Download The following problems are presented in the Week 5 videos

Week 5: The Definition of a Derivative and
Horizontal Asymptotes
Welcome to the Weekly Review for MATH 2413. This week’s review talks about the definition of a derivative
and finding horizontal asymptotes. We would like to thank Patrick Bourque and the Fall 2014 MATH 2413
students for allowing us to film the Weekly Reviews.
The following problems are presented in the Week 5 videos. Thank you!
1. The Definition of a Derivative.
2. Use the definition of a derivative to find f 0 (x) of f (x) = x2 + 10x.
3. Find the equation of the tangent line to f (x) =
√
1
x + 4 when x = 5.
4. Use the definition of a derivative to find f 0 (x) for f (x) =
5. Special Trigonometric Limits
(a) limπ
x→ 4
cos(x) − sin(x)
1 − tan(x)
=
(b) Theorems
2
2x
x+1 .
(c) lim
x→2
(d) lim
x→3
(e) lim
x→1
sin(x2 − 4)
x−2
sin(x − 3)
sin(x2 − 9)
=
=
sin(x3 + x − 2)
x−1
=
(f) Trigonometric Identities
3
6. Use the definition of a derivative to find f 0 (x) for f (x) = sin(2x).
7. Use the definition of a derivative to find f 0 (x) for f (x) = cos(3x).
4
2
8. Show f 0 (2) Does Not Exist for f (x) = (x − 2) 3
9. Show that f 0 (1) Does Not Exist for f (x) = |x − 1|.
10. Find 2 points on f (x) = x2 where the tangent line passes through (1, −3).
5
11. Find a g(x) that will make f (x) continuous and use the definition of the derivative to find f 0 (0).
f (x) =

 x2 sin( x1 ) x 6= 0
g(x)

12. Find all horizontal asymptotes of f (x) =
2x3 +x
x3 +1 .
√
13. Find all horizontal asymptotes of f (x) =
4x2 +1
x+1 .
6
x=0
Want some more practice? The following problems were provided to you by the Math Lab Learning Specialists. Please feel free to come and visit the UT Dallas Math Lab if you need any help. Thank you!!
1. Use the definition of a derivative to find f 0 (x) for f (x) = 3x2 − 5x + 4.
2. Use the definition of a derivative to find f 0 (x) for f (x) =
6x
x+1 .
3. Use the definition of a derivative to find f 0 (x) for f (x) = sin(5x).
4. Use the definition of a derivative to find f 0 (x) for f (x) = cos(4x).
5. Find the equation of the tangent line to f (x) =
√
6x + 1 when x = 4.
6. Show that f 0 (2) Does Not Exist for f (x) = |x − 2|.
7. Find 2 points on f (x) = 4x − x2 where the tangent line passes through (2, 5).
8. Find values α and β to make the function differentiable for all real numbers.
f (x) =

 α − βx2
x2

9. Find all horizontal asymptotes of f (x) =
2x2 +1
3x−5 .
7
x<1
x≥1