(Calculus I)
2024-1 GEST1004
Assignment 1
Problem 1:
(4 points)
Limit evaluation (4 points).
Evaluate the given limit if it exists:
(a)
x2 − 4
;
x→2 3x2 − 2x − 8
lim
(b)
x2 − 4
;
x→−2 3x2 − 2x − 8
lim
(c)
lim
x→2+
2−x
;
|x − 2|
(d)
x2 − 16
√ .
x→4 2 −
x
lim
Solution:
Problem 2:
One-sided limit (2 points).
(2 points) There is exactly one point a where both the right-hand and left-hand limits fail to exist
for the function
x−1
f (x) = 2
.
x − 3x + 2
Describe the changing behavior of f (x) for x near a by finding the one-sided limits.
Problem 2 continues. . .
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(Calculus I)
2024-1 GEST1004
Problem 2 (continued)
Solution:
Problem 3: Meaning of derivative in three steps (5 points).
(5 points) (a) Give the definition of derivative f ′ (x) for y = f (x);
(b) Use the definition of derivative in three steps to find f ′ (x). Give also the meaning of each
step.
3
f (x) = x2 + .
x
(c)Then write an equation for the line tangent to the curve y = f (x) at the point where x = 2.
Solution:
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2024-1 GEST1004
(Calculus I)
Problem 4
Problem 4: Definition of derivative (2 points).
(2 points)
Apply the definition of the derivative to find f ′ (x).
(a)
f (x) =
1
;
2x + 1
f (x) =
√
x + 1.
(b)
Solution:
Problem 5:
Average rate of change and Instantaneous rate of
change (2 points).
(2 points) A water bucket containing 10 gal of water develops a leak at time t = 0, and the volume
V of water in the bucket t seconds later is given by
t
V (t) = 10 1 −
100
2
until the bucket is empty at time t = 100.
(a) At what rate is water leaking from the bucket after exactly 1 min has passed?
(b) When is the instantaneous rate of change of V equal to the average rate of change of V from
t = 0 to t = 100?
Solution:
Problem 5 continues. . .
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2024-1 GEST1004
(Calculus I)
Problem 5 (continued)
Problem 6: Percentage change and percentage rate of change (4
points).
(4 points)
The population (in thousands) of the city Metropolis is given by
P (t) = 100[1 + (0.04)t + (0.003)t2 ].
with t in years and with t = 0 corresponding to 1980.
(a) What is the rate of change of P in 1986 ;
(b) What is the actual of change of P from 1986 to 1987 ;
(c) What is the percentage change of P from 1986 to 1987 ;
(d) What is the percentage rate of change in 1986 .
Solution:
Problem 6 continues. . .
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(Calculus I)
2024-1 GEST1004
Problem 6 (continued)
Problem 7: Evaluate limit by relating to a derivative (1 point).
(1 point)
Evaluate the limit by relating it to a derivative.
(1 + x)999 − 1
;
x→0
x
lim
Solution:
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(Calculus I)
2024-1 GEST1004
Problem 8
Problem 8: (Extra exercise) Study of differentiability (not graded).
(not graded)
Investigate the differentiability of the function f defined by
f (x) =
(
2x + 1
4x − x
if x < 1,
2
if x ≥ 1.
Solution:
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