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1 Math 121 Final EF: December 10, 2013 Name Directions: 1 /15 2 /15 3 /15 4 /15 5 /10 6 /10 7 /20 8 /15 9 /15 10 /10 11 /15 12 /20 13 /15 14 /10 Total / No book, notes, or ouiji boards. You may use a calculator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. 2. You should show your work, and explain how you arrived at your answers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. 3. You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. 4. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam. Have a Safe and Happy Break
2 1. (15 points) Find the following limits: 3 x + 9 (a) lim x 0 x (b) lim x 4 x2 8x + 16 x 4 (c) lim x 0 x sin x x 3
3 2. (15 points) (a) If f(2 + x) f(2) = xe 2x + 3x 2 + 3x, what is f (2)? (b) Find f (x) for f(x) = sin(2x) + sinh(arctan x) cos(2x) (c) Find g (3) if g(x) is the inverse of f(x) = x 5 + x 3 + x.
4 3. (15 points) Find dy dx for (a) y = x (ex). (b) y = sin(3x + 4y). (c) y = e x ln( 5x 2 + 1).
5 4. (15 points) For y = x 4 x 2 ( y = 2(2 x2 ) 4 x 2 and y = 2x(x2 6) (4 x 2 ) 3/2 ) find: (a) Domain: (b) Range: (c) x-intercepts: (d) y-intercepts: (e) Where y is increasing: (f) Where y is decreasing: (g) Critical points: (h) Where y is concave up: (i) Where y is concave down: (j) Inflection points: (k) Vertical asymptotes: (l) Horizontal asymptotes: (m) Sketch the graph of y
6 5. (10 points) You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 3 mph and walk at 5 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time? 1 Mile Campground 1 Mile River You
7 6. (10 points) A rocket is rising straight up from the ground at the rate of 1000 miles per hour. An observer 3 miles from the launching site is photographing the rocket. How fast is the angle θ of the camera with the ground changing when the rocket is 3 miles above the ground?
8 7. (20 points) (a) We know the following about g(x): x g(x) g (x) i. Use a linear approximation to estimate g(2.1) ii. If you were using Newton s method to find where g(x) = 0 starting at x 0 = 2, what is x 1? (b) Find the maximum and minimum of f(x) = 6x 4/3 3x 1/3 on the interval [ 1, 1]. n (c) Find n lim (sin x j ) x on [0, π/2], where x j is the right hand j=1 π end point of the j-th subinterval and x = 2 0 n.
9 8. (15 points) (a) x 2 + x + 1 dx x 2 (b) x2 dx (c) Find lim x 0 x 0 arctan t dt x 2
10 9. (15 points) (a) 9 x 9 x + 1 dx (b) π 0 x sin x 2 dx (c) 8 0 x 1 + x dx
11 10. (10 points) A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many will there be 5 hours from the initial time given? 11. (15 points) (a) Find the area of the region bounded by the curves y = x 2 3x and y = x (b) Find the average value of f(x) = x sin x 2 on the interval [0, π].
12 12. (20 points) (a) Find the volume of the solid generated when the region bounded by y = x 2, y = 1 and x = 2 is revolved about the x-axis (b) Find the volume of the solid generated when the region bounded by y = x 2, y = 1 and x = 2 is revolved about the y-axis
13 13. (15 points) A right circular conical tank of height 6 ft and radius of base 2 ft has its vertex at ground level and axis vertical. The French Silk pie filling weighting 30 lb/ft 3 in the tank is 4 feet deep. Find work done in pumping all the French Silk pie filling over the top of the tank.
14 14. (10 points) Indicate whether the following statements are true or false by circling the appropriate letter. A statement which is sometimes true and sometimes false should be marked false. a) If f(c) = L, then lim x c f(x) = L. T F b) 25 j=1 25 = 25 T F c) If b a f(x) dx > 0 then f(x) > 0 on [a, b] T F d) If f is continuous on (a, b), then f(x) has a maximum on (a, b). T F e) = 2 T F
15 FORMULA PAGE sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ sin 2 1 cos 2x x = 2 cos cos 2x x = 2 sin 2x = 2 sin x cos x (sin x) = cos x (cos x) = sin x (tan x) = sec 2 x (sec x) = sec x tan x (csc x) = csc x cot x (cot x) = csc 2 x (e x ) = e x (ln x) = 1 x (arcsin x) 1 = 1 x 2 (arctan x) = x 2 (arcsec x) 1 = x x 2 1 du a 2 u = arcsin u 2 a + C du a 2 + u 2 = 1 a arctan u a + C du u u 2 a = 1 u arcsec 2 a a + C (sinh x) = cosh x (cosh x) = sinh x sinh x = ex e x 2 cosh x = ex + e x 2 x n+1 = x n f(x n) f (x n ) f (c) = f(b) f(a) b a f(c) = 1 b f(x)dx b a a n n(n + 1) j = 2 j=1 n j 2 n(n + 1)(2n + 1) = 6 j=1 n [ ] n(n + 1) 2 j 3 = 2 j=1 x = b a n x j = a + j x b n f(x)dx = lim f(x j ) x n a j=1 f(x + x) f(x) + f (x) x V = V = b a b a W = π (f(x)) 2 dx 2π x f(x) dx b a F (x) dx = 2
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