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37. 39. CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS tanh x ln ( cosh x) dx. sech2 x dx. 1 + tanh x 38. 40. 1 + tanh x dx. cosh2 x tanh x sech x dx.
5 2 (a) Show that v (t ) = mg tanh k gk t m In Exercises 41–43, verify the given integration formula. In each case, take a > 0. 41. 42. √ √ 1 a2 + x2 1 x 2 − a2 dx = sinh−1 dx = cosh
−1 x + C. a x + C. a is a solution of the equation which satisﬁes v(0) = 0. (b) Find lim v(t ).
t →∞ This limit is called the terminal velocity of the body. c 45. Use a CAS to ﬁnd a one-term expression for f .
(a) (b) f (x) dx = − f (x) dx = sech x(2 + tanh x) . 3(1 + tanh x)2 1 tanh−1 x + C If |x| < a. a 1 a 43. dx = 1 a2 − x 2 coth−1 x + C If |x| > a. a a 44. If a body of mass m falling from rest under the action of gravity encounters air resistance that is proportional to the square of its velocity, then the velocity v(t ) of the body at time t satisﬁes the equation m dv = mg − kv2 dt ln (x4 − 1) x2 tanh−1 x2 + . 2 4 Verify your results by differentiating the right-hand sides. c 46. Use a CAS to ﬁnd a one-term expression for f . x x (a) f (x) dx = 16 + x2 − 8 sinh−1 . 2 4
x2 + 1 x2 −1 x2 cosh−1 x2 + . 2 x2 + 1 2 Verify your results by differentiating the right-hand sides. (b) f (x) dx = − where k > 0 is the constant of proportionality and g is the gravitational constant. CHAPTER HIGHLIGHTS
7.1 One-to-One Function; Inverses cot x dx = ln | sin x| + C , csc x dx = ln |csc x − cot x| + C . logarithmic differentiation (p. 394)
7.4 The Exponential Function one-to-one function; inverse function (p. 371) one-to-one and increasing/decreasing functions; derivatives (p. 374) relation between graph of f and the graph of f −1 (p. 375) continuity and differentiability of inverse functions (p. 375) derivative of an inverse (p. 376)
7.2 The Logarithm Function, Part 1 deﬁnition of a logarithm function (p. 381) x dt , x > 0; natural logarithm: In x = t t domain (0, ∞), range (− ∞, ∞) basic properties of ln x (p. 385) graph of y = ln x (p. 385)
7.3 The Logarithm Function, Part II The exponential function y = e x is the inverse of the logarithm function y = ln x. graph of y = e x (p. 398); domain (− ∞, ∞), range (0, ∞) basic properties of the exponential function (p. 398) du du (e ) = e u , dx dx
7.5 eg (x) g (x) dx = eg (x) + C Arbitrary Powers; Other Bases d 1 du (ln |u|) = , dx u dx g (x ) dx = ln |g (x)| + C , g (x ) tan x dx = ln | sec x| + C , sec x dx = ln | sec x + tan x| + C , xr = er ln x for all x > 0, all real r logp x = ln x ln p du du (p ) = pu ln p (p a positive constant), dx dx d 1 du ( logp u) = dx u ln p dx 1+ 1 n
n ≤e ≤ 1+ 1 n n+1 e ∼ 2. 71828 = CHAPTER HIGHLIGHTS 7.6 Exponential Growth and Decay 445 All the functions that satisfy the equation f (t ) = kf (t ) are of the form f (t ) = C ekt (p. 413) population growth (p. 414) radioactive decay, half-life (p. 417) compound interest, continuous compounding (p. 418) rule of 72 (p. 420)
7.7 The Inverse Trigonometric Functions 7.8 dx 1 x = tan−1 +C 2 +x a a 1 d ( sec−1 x) = √ dx |x| x2 − 1 a2 x √ dx x2 − a2 = 1 sec−1 a |x| a +C (a = 0) (a > 0) deﬁnition of the remaining inverse trigonometric functions (p. 431)
The Hyperbolic Sine and Cosine The inverse sine, y = sin−1 x, is the inverse of y = sin x, x ∈ [− 1 π , 1 π ]. 2 2 The inverse tangent, y = tan−1 x, is the inverse of y = tan x, x ∈ (− 1 π , 1 π ). 2 2 The inverse secant, y = sec−1 x, is the inverse of y = sec x, x ∈ [0, 1 π ) ∪ ( 1 π , π ]. 2 2 sinh x = 1 (e x − e−x ), 2 d ( sinh x) = cosh x, dx graphs (p. 436)
*7.9 cosh x = 1 (e x + e−x ), 2 d ( cosh x) = sinh x. dx basic identities (p. 437) graph of y = sin−1 x (p. 423) graph of y = tan−1 x (p. 426). graph of y = sec−1 x (p. 429) d 1 ( sin−1 x) = √ dx 1 − x2 dx x = sin−1 +C (a > 0) √ 2 − x2 a a d 1 ( tan−1 x) = dx 1 + x2 The Other Hyperbolic Functions cosh x sinh x , coth x = , cosh x sinh x 1 1 , csch x = . sech x = cosh x sinh x derivatives (p. 440) hyperbolic inverses (p. 441) derivatives of hyperbolic inverses (p. 443). tanh x = ...
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.
- Spring '10