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MATH1141: Calculus I A2-1

TRU Open Learning

Assignment 2 (8%)

Assignment 2 is graded with a total of 100 marks and it contributes 8 percent

towards your course grade.

[10 marks] 1. A roast turkey is taken from an oven when its temperature has

reached 85 OC and then placed on a cooling rack. Over time, its

temperature is recorded at 30-minute intervals with the

following results:

t mins 0 30 60 90 120 150

Temp OC 85 69 52 40 36 33

(6 marks) a. Find the average rate of change of temperature over each of

the intervals: [30, 60], [60, 90], and [30, 90]. Include the

units.

(1 marks) b. Estimate the instantaneous rate of change of temperature

after 1 hour.

(3 marks) c. Draw a graph by hand and use it to estimate the

instantaneous rate of change of temperature after 1 hour.

[10 marks] 2.

(5 marks) a. Make a table of values of 𝑓(𝑥) = 𝑥−4

√𝑥−2 and 𝑥 =

4.1, 4.01, 4.001,⋯and 𝑥 = 3.9, 3.99, 3.999,⋯ to guess the

value of lim 𝑥→4

𝑥−4

√𝑥−2

(5 marks) b. Use an appropriate factorization to find lim 𝑥→4

𝑥−4

√𝑥−2 .

A2-2 Assignment 2

TRU Open Learning

[20 marks] 3. Find the limit:

(4 marks) a. lim ℎ→0

[ (3+ℎ)−2−3−2

ℎ ]

(4 marks) b. lim 𝑥→2

𝑥2+𝑥−6

𝑥2−2𝑥

(4 marks) c. lim 𝑥→−2

3𝑥2−√9𝑥4+𝑥+2

𝑥3+2𝑥2

(4 marks) d. lim 𝑥→∞

2−3𝑥2

6𝑥2+5

(4 marks) e. lim 𝑥→0

𝑥4 sin ( 1

2𝑥2+3𝑥 )

[12 marks] 4. Consider the function f(x) given by

𝑓(𝑥) =

{

0 𝑥 < −5

√25 − 𝑥2 −5 ≤ 𝑥 < 0

−√25 − 𝑥2 0 < 𝑥 ≤ 5 𝑥 − 5 5 < 𝑥 < 10 2 𝑥 ≥ 10

(4 marks) a. Draw the graph of f by hand.

(6 marks) b. For each of x = 0, 5, 10, say whether f is continuous from

the right, continuous from the left, or continuous at the

number. Justify your answer using the definition of

continuity at a point.

(2 marks) c. For which values of x is f continuous at x? Give your

answer in interval notation.

MATH1141: Calculus I A2-3

TRU Open Learning

[12 marks] 5. Let 𝑓(𝑥) = |𝑥| + ⟦𝑥⟧ where |𝑥| is the absolute value of x and

⟦𝑥⟧ is the greatest integer function in x.

(4 marks) a. Draw the graph of f by hand for x in [0, 2].

(4 marks) b. For which values of a in (0, 2) does lim 𝑥→𝑎

𝑓(𝑥) exist?

(4 marks) c. For which values of x in (0, 2) is f continuous at x? Explain

using the definition of continuity.

[8 marks] 6.

(4 marks) a. State the domain of 𝑓(𝑥) = √𝑥2−4

𝑥2+𝑥−12 in interval notation.

(4 marks) b. Use any of Theorems 4, 5, 7 and 9 on continuity to show

that the function f above is continuous at each point of its

domain.

[8 marks] 7.

(4 marks) a. Use the intermediate value theorem to show that there is a

root of 𝑓(𝑥) = 𝑥4 + 1 − 1

𝑥 in (

1

2 , 1). Note that you must

show the hypothesis of the theorem are satisfied before

you can apply the theorem.

(4 marks) b. By using the theorem repeatedly find the root to two

decimal places.

[10 marks] 8.

(7 marks) a. Find 𝑓’(𝑥) for 𝑓(𝑥) = √4𝑥 + 1 using the definition of the

derivative.

(3 marks) b. Find an equation of the tangent line to f at 𝑥 = 1

A2-4 Assignment 2

TRU Open Learning

[10 marks] 9.

(5 marks) a. Copy the graph below of the function f by hand and directly

below it, sketch its derivative by hand.

(5 marks) b. Sketch the graph of a single function f that satisfies all of the

following conditions:

i) lim 𝑥→−∞

𝑓(𝑥) = −2 ii) lim 𝑥→∞

𝑓(𝑥) =

0 iii) lim 𝑥→−3+

𝑓(𝑥) = ∞

iv) lim 𝑥→−3−

𝑓(𝑥) = −∞ v) lim 𝑥→3+

𝑓(𝑥) = −2 vi) f is

continuous from right at 3.