sol2-0607 - MAT 137Y, 20062007 Winter Session, Solutions to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT 137Y, 20062007 Winter Session, Solutions to Term Test 2 1. (9%) (i) Find the equation of the tangent line to the graph of y = sec x- 2cos x at the point ( 3 , 1 ) . y = sec x tan x + 2sin x , so the slope of the tangent line is y ( 3 ) = sec 3 tan 3 + 2sin 3 = 2 3 + 2 3 2 = 3 3. Thus the equation of the tangent line is y- 1 = 3 3 ( x- 3 ) or y = 3 3 x- 3 + 1. (9%) (ii) For the equation x 2 + 4 xy + y 3 + 5 = 0, find d 2 y dx 2 at the point ( 2 ,- 1 ) . Differentiating implicitly, we have 2 x + 4 x dy dx + 4 y + 3 y 2 dy dx = . Plugging in x = 2 , y =- 1 gives us 4 + 4 dy dx- 4 + 3 dy dx = = dy dx ( 2 ,- 1 ) = . Differentiating implicitly again, we have 2 + 4 x d 2 y dx 2 + 4 dy dx + 4 dy dx + 6 y dy dx 2 + 3 y 2 d 2 y dx 2 = . Plugging in x = 2 , y =- 1 , dy dx = 0, we have 2 + 8 d 2 y dx 2 + 3 d 2 y dx 2 = = d 2 y dx 2 =- 2 11 . 2. (10%) (i) A balloon is rising at a constant speed of 5 3 meters 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 y be the height of the balloon and x be the horizontal distance between the balloon and the bicycle. If z is the distance between the boy and balloon, then x 2 + y 2 = z 2 = 2 x dx dt + 2 y dy dt = 2 z dz dt = x dx dt + y dy dt = z dz dt ....
View Full Document

Page1 / 4

sol2-0607 - MAT 137Y, 20062007 Winter Session, Solutions to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online