APPM 1350 Final Exam Fall 2016
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1 APPM 350 Final Eam Fall 06. (5 pts) Evaluate the following integrals. (a) 0 4 ( d (b) + ) 5 t 5 t dt (c) 0 / 6 d (a) (9 pts) Let u +, du d [ ( + ) d u du ] 5 ( 5 ) u (b) (8 pts) Let u 5 t, du 5 t (ln 5) dt. (c) (8 pts). A drone is sitting.5 meters from its operator who is sitting on the 5 t du 5 t ground as well. The drone takes o and flies directly at, and over, its operator at a height of h p,where is the distance from its take o point. How close does the drone come to the sitting location of its operator? 5 t dt ln 5 u ln ln u + C ln 5 ln 5 + C or log 5 5 t + C. 0 A piece of6 cardboard d measures 6 arcsin 0 in. 0 by 5 in. Two equal squares are removed from the corners of a 0 in. side as shown in the figure. Two equal rectangles 6(0 + areπ/6) removed from π the other corners so that the tabs / / can be folded to form a rectangular bo with lid. Find the maimum volume and the of that gives it.. (6 pts) The following problems are not related. (a) Show that the function h() cos + + has at least one real root. Indicate an interval where the root can be found. 0 BASE LID (b) Find an equation for the line tangent to y sin 3 () at π/6. Epress your answer in the form y m + b. (c) Use the graphs of f() and 5 g(), shown below, to find the following values. (No justification is required.) 3. Consider the graphs of f() and g() below (No work required). i. f ( 3) + f () ii. f (g()) iii. (fg) () (a) Find f 0 ( 3) + f 0 () (b) Find f 0 (g()) (c) Find (fg) 0 () 6 4 y f() 5 0 y g()
2 (a) (8 pts) By the Intermediate Value Theorem, since h( π/) π + < 0, h(0) > 0, and h is continuous, there is a root in the interval ( π/, 0). (b) (9 pts) y sin 3 () y(π/6) sin 3 (π/3) ( 3/ ) 3 3 3/8 y 6 sin () cos() y (π/6) 6 sin (π/3) cos() The tangent line is y ( π/6) or y π 8. (c) (9 pts) 3. (8 pts) i. f ( 3) + f () / / 0 ii. f (g()) f (5) iii. (fg) () f()g () + g()f () (0) + 5( /) 5/ (a) Find the value of arctan ( 0 ). (b) Use the Squeeze Theorem to find the value of arctan ( 0 ). arctan(/ ) (c) Does eist? Justify your answer. arctan(/) (a) (6 pts) arctan ( 0 ) does not eist because π. (b) (6 pts) arctan ( 0 + π < arctan < π π < arctan ( ) < π ) π Since π 0 π 0 0, then arctan ( 0 ) 0. (c) (6 pts) arctan(/ ) LH arctan(/) ( ) and 0 arctan ( )
3 4. (6 pts) For this problem let f() (ln ). (a) Find the domain of f. (b) Find the instantaneous rate of change of f with respect to. (ln(e + h)) e+h (c) Find the value of. h 0 h (a) (4 pts) Domain: or [, ). (b) (6 pts) Alternate y (ln ) ln y ln(ln ) ln y ln(ln ) dy y d dy (ln ) d ln + ln(ln ) ( + ln(ln ) ln ) y (ln ) ln(ln ) e y e ln(ln ) d d (ln(ln ) ) (ln ) + ln(ln ) ln (ln(e + h)) e+h (c) (6 pts) By the definition of the derivative, f (e) ( + 0). h 0 h 5. (0 pts) Bug A is moving along the curve y + / and Bug B is moving along the curve y ln so that the bugs are always vertically aligned (one directly above the other). (a) The distance between the two bugs is minimized at what -coordinate? (b) As Bug A reaches, its y-coordinate is decreasing at a rate of 0. unit/sec. How fast is Bug B s y-coordinate changing then? (a) (0 pts) Let d equal the distance between the bugs: ( d + ) ( ln ) + ln.
4 Then d + + and d 0 at /. Since d 3 and d (/) 6 8 > 0, the distance is minimized at /. (b) (0 pts) Let y + / and y ln. We are given that dy /dt 0. unit/sec when. We wish to find dy /dt. First solve for d/dt. y + dy dt d dt 0. d 4 dt d dt Bug B s -coordinate will be changing at the same rate. Solve for dy /dt. 0.4 unit/sec y ln dy dt d dt 6. (5 pts) The following problems are not related. dy dt (0.4) 0.4 unit/sec (a) Find all asymptotes (if any) of the function h() e. Justify your answer using its. e (b) Let y 5. Find (i) the domain of the function, and (ii) the intervals of increase and decrease. Epress your answers in interval notation. (a) ( pts) Horizontal asymptotes: e e LH e and e There are horizontal asymptotes at y and y 0. e e Vertical asymptotes: Check ln where the denominator equals zero. (ln ) + e e or 0 There is a vertical asymptote at ln. (ln ) e e 0 +
5 (b) (3 pts) (i) Domain: (, 5]. (ii) Solve y 0. y 5 y y + 4(5 ) ( + 4) 5 0 0, 4 Since y ( ) < 0, y () > 0, and y (9/) < 0, y is decreasing on (, 0), (4, 5) and increasing on (0, 4). 7. (0 pts) The following problems are not related. (a) Find the sum of (b) Let g() 0 n 0 (ln(3n) ln(n)). Simplify your answer. tanh(t t ) dt. Find g () and g (). (c) A sample of radioactive cesium-37 with an initial mass m of 50 mg decays at the rate of dm dt (ln )m 30 mg/year. Find an epression for m(t), the mass remaining after t years. Simplify your answer. (a) (6 pts) 0 n (ln(3n) ln(n)) 0 n ln 3 0 ln 3.
6 (b) (8 pts) g () tanh ( ) by FTC-. g () ( ) sech ( ) (c) (6 pts) m(t) 50e ln 30 t t ln e 50 t 30 or 5 t 30 mg.
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