Sample Exam #1
Calculus 1
Instructions. No books, notes, or calculators are allowed.
Mastery Test Section
Directions: Compute the derivatives of the following functions. You do not need
to simplify your answers. Remember that no partial credit is given. Good luck!
0
√
x3
1. (5 points) 6 x −
+2 =
3
Solution:
2. (5 points)
3
√ − x2
x
x−2 + 2x1/3
3x3 − x − 1
0
=
Solution:
(3x3 − x − 1)(−2x−3 + 3x22/3 ) − (x−2 + 2x1/3 )(9x2 − 1)
(3x3 − x − 1)2
0
3. (5 points) (x5 − 4x + 8)7 =
Solution:
7(x5 − 4x + 8)6 (5x4 − 4)
4. (5 points)
h
ex + e−2x
6 i0
=
Calculus 1
Sample Exam #1
Solution:
6 ex + e−2x
5
ex − 2e−2x
0
5. (5 points) arctan(x3 − 1) =
Solution:
3x2
1 + (x3 − 1)2
0
6. (5 points) −(3x + 2)−5 =
Solution:
15(3x + 2)−6
7. (5 points)
h√
i0
2x − x4 =
Solution:
(2x − x4 )−1/2 (x − 2x3 )
0
8. (5 points) ln(x3 ) + ln(x) =
Solution:
9. (5 points)
4
x
0
1
x tan
=
x
2
Calculus 1
Sample Exam #1
Solution:
1
1
2
− sec
2x tan
x
x
0
10. (5 points) 3 sin2 (πx) =
Solution:
6π sin(πx) cos(πx)
The Rest of the Exam
2x + 6
(or show it doesn’t exist). Show
+x−6
11. (25 points) Evaluate the limit lim
x→−3 x2
all work.
Solution:
lim
x→−3
2(x + 3)
(x + 3)(x − 2)
2
2
−2
= lim
=
=
x→−3 x − 2
−5
5
2x + 6
=
+x−6
x2
lim
x→−3
1
12. (25 points) Find the derivative of f (x) = √ using the definition of the derivax
tive.
Calculus 1
Sample Exam #1
Solution:
f (x + h) − f (x)
h→0
h
√1
√1
−
x
x+h
lim
h→0
h√
√
x− x+h
lim √ √
h→0 h x x + h
√
√
√
√
x− x+h
x+ x+h
√
lim √ √
√
h→0 h x x + h
x+ x+h
x − (x + h)
√
lim √ √
√
h→0 h x x + h( x +
x + h)
−1
√
lim √ √
√
h→0
x x + h( x + x + h)
−1 −3/2
−1
√
√ =
x
2
x( x + x)
f 0 (x) = lim
=
=
=
=
=
=
Extra Credit
sin(x2 − 1)
.
x→1 sin(x − 1)
13. (20 points) Find lim
Solution:
sin(x2 − 1)
x−1
1
sin(x2 − 1)
2
= lim
lim
(x − 1)
x→1
x→1 sin(x − 1)
x2 − 1
sin(x − 1)
x−1
−1 2
2
sin(x − 1) sin(x − 1)
x −1
= lim
x→1
x2 − 1
x−1
x−1
−1
2
sin(x − 1) sin(x − 1)
(x + 1)(x − 1)
= lim
2
x→1
x −1
x−1
x−1
−1
sin(x2 − 1) sin(x − 1)
= lim
(x + 1)
x→1
x2 − 1
x−1
= 1(1−1 )(1 + 1)
= 2