas

table ofvalues x y

x2exat x

derivatives ofexponential

lim h0

value of1

hbbxfhhx 1lim0ln bbxf