MAT 137Y, 2006–2007 Winter Session, Solutions to Term Test 2
(9%) (i) Find the equation of the tangent line to the graph of y=secx−2cosxat the point (π
y0=secxtanx+2sinx, so the slope of the tangent line is
3) = sec π
Thus the equation of the tangent line is y−1=3√3(x−π
(9%) (ii) For the equation x2+4xy +y3+5=0, ﬁnd d2y
dx2at the point (2,−1).
Differentiating implicitly, we have 2x+4xdy
dx =0.Plugging in x=2,y=−1 gives
Differentiating implicitly again, we have
Plugging in x=2,y=−1,dy
dx =0, we have
(10%) (i) A balloon is rising at a constant speed of 5
3meters per second. A boy is cycling along a straight
road at a speed of 5 meters per second. When he passes under the balloon, it is 15 meters above
him. How fast is the distance between the boy and the balloon changing three seconds later?
Let ybe the height of the balloon and xbe the horizontal distance between the balloon and the
bicycle. If zis the distance between the boy and balloon, then
Three seconds after the bicycle passes the balloon, we have x=15 and y=20, so by the
Pythagorean Theorem z=25. Since dx
dt =5 and dy
3, we get
15(5) + 20(5
3) = 25dz
so the distance between the cyclist and the balloon is increasing at 13
3meters per second.
(10%) (ii) Consider the function f(x) = (x2sin 1
Prove that fis differentiable at 0, and ﬁnd f0(0).