Fall 2014
Math 151
WIR5: 3.2 - 3.6
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1. Find the first derivative of the following functions/equations:
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a. 𝑓 𝑥 = 3𝑥 − 2 1 − 𝑥 +
2 1−𝑥
b. 𝑓 𝑥 = 𝑥 sin2 𝑥 cos 𝑥
c. 𝑓 𝑥 =
cos 𝑥
sin 2𝑥 + cos 𝑥
d. 𝑓 𝑥 = 𝑥 2 + 3𝑥 + 1 − 𝑥
e. 𝑓 𝑥 =
f. 𝑓 𝑥 =
1 1
−
𝑥 3
𝑥−3
, 2𝑥 − 3 . What is the domain of 𝑓 𝑥 and 𝑓 ′ 𝑥 ?
2𝑥
4−𝑥 2
g. 𝑓 𝑥 = 𝑥 3 − 5
h. 𝑓 𝑥 =
10
2𝑥+3 3
4𝑥 2 +1 8
i. 9𝑦 4 − 12𝑥 2 𝑦 2 + 5𝑥 2 = 11𝑥
j.
𝑦 cos 𝑥 + sin 3𝑦 − cot 2 3𝑥 = 1
2. Find the following limits:
a. lim𝑥→0
b. lim𝑥→0
c. lim𝑥→0
d. lim𝑥→0
e. lim𝑥→0
3𝑥
sin 9𝑥
tan 5𝑥
sin 3𝑥
tan 3𝑥
𝑥 sin 2𝑥−𝑥
tan 5𝑥
sin 3𝑥
cos 𝑥−1
𝑥2
3. Find the slope of the line tangent to the curve sec 𝑥 + 𝑦 − tan 𝑥 − 𝑦 = 1 at (𝜋, 𝜋)
Fall 2014
Math 151
WIR5: 3.2 - 3.6
2
4. Find all values of a where the lines tangent to the curve 𝑓 𝑥 = sin2 𝑥 + cos 𝑥 at 𝑥 = 𝑎
are horizontal given that 0 ≤ 𝑎 ≤ 2𝜋
5. Show that the curves 𝑥 2 + 𝑦 2 = 4𝑥 and 𝑥 2 + 𝑦 2 = 2𝑦 are orthogonal.
6. If 𝑟 𝑡 = 4 sin 𝑡, 4 cos 𝑡 is the position vector of a moving particle at time t, find the
velocity and speed of the particle at the point (2, −2 3)
7. A ball is thrown vertically up from the ground with a velocity of 128 m/s. It’s height after
t seconds is given by the equation 𝑕 𝑡 = 128𝑡 − 16𝑡 2 .
a. After how long will the ball reach its maximum height?
b. Find the maximum height reached by the ball.
c. What is the velocity of the ball when it is 240 m above the ground?
d. After how long and with what velocity will the ball hit the ground?
8. The equation 𝑥 2 − 𝑥𝑦 + 𝑦 2 = 3 is an ellipse whose axes are not parallel to the
coordinate axes. Find the points where the ellipse crosses the x-axis and show that the
tangent lines at these points are parallel.
a. Where does the normal line to the above curve at the point (1, 1) intersect the
ellipse a second time? What is the slope of the tangent at this intersection point?
9. An object is placed in front of a lens at a distance p from the lens and its image appears
at a distance q from the lens. If the focal length of the lens is f, then from the lens
1
1
1
equation, we can say that = + . Find the rate of change in p wrt q.
𝑓
𝑝
𝑞
10. Given the graph below, if 𝑢 𝑥 = 𝑓 𝑔 𝑥 , 𝑣 𝑥 = 𝑔 𝑓 𝑥
find 𝑢′ 1 , 𝑣 ′ 1 and 𝑤 ′ 1 .
and 𝑤 𝑥 = 𝑔 𝑔 𝑥 ,