MAT 137Y, 2004-2005, Solutions to Term Test 2

1. For the following, simplify your answers unless otherwise instructed.

(8%) (i) Evaluate lim

x→0

cos5x−cos3x

x2.

Let Lbe the limit above. Then by L’Hˆ

opital’s Rule,

LH

=lim

x→0−5sin5x+3sin3x

2xH

=lim

x→0−25cos5x+9cos3x

2=−8.

(8%) (ii) Find the derivative of f(x) = √−x, where x<0, using the deﬁnition of derivative.

f0(x) = lim

h→0

f(x+h)−f(x)

h=lim

h→0p−(x+h)−√−x

h·p−(x+h) + √−x

p−(x+h) + √−x

=lim

h→0−(x+h)−(−x)

h(p−(x+h) + √−x)=lim

h→0−h

h(p−(x+h) + √−x)=lim

h→0−1

p−(x+h) + √−x

=−1

2√−x.

(8%) (iii) Find the equation of the tangent line to the curve 2(x2+y2)2=25(x2−y2)at the point (3,1).

Implicitly differentiating gives us

4(x2+y2)·(2x+2yy0) = 50x−50yy0.

Sticking in x=3, y=1 gives 40(6+2y0) = 150−50y0or 130y0=−90, so y0=−9

13. Hence the

equation of the tangent line is y−1=−9

13(x−3).

(8%) (iv) Use Newton’s Method with x1=2 to ﬁnd the next approximation x2to the root of x4−20 =0.

Applying Newton’s Method to f(x) = x4−20, we have

x2=x1−f(x1)

f0(x1)=2−24−20

4·23=2+4

32 =17

8.

(12%) 2. A car leaves an intersection at noon and travels due south at a speed of 80 km/h. Another car has been

heading due east at 60 km/h and reaches the same intersection at 1:00 p.m. At what time were the cars

closest together? Verify that your answer is correct by showing that at that given time the distance is

minimized.

Let tbe the number of hours past noon. Let the intersection be the origin on the coordinate axes. Then

at any given time t, the car travelling south is at the point (0,−80t), and the car travelling east is at the

point (−60+60t,0). The distance Dbetween the two cars at any given time tis given by the equation

D2(t)=(−60+60t)2+ (−80t)2=10000t2−7200t+3600,t∈[0,1].

To minimize D(t), it is sufﬁcient to minimize D2(t) = f(t) = 10000t2−7200t+3600. f0(t) =

20000t−7200 =0 when t=9

25. Since f00(t) = 20000 >0, then t=9

25 is a local minimum, and

hence an absolute minimum (since it is the only critical point). Therefore the cars are closest together

9

25 hours past noon, or at 12:21:36 p.m.

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