Math 151 “FREE” Final Exam Review
Disclaimer: This review by no means covers every concept covered in Math 151 and should not be used
as your sole source of study.
Solutions will be posted at www.math.tamu.edu/∼baurispa/math151/homepage.html
1. Compute each of the following.
Z (a)
Z 2
(b)
3
x 3
√
+ +
2
3 x
1−x
dx
(ex + x2 − 4) dx
0
(c)
Z 4 2
x +x
√
1
dx
x
2. Find the value(s) of c and d that make the function below continuous for all x.
f (x) =

2

 cx − 1 for x < 3
d

 5 − cx
for x = 3
for x > 3
3. Given f (x) = x2 e−x , find the intervals where f is increasing/decreasing and identify any local extrema.
4. A closed can is to be constructed in the shape of a cylinder. If the surface area of the can (including
the top and bottom of the can) is fixed at 60 square feet, find the radius and height of the can with
maximum volume.
d
5. Find
dx
Z e3x p
4 + 5t4 dt
0
Z 4
6. Approximate
−2
(16 − x2 ) dx using 3 equal subintervals and taking x∗i to be the midpoint.
7. Find the inflection points and intervals of concavity for the function f (x) =
8. Calculate the following limits.
(a) lim
x→3−
|x − 3|
1
1
x − 3
2
(b) lim (cos x)1/x
x→0
x2 − 4x − 5
x−4
x→4+
√
√
4
x+h− 4x
lim
h→0
h
ln(1 + x) − x
lim
x→0
x2
!
2x3 − 5x
lim arccos
x→∞
−4x3 + 1
√
x2 + 4
lim
x→−∞ 4x + 1
(c) lim
(d)
(e)
(f)
(g)
(h) lim e
x→∞
3−x2
x+1
x4
1
.
+2
9. Express tan(sin−1 x) without using trig or inverse trig functions.
10. Sand is being dumped at a rate of 100 cm3 per second onto a cone-shaped pile whose diameter is
one-half its height. How fast is the height of the pile increasing when the height is 75 cm?
D
2 −1
11. Find the equation of the tangent line to the curve r(t) = et
E
, (2t + 4)3 at the point (1, 8).
12. Find f 0 (x) for f (x) = (sec 2x)arctan 5x .
13. Find the linear approximation of f (x) =
14. Find
dy
dx
√
3
x at x = 27 and use it to approximate
√
3
26.9.
3
for the equation x3 y 5 + cot(3y) + ey = 12.
15. Find f (x) given that f 0 (x) = x − 3 sin x + ex and f (0) = −2.
16. Find an equation of the tangent line to the graph of f (x) = tan2 (x) at the point where x = π4 .
17. Given the points P (2, 6), Q(−1, 2), and R(1, −3), find the following.
(a) A unit vector orthogonal to the vector from P to Q.
(b) The angle at the point Q in the triangle formed by these points.
−−→
−→
(c) The vector projection of P Q onto P R.
(d) A vector equation and parametric equations of the line passing through P and Q.
−−→
−→
(e) What is the slope of the vector P Q − 2P R?
(f) Find parametric equations of the line that passes through R and is perpendicular to the line with
parametric equations x = 4 − 5t, y = 3 + 7t.
18. A boat heads in a direction 330◦ from the positive x-axis at 18 mph. The water is flowing due north
at 5 mph. Find the true velocity and speed of the boat.