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Review Exercise
1. Differentiate each of the following:
a. y 6 e x
d.
b. y 2x 3e x
e.
c. y e 2x3
f.
dy
2. Determine for each of the following:
dx
a. y 10 x c. y 15x215 x 2 e.
y e 3x2 5x
y xe x
s et 1
e t 1
y 4x
4 x
b. y 4 3x2 d. y 1x 4 2 2 x f. y 5Vx
x
3. Differentiate each of the following:
a. y 3 sin 2x 4 cos 2x d. y x tan 2x
b. y tan 3x
e. y 1sin 2x2e 3x
1
c. y
f. y cos 2 2x
2 cos x
4. a. Given the function f 1x2 ex solve the equation f ¿1x2 0.
x ,
b. Discuss the significance of the solution you found in part a.
5. a. If f 1x2 xe2x , find f ¿Q 1 2 R.
b. Explain what this number represents.
6. Determine the second derivative of each of the following:
a. y xe x e x b. y xe 10x
dy
7. If y e2x 1
prove that
dx 1 y2 .
e 2x 1 ,
8. Determine the equation of the tangent to the curve defined by y x e x
that is parallel to the line represented by 3x y 9 0.
9. Determine the equation of the tangent to the curve at the point
where x p y x sin x
2 .
10. An object moves along a line so that, at time t, its position is s
sin t
3 cos 2 t ,
where s is the displacement in metres. Calculate the object’s velocity at t p 4 .
NEL
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