l. Evaluate and simplify the following derivatives. do (tan (sino) 2. Evaluate t

Question

l. Evaluate and simplify the following derivatives. do (tan (sino) 2. Evaluate the following integrals using U-substitution. Fully simplify your answers sin x e. J ecos sin x dx 3. A confused bug walks back and forth along a straight line.The bug's velocity is given by v(t) 6sint cm/s for t20. (a) Find the displacement of the bug over the interval 0StS2r. (b) Find the total distance traveled by the bug over the interval 0 sts2r. (Hint: consider the graph of v(t)). (c) Dete the position function, s(t), for t20provided s(0) 4. rmine 4. When records were first kept (t 0) the population of a rural town was 300 people. During the following years, the population grew at a rate Pin 400M +r a. Find the population Pt) at any time t20. b. What is the population after 30 years? (Round to the nearest whole number as needed.) 5. Consider the region bounded by the functions y x +2x 1 and y 2x 5. a. Find the points of intersection of the curves. b. Find the area of the region bounded by the above curves.

Explanation / Answer

1.a. d(2x sqrt(x2-2x+2)) = 2x * d(sqrt(x2-2x+2))/dx + sqrt(x2-2x+2) d(2x)/dx

= 2x*(2x-2)/2 sqrt(x2-2x+2) + 2 sqrt(x2-2x+2) = x(2x-2)/sqrt(x2-2x+2) + 2 sqrt(x2-2x+2)

= (2x2-2x +2x2-4x+4)/sqrt(x2-2x+2)

= (4x2-6x+4)/sqrt(x2-2x+2)

b. d (tan(sin theta))/d theta

let din theta=u

so we get

sec2u * cos x= sec2(sin x) * cos x

c. d(ln 2sqrtx)/dc

let 2 sqrt x=t

So we get

1/t * (2/ 2sqrtx)= 1/ (2 sqrt x) * 1/sqrt x= 1/2 x

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