MATH 151 Engineering Mathematics I
Week In Review
Fall, 2015, Problem Set 11 (Final Review)
JoungDong Kim
1. Find the parametric equations for the line passing through the point (4, −7) and perpendicular
to the line 3x + 4y = −8.
t2 − 16
2. lim √
t→4
t−2
3. Consider the triangle with vertices A(1, 3), B(2, 1) and C(−2, 0). Find the angle ∠BAC.
1
4. Find the distance from the point P (5, 1) to the line y = 2x − 1.

2

ax + 2x
5. Consider f (x) = k

 3
x + ax − 3
if x < 2
if x = 2 . For what values of k and a is f (x) continuous?
if x > 2
6. Find the equation of tangent line to the curve x3 − 3xy + y 3 = 3 at the point (2, 1).
2
7. Find the linear and quadratic approximation for f (x) = xe3x at x = 0.
8. Find all point(s) on the curve x = t3 − 3t − 1 and y = t3 − 12t + 3 where the tangent line is
vertical.
2e−3x − 3e3x
.
x→−∞ 4e−3x + 2e3x
9. lim
3
10. For the equation y = e2x + e−3x , find y ′′ + y ′ − 6y.
11. If f ′ (x) = 2 sin x + 4 cos x − ex and f (0) = 5, what is f (π)?
23x − 5x
x→0
4x
12. lim
4
13. Find the absolute extrema for f (x) = x2 +
2
over the interval [ 21 , 2].
x
14. If f (x) = arctan(x2 + 2x + 3) + arccos(3x), find f ′ (0).
15. Give f (x) = 2xe3x , find the interval of increasing, decreasing, concavity, and find local extrema
and inflection point(s).
5
n 1
P
16. lim
n→∞ i=1 n
2
i
n
17. Estimate the area under the graph of f (x) = x3 + 2, [−1, 2] using 6 approximating rectangles and
left endpoints.
18. For the given function f (x) = x3 + 2x, set up the limit of Riemann Sum that represents the area
under the graph of f (x) on the interval [0, 2] using right endpoints.
6
19. Write as a single integral
Z
5
−3
f (x)dx −
Z
0
f (x)dx +
−3
20. Find the derivative of the given function.
Z x
(a) f (x) =
(t2 − 1)20 dt
1
(b) g(x) =
Z
2
cos(t2 )dt
x
(c) y =
Z
sin x
t cos(t3 )dt
25
7
Z
6
f (x)dx
5
21. Evaluate the integrals.
Z 2
(a)
(5x2 − 4x + 3)dx
1
(b)
(c)
(d)
(e)
Z
Z
Z
Z
2 6
t − t2
dt
t4
1
π
2
(cos θ + 2 sin θ)dθ
0
−e
−e2
3
dx
x
(
x
f (x)dx where f (x) =
sin x
−π
π
if − π ≤ x ≤ 0
if 0 ≤ x ≤ π
8