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Assignment 4: Exponential and Logarithmic Functions (80 marks total) Your mark will be given as a percentage. [10 marks] 1. {5 marks} {5 marks} [10

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Assignment 4: Exponential and Logarithmic Functions (80 marks total) Your mark will be given as a percentage. [10 marks] 1. {5 marks} {5 marks} [10 marks] 2. {5 marks}
(5 marks} [Iii marks] 3. {4 marks} (4 marks}
(4 marks) {4 marks} Consider the fimction Ht] = 100(1 — 2'“). a. Find an equation of the tangent line at t = 1. Give exact
values. b. If 1’ represents the percentage of people who have heard a
rumour in 1: months, when will 9ﬂ% of people have heard
the £11310th“? Consider the fimcﬁon 3(1) = e'2fln (3x).
a. Find an equation of the tangent line atx = 1.
h. Find and simplify g&quot;(x). Differentiate the following functions. You need not simplify,r
h. JMir] = x“ — 9‘“ + em + 23. c. Mr] = e'sxln (5x). :1. Hr) =1n[::;§).

[8 marks] 4. Suppose that a company finds that the percentage of people
who bought their product after t advertisements had been run on television is modelled by PR) = mu 1+12e—0‘3&quot;
{2 marks} a. What percentage of people bought the product before seeing
{3 marks} b. Find and interpret P’[t).
the ﬁmction P? TRU Open Learning A42 Assignment 4: Exponential and Logarithmic... [10 rims-ks] 5. Find the maximum and minimum values, giving exact values,
of {5 marks} a. f(x] = 4152s.&quot;jr over [[1,5]. ('5 marks} b. ﬁx) = xlnx over [1, 11-].

[10 marks] 6. {3 marks}
('3 marks) ('3 marks) (I mark) [8 marks] F. ('3 marks} {2 marks}
('3 marks) Suppose that the total cost, in millions of dollars, for a company
is given by CE) = 500 — Title-'7-t where t is the time in years
since its start—up date. a. Find and interpret C(t). b. On which interval(s) is C is increasing and on which C is
decreasing. c. On which interval(s} is C concave up and on which interval{s) it is concave donut? d. Sketch the graph of C. Suppose that the population in a certain country was 150
million in 1990 and 146‘ million in 1995. Assuming an
exponential decay model, a. Find the value ofk and write the model, using tas the
number of years since 1990. 1:. Estimate the population in 1998.
c. In what year will the population be 136 million?

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