Fall 2014
Math 151
Exam 2 Review (3.1 - 4.1)
1
1. A boat is pulled into a dock by a pulley that is fixed 1 meter above the water level at a
rate of 1 m/s. How fast is the boat approaching the dock when it is 8 meters away?
2. A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the
angle between the string and the horizon decreasing when the string is 200 ft long?
3. Two sides of a triangle have length 12 m and 15m and the angle between them is
increasing at a rate of 2 degrees per minute. How fast is the length of the third side
increasing when the angle between the sides of fixed length is 60 degrees?
4. Find the linear approximation for the function 𝑓 𝑥 =
a. Use the above to approximate the value of
1
1
2+𝑥
at 𝑎 = 0
2.01
5. Find the linear and quadratic approximation to 𝑓 𝑥 = cos 𝑥 at 𝑥 = 𝜋/6
6. Use linear approximation to estimate the value of
3
8.012
7. Evaluate the following limits:
a. lim𝑥→0−
b. lim𝑥→∞
1
c. lim𝑥→−∞
2+𝑒 cot 𝑥
4𝑒 5𝑥 −3𝑒 −𝑥
7𝑒 3𝑥 +5𝑒 −2𝑥
4𝑒 5𝑥 −3𝑒 −𝑥
7𝑒 3𝑥 +5𝑒 −2𝑥
8. Find the third derivative of 𝑓 𝑥 = 𝑒 𝑥
3 +sin
𝑥
9. Find the 25th derivative of 𝑓 𝑥 = 𝑥𝑒 −𝑥
10. Find the slope of the tangent line to the curve 3𝑦 3 − 𝑥𝑦 2 + 3 = 0 at the point (0, −1)
11. Find the points on the curve given by 𝑥 = 𝑡 3 − 3𝑡 2 − 9𝑡 + 1, 𝑦 = 𝑡 3 + 3𝑡 2 − 9𝑡 + 1
where the tangent lines are vertical or horizontal
a. Find the equation of the tangent line when 𝑡 = 2
𝜋
12. If 𝑓 𝑥 = sin4 𝑥 , what is 𝑓′( ) ?
3
13. Compute the following limits:
a. lim𝑥→0
sin 2 6𝑥
4𝑥 2
b. lim𝑥→0
3 cos 𝑥−3+4 sin 𝑥
5𝑥
2
14. Find the value of c so that the function 𝑓 𝑥 = 𝑥𝑒 𝑐𝑥 has a horizontal tangent at 𝑥 = 3.
Fall 2014
15. Given 𝑓 𝑥 =
Math 151
𝑥 3 +1
𝑥 2 +1
Exam 2 Review (3.1 - 4.1)
2
, find the equation of the tangent line at the point where 𝑥 = −1
16. Find the velocity, speed and acceleration of a particle
a. at time 𝑡 = 2, if its position is given by the vector 𝑟 𝑡 = 𝑡 2 + 5, 𝑡
b. at time 𝑡 = 𝜋/3, if its position is given by 𝑟 𝑡 = 4 cos 2𝑡 𝑖 + 3sin⁡
(2𝑡)𝑗