Exam 1 Review
Fall 2011, Math 1210-007
1. Evaluate lim x
→− 2
2. Evaluate lim x → 0+ x
2 − 4 x 2 − x − 6
.
| x | x and lim x → 0 −
| x |
.
x
3. Evaluate lim x
→ 3+
−
[[ x ]] x and lim x
→ 3
−
−
[[ x ]] x
.
4. Let f ( x ) =
2 x − 5 , x < 2
. Evaluate
( x − 5)
2
, x ≥ 2 lim x → 2 f ( x ) .
5. Find the value of k if lim
− x → 0
k +
| x |
x
= 7 .
6. Evaluate the following limits.
(a)
(b)
(c) lim x
→ 0 lim x → 0 x + sin ( x
2 ) x x tan 3 x lim x
→
π/ 2 sin x x
(d)
(e)
(f) lim x
→∞ lim x →−∞
7 x
3 − 4 x + 15
3 x 3
8 x
2
− 14 x
+ 4 x + 2
9 x 3 + 5 x 2 + 10 lim x
→ 4+
− 2 x − 4
+ 2
(g)
(h)
(i) lim x → 7+
− 2( x 2 x − 4
− 8 x + 7) x lim x
→ 0
1 − cos x sin 3 x lim x
→
π/ 6 cos 2 x
7. Determine all points at which the following functions are discontinuous. Specify whether these points have a removable discontinuity.
√ x, x > 0
(a) f ( x ) = x
2 − x, x < 0
(b)
f ( x ) =
6 , x 2 x
2 − 9
− 2 x − 3
, x x
= 3
= 3
8. The function g ( x ) = sin x
2 x is continuous at every point except at x = 0 . Define g (0) so that g is continuous at every real number.
9. Let f ( x ) = 4 x
2 − 10 x + 65 . Find f ′ (2) using the limit definition of the derivative.
10. Use the limit definition of the derivative to show that d d t
| t − 1 | does not exist at t = 1 . (Hint: consider what happens for t < 1 and for t > 1 separately.)
11. Find the slope of the curve y = x
3 − x
2 + 1 at the point (2 , 5) .
12. An object travels on a horizontal track starting at 0 with a directed distance in feet of s =
1
3 t
3 − 6 t
2 + 27 t , where t measured in seconds. When is the object moving to the right?
13. Find the equation of the tangent line to the curve y = 2 x
3 − 4 x
2 + x − 8 at the point (2 , − 6) .
14. Find the specified derivative of the given expression.
(a) d g d u
, where g =
√
2 u + ( u − 1) 4 −
1 u
(b)
(c)
(d)
(e)
(f) f
D
′ ( x t
[
) f
, where
( x )] f ( t
, where
) = f ( x t 2 t
3 − 1
+ 3 t − 2
) =
1 + x
1 − x
− 4 x
3
D t s for s = − 5 t csc t f ′ ( x ) , where f ( x ) = sin 2 ( x )cos ( x ) x 2 d H
, where d s
H ( s ) =
1 + s
4
1 − s
(g)
(h)
D x y for y = x cos (sin ( x )) d y d x
, where y = x
2 cos (2 x )

15. Let f ( x ) =
1 + x
. Find x f ′ ( x ) , f ′′ ( x ) , and f ′′′ ( x ) .
x
2 + 3
16. Find the equation of the tangent line to the graph of y = at the point (0 , 3) .
17. Find the equation of the tangent line to f ( x ) = x
√
1 + x 3
1 − x at the point where x = 2 .
18. An object thrown directly upward has a height of s = − 16 t
2 + 32 t + 16 after t seconds.
(a) At what time does it reach its maximum height?
(b) What is the maximum height obtained?
19. A water balloon is launched straight up in the air with initial velocity 160 ft/sec. Its height above ground is given by s = − 16 t
2 + 160 t .
(a) What is the velocity of the balloon t seconds after it’s thrown?
(b) At what time t does the balloon reach its maximum height?
(c) If the balloon is launched from an initial height of 25 ft, what is the new expression s for its height above the ground at any time t ?
20. Use implicit differentiation to find d y d x if
(a) 5 x
3 + 2 xy
2 + y
3 = 10
(b) y = x sec y
(c)
(d) y
2 − x
2 = cot(
√ x + y = x xy ) (e) x
3 − 2 + xy + y
3 + 24 = 62
(f) sin 2 (2 y ) − 3 x
2 = y
2
21. A spherical balloon is inflated at a rate of 4 ft
3
/min. How fast is the radius increasing when the radius is 2 feet? (The volume of a sphere is given by V = 4
3
πr
3
.)
22. Little Mikey starts to cry when he lets go of his balloon animal at the fair and it rises directly above his head at a rate of 2 ft/sec. A sullen teenager sitting 300 ft from Little Mikey watches apathetically as the ballon rises above Little
Mikey’s head. At what rate is the distance between the balloon and the teenager changing when the height of the balloon is 400 ft?