Question

Question 3

(a) [10 marks] FAEN102 students must attend t hours, where t ∈ [0,H], of lectures and pass two quizzes to be in good standing for the end of semester examination. The number of students who attended between t1 and t2 hours of lectures is de- scribed by the integral

? t2

20t2 dt, 0≤t1 <t2 ≤H.

t1

As a result of COVID-19, some students attended less than H2 hours of lectures before the university was closed down and passed the quizzes so they are qualified to take the examination. Find the ratio of students who attended less than H2 hours of lectures

(b) [10 marks] The final score each student got is proportional to the amount of lectures he/she had attended which is given by the function

L(t) = 100t. H

What was the average score in the class?

FAEN102 - Calculus I

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Question 4

(a) [10 marks] Find the values of α and β such that the function

?αcosx+2 if x<0 f(x)= βe3x+αx2 if x≥0

is continuous on the interval (−∞, ∞).

(b) [10 marks] For which values of α and β is the function f(x) differentiable?

Question 5

(a) [6 marks] How many solutions does the equation e−x2 = x have?

− x has (c) [8 marks] Use the Newton-Raphson method to find the approximate solution of

(b) [6 marks] Use the intermediate value theorem to show that f (x) = e at least one root in the interval [0, 1].

the function f (x) = e decimal places.

−x2

− x . Start from x0 = 0.5 and correct your answer to 4

Question 6

(a) [10 marks] Find an equation of the curve y = f(x) if y′′ = 6x and the tangent

line to the curve at (1, 2) is horizontal.

(b) [10 marks] A straight line from the origin passes through the point P(x,y) in the first quadrant and is inclined at θ = π3 from the positive x-axis. Write an equation for the line.

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Answer 1