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Calculus 6.56.5-6.7, 6.9 Review o Find Name:____________________________________ Mitchell dy . dx 1) y = 7 5 x 4 2) y = log 3 5x −3 x 3) y = x sin x 4) y = arctan(2x) + 2 arctan x + (arctan x)2 5) y = cos −1 (ln x) 6) y = arc cot 2x + 1 2 7) y = csc −1 (e x ) 8) y = e 4 x sec −1 (4x) o Integrate for problems 99-18. 9) dx ∫ 4 + 25x 2 10) 2xdx ∫ 4 + 25x 2 1 11) x 2 dx ∫ 4 + 25x 2 12) ∫ 13) 25x 2 + 4 ∫ x 2 dx 14) ∫x 15) ∫ cot(8x)dx 16) ∫ 18) sec 2 5x ∫ 9 + tan2 5x dx 17) ∫ csc 2 (2t) 1 − cot 2 (2t) dt 4 − 25x 2 dx dx x8 − 9 sec(ln x) dx x 19) Find the the area of the region bounded by the x graphs of y = 4 , x = 1 , and y = 0 . x +1 20) If y > 0 and y = 4 when x = 0 , solve for y. dy = 7y dx 21) The rate of change in the number of bacteria in a culture is proportional to the number present. The culture had 10,000 bacteria initially. Ten minutes later, there were 50,000 bacteria present. a) In terms of t only, find the the number present at time t minutes where t ≥ 0 . b) How many bacteria are there after 20 minutes? c) At what time are there 20,000 bacteria? 22) A radioactive substance has a halfhalf-life of 5 days. How long will it take for an amount A to disintegrate disintegrate to the extent that only 1% of A remains? 23) Use your calculator to find the length of the arc on y = sin−1 x from x = − 4π to x = 4π to the nearest thousandth. x2 (arc length = ∫ x1 dy 2 1 + ( dy dx ) dx ) 24) A balloon is released from ground level, 500 meters away from a person who observes its vertical ascent. If the balloon rises at a constant rate of 2 m/s, use inverse trig functions to find the rate at which the angle of elevation of the observer’s line of sight is changing at the instant the balloon is 100 meters above the ground. (disregard the observer’s height) 25) Consider the region bounded by y = 2 x , x = 0 , y = 0 , and x = 2 . a) Calculate the volume of the solid formed when the region is rotated about y = −1 . b) Set up an integral that represents the volume of the solid formed when the region is rotated about x = 2 . 26) How long does it take to double your money if it is deposited in an account at 12% compounded a) monthly? b) continuously? 27) $1000 is put in a bank account for 5 years at an annual rate of 6.5%. How much will be in the account if it is compounded compounded a) annually? b) monthly? c) daily? d) continuously? o Find each limit, if it exists. ln(2 − x) 28) lim 2x x →0 1 + e 29) lim e 2x − e −2x − 4x x →0 x3 31) lim 30) lim sin2 x 32) lim x → 0 1 − cos x x →0 x →0 sin 2x − tan 2x x2 4 − x2 − 2 x e x − ln(1 + x) − 1 33) lim x →0 x2 ANSWERS TO 6.56.5-6.7,6.9 REVIEW 1) 7 5 x 4) 7) 10) 4 −3 x 2 1+ 4 x 2 (ln 7)(20x 3 − 3) + 2 arctan x (ln 2) ( 1+1x 2 ) +2 arctan x ( 1+1x 2 ) −2xe x ex 2 1 25 2 2 e2x − 1 ln 4 + 25x 2 + C 13) 25x − 4 +C x 16) ln sec(ln x) + tan(ln x) + C 19) 22) π 8 1 5ln( 100 ) ≈ 33.219 days 1 ln( 2 ) 27 π ≈ 61.187 2ln 2 25) (a) 2 (b) ∫ 2 π(2 − x)(2 x )dx 0 28) ln 2 2 31) 0 2) 1 1 or 2xln3 x ln9 5) − 1 2 x 1 − (ln x) 8) sin x 3) y + ln x(cos x) x 6) − 1 (2x + 2) 2x + 1 9) tan−1 ( 25 x) + C 1 10 1 4e 4 x sec −1 (4x) + 4x 16x 2 − 1 2 x − 125 tan−1 ( 25 x) + C 12) 1 5 sin−1 ( 25 x) + C x4 sec −1 + C 3 15) 1 8 ln sin8x + C 17) − 21 sin−1 (cot 2t) + C 18) 1 15 20) y = 4e 7 x 21) (a) N = 10,000e 10 t (b) 250,000 10 ln 2 min ≈ 4.307 min (c) ln5 1 24) radians per second 260 11) 1 25 14) 1 12 23) 2.399 tan−1 ( 31 tan5x) + C ln 5 t 26) (a) 69.66 months ≈ 5.805 yrs 27) (a) $1370.09 (b) 69.31 months ≈ 5.776 yrs (b) $1382.82 (c) $1383.99 (d) $1384.03 29) 0 32) 2 30) 8 3 33) 1