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The gure shows a mathematical model for the problem The angle is the angle less the angle Thus we have = = Cot 1 (x/12) Cot 1 (x/2) Notice that when the viewer is standing with her face against the wall then = = /2 so that = 0 Also when the viewer is far from the tapestry then is quite small So the maximum value for will occur for some nite, positive value of x That value can be found by differentiating with respect to x, setting the derivative equal to zero, and solving for x
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We leave it to you to perform the calculation and discover that optimal distance at which the viewer should stand
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You Try It: Redo the last example if the tapestry is 20 feet high and the bottom of the tapestry is 6 inches above eye level
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1 Simplify these logarithmic expressions (a) ln (b) a 2 b 3 c4 d
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log3 (a 2 b) log2 (a 2 b)
(c) ln[e3x z4 w 3 ] (d) log10 [100w 10] 2 Solve each of these equations for x (a) 3x 5 x = 2x e3 (b) 3x = 10x 102 5 x 42x
(c) 23x 34x 45x = 6 5 2 (d) = x x 3x e5x 2 3 2 3 Calculate each of these derivatives d ln[sin(x 2 )] (a) dx d x2 (b) ln dx x 1 d sin(ex ) e (c) dx d sin(ln x) (d) dx 4 Calculate each of these integrals (a) e x x 3 dx [Hint: Guess p(x) e x , p a polynomial]
CHAPTER 6 Transcendental Functions
(b) x 2 ln2 x dx [Hint: Guess p(x) ln2 x + q(x) ln x + r(x), p, q, r
polynomials] e ln x dx (c) x 1
ex dx ex + 1
5 Use the technique of logarithmic differentiation to calculate the derivative of each of the following functions (a) x 3 (b) x2 + 1 x3 x
sin x (x 3 + x) x 2 (x + 1)
7 8
(c) (x 2 + x 3 )4 (x 2 + x) 3 (x 1) x cos x (d) ln x ex There are 5 grams of a certain radioactive substance present at noon on January 10 and 3 grams present at noon on February 10 How much will be present at noon on March 10 A petri dish has 10,000 bacteria present at 10:00 am and 15,000 present at 1:00 pm How many bacteria will there be at 2:00 pm A sum of $1000 is deposited on January 1, 2005 at 6% annual interest, compounded continuously All interest is re-invested How much money will be in the account on January 1, 2009 Calculate these derivatives d (a) Sin 1 (x ex ) dx x d Tan 1 (b) dx x+1 (c)
d Tan 1 [ln(x 2 + x)] dx d Sec 1 (tan x) (d) dx 10 Calculate each of these integrals (a) 2x x dx 1 + x4
(b) (c)
3x 2 dx 1 x6
Transcendental Functions
2 sin x cos x
1 sin4 x dx 5 + 2x 2
Methods of Integration
71 Integration by Parts
We learned in Section 45 that the integral of the sum of two functions is the sum of the respective integrals But what of the integral of a product The following reasoning is incorrect: x 2 dx = x x dx = x dx x dx
because the left-hand side is x 3 /3 while the right-hand side is (x 2 /2) (x 2 /2) = x 4 /4 The correct technique for handling the integral of a product is a bit more subtle, and is called integration by parts It is based on the product rule (u v) = u v + u v Integrating both sides of this equation, we have (u v) dx = u v dx + u v dx
The Fundamental Theorem of Calculus tells us that the left-hand side is u v Thus u v = u v dx + u v dx
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