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QUESTION 9 The displacement of a particle (in metres) at time ¢ (in seconds) is represented by the function s@)=tln@t)—t 0<t<4 Determine the approximate acceleration of the particle at time ¢ = 3. (A) 0.66ms > (B) 0.33ms> (C) -033ms™ (D) —0.66m s
QUESTION 14 (4 marks) Consider the function f(x)=In(3x+4) for x > _?4 a) Determine f`(x). [1 mark] b) Determine the x-intercept of the graph of f/(x). [2 marks] c) Determine the gradient of the tangent to the graph of f(x) at the x-intercept. [1 mark]
QUESTION 13 (9 marks) a) Determine the derivative of f`(x)= 32 [1 mark] : ln(x) : : b) Given that g(x)=——= determine the simplest value of g`(e). [3 marks] X ¢) Determine the second derivative of 4(x) = xsin(x). (Give your answer in simplest form.) /5 marks]
QUESTION 5 The slope of the tangent to the graph of y = In(x?) atx = e? is 1 (A) 2 2 (B) 2 4 ©) i 1 (D) —
QUESTION 11 (5 marks) Determine: 4 3In(5x) 2) <= 3In(5Y) d b) — (x3cos(x* + 1)) dx c) [3e*dx [1 mark] (You do not need to simplify.) /3 marks] [1 mark]
QUESTION 1 el (*) is equivalent to (A) 0 (B) 1 (C) x (D) —
QUESTION 16 (5 marks) Solve for x in the equation 4+ 7e?* = 3e”.
QUESTION 19 (7 marks) Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days. The number of daily customers for each shop can be modelled by the equation y = A In( Bx) where x is the day and y is the number of customers. The constants A and B are different for each shop. Determine algebraically whether the total number of customers for Jaxon and Shari’s shops will be the same every day in the future.
QUESTION 6 Sa | ; | dx a#0is a x+a (A) 1.7918 (B) 1.6094 (C) 1.3863 (D) 1.0986
QUESTION 8 The number of koalas in a conservation park is modelled by N= 15 In(7t+ 1) t = 1 where ¢ represents the time (years) since the park opened. There were 20 koalas in the park when it opened. Determine the approximate rate of change in the number of koalas when f= 3. (A) 46 (B) 26 (C) 25 (D) 5
QUESTION 15 (4 marks) Determine the derivative of f (x) =Inx? + In(x —35 y. Express the derivative as a single fraction in its simplest and factorised form.
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