Calculus 1321 Final Practice Exam - MATH 1321 Calculus I...

MATH 1321 – Calculus I
Sections 10 and 14 Fall 2004
Final Exam – Preparation
1. Find the following limits:
(a) lim
x → 2 -
x
2
- x - 2
| x - 2 |
(b) lim
x → 7
√
x + 2 - 3
x - 7
2. Use the Squeeze Theorem to find lim
x → 0
e
x
2
sin(1 /x )
.
3. Find the following limits:
(a) lim
x →∞
1 - x - 3 x
2
2 x 2 - 7
(b) lim
x →∞
√
9 x 6 - x
x 3 + 1
(c) lim
x →∞
(
√
x
2
+ x - x )
4. Which of the following functions f has a removable discontinuity at a ? If the discon-
tinuity is removable find an everywhere continuous function F such that f ( x ) = F ( x )
for all x = a .
(a) f ( x ) =
x 2 - 2 x - 8
x + 2
, a = - 2 (b) f ( x ) =
x - 7
| x - 7 |
, a = 7
(c) f ( x ) =
x 3 + 64
x + 4
, a = - 4
5. Find the vertical and horizontal asymptotes of the curve y =
√
x 2 - 9
2 x - 6
, x > 3.
6. Use the definition of the derivative to find f ( x ) if f ( x ) =
√
1 + x .
7. Use the definition of the derivative to find f ( x ) if f ( x ) =
x
x + 1
.
8. Let f ( x ) = sin 2 x . Use the definition of the derivative to find f (0).
9. Let
f ( x ) =
x 2 if x ≤ 2
mx + b if x > 2
(a) Find the values of m and b that make f continuous everywhere.
(b) Find the values of m and b that make f differentiable everywhere.
10. Use the definition of differentiability to show that f ( x ) = | x - 6 | is not differentiable
at a = 6.
11. If f ( x ) = e
x
g ( x ), where g (0) = 2 and g (0) = 5, find f (0).
12. Let f ( x ) =
h ( x )
x
, x = 0. If h (2) = 4 and h (2) = - 3, find f (2).

13. Let f ( x ) = g ( h ( x )). If h (3) = 6, h (3) = 4, g (3) = 2 and g (6) = 7, find f (3).
14. Find the derivative y :
(a) y =
1
√
2 π
e
- x
2
/ 2
(b) y =
sin(3 x )
x
(c) y = (tan x )(2
sin x
)
15. If y = ln( x
2
+ y
2
), find y .
16. Find equations of the tangent lines to the curve y = x/ ( x + 1) that are parallel to the
line x - 2 y = 2.
17. Find an equation of the tangent line to the curve y = e x that passes through the origin.
18. Find the point on the parabola y = x
2
at which the slope of the tangent line is equal
to the slope of the secant line between A ( a, a
2
) and B ( b, b
2
).
19. Find the slope of the tangent line to the curve x 3 y 2 - 2 y = e y - 1 at the point (1 , 0).
20. Let f ( x ) = x
2 / 3
.
(a) Use the definition of the derivative to show that f (0) does not exist.
(b) Find f ( x ) for x = 0. (Here you don’t have to use the definition of the derivative.
You are allowed to use the power rule.)
(c) Show that the graph of f has a vertical tangent line at (0 , 0).
(d) Illustrate part (c) be sketching the graph of f .