MATH 180 Study Guide - Final Guide: Product Rule, Maxima And Minima, Scilab

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13 Dec 2018

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Problem 1 solution: di erentiate the following functions: (a) x2 ln x (b) sin(a + bx) (c) arctan(3x) Solution: (a) use the product rule. (b) use the chain rule. (c) use the chain rule. (x2 ln x) = (x2)(ln x) + (x2) (ln x) [sin(a + bx)] = cos(a + bx) (a + bx) . Problem 2 solution: di erentiate the following functions: (a) e2 x. Solution: (a) use the chain rule. (b) use the product rule. (cid:16)e2 x. 0 y3 y + x = 0 d d y3 x = y + dx dx. At the point (6, 2), the value of dy dx is: dy dx. 3y2 1 (b) the value of from part (a) is the slope of the tangent line at the point (6, 2). An dy dx equation for the tangent line is then: dy dx(cid:12)(cid:12)(cid:12)(cid:12)(6, 2)

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3) Find the derivatives of the following functions and simplify your answers (they should be pretty simple).

a) y=18arcsin(x/6) + x/2sqrt(36-x^2) +24 (show your work)

b). y=xarctan(3x)-1/6ln(9x^2 +1) + 1 (show your work).

c). y=be^(ax)sin(bx)+ae^(ax)cos(bx) (where a and b are constants. show your work)

Use analytical methods to evaluate the following limits.

1) limit as x approaches infinity of x^2ln(cos(1/x))

2) limit as x approaches pi/2 of (pi-2x)tanx

3) limit as x approaches 0+ of x^(1/lnx)

4) limit as n approaches infinity of (1+2+...+n)/n^2

5) Suppose a and b are real numbers. Evaluate the limit as x approaches infinity of (ax-sqrt(a^2x^2-bx))

3. (21 pts) Evaluate the limit or say that the limit does not exist. If a limit does not exist, explain why. Show your work. You may NOT use a table of values, a graph, or L'Hospitals's Rule to justify your answer. (Note: Possible answers include + or -..) (a) (7 pts) x2 - 2x - 3 - lim x+3 Vx+1 - 2 x2 - 2x - 3 Vx+ 1 + 2 lim x+3 Vx+1 -2 Vx+1 +2 (x - 3) (x + 1) (Vx+ = lim x+3 x + 1 - 4 = lim x+3 (x - 3) (x + 1) (Vx+1 +2). -- x-3 = lin (x + 1) (Vx+1 +2) = (3 + 2) (V3+1 +2) = = 4 (V4 +2) = 16 (b) (7 pts) lim xsin (In x) X >0+ NOTE: We cannot apply the Product Law, since lim sin (1n x) DOES NOT EXIST. On the other hand, -1s sin (in x) s 1 and , therefore, -xs x sin (in x) sx, (since x > 0). Since lim X = lim (-X) = 0, x0+ X0 the inequality above implies that lim xsin (in x) = 0, by the Squeeze Theorem . x0+ EXPLANATION : Since lim ln x = -0, x →0+ sin (lnx) oscillates between 1 and -1 as x + 0*, and, therefore lim sin (In x) DOES NOT EXIST. x >0+ (c) (7 pts) in x+2- (lnx) +In 20 -= -00 (x - 2) 70-

MATH 180 Study Guide - Final Guide: Product Rule, Maxima And Minima, Scilab

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