mathupdated.pdf
1) Find the average rate of change of 𝑓(𝑥) = −2𝑥3 + 11𝑥2 + 3𝑥 from 𝑥 = −6 to 𝑥 = 2. {10 pts}
2) Determine the values of x for which the following functions are continuous: {5 pts each}
a) 𝑔(𝑥) = 𝑥 2 + 16
𝑥3 − 6𝑥2 + 8𝑥
b) ℎ(𝑥) = 𝑥 2+ 7𝑥 − 98
2
3) Find the derivatives of the following functions (DO NOT SIMPLIFY!): {5 pts each}
a) 𝑓(𝑥) = 18𝑥5/2 + 8𝑥3/4 − 9𝑥−4/3 b) 𝑔(𝑥) = (2𝑥4 − 5𝑒𝑥)(6𝑥7 − 4𝑥−2) c) 𝑓(𝑡) = 𝑒𝑡
7 + 3𝑡5 − 2 + 50 ln 𝑡 d) 𝑔(𝑥) = 4√𝑥2 − 49
e) 𝑧(𝑥) = √𝑒4 + ln 36 f) 𝑔(𝑥) = 𝑒𝑥 ln(8𝑥 + 21)
g) 𝑓(𝑥) = (𝑥4 + 5)3
6𝑥 − 7
h) ℎ(𝑥) = 𝑥 20− 𝑥
64 − 𝑥3
i) 𝑓(𝑥) = 3 ln� 𝑥10 − 6√𝑥3 �
13
j) 𝑦 = 10𝑒0.5𝑥 + ln(2𝑥9 − 12𝑥2 + 𝑥)
4) The population of the town of Somewhereville can be modeled by the function 𝑃(𝑡) = 𝑡4 − 15𝑡3 − 24𝑡2 + 371𝑡 + 6009 for 0 ≤ 𝑡 ≤ 15 where t is in years and 𝑡 = 0 corresponds to the year 1999. {10 pts total}
a) What was the population of Somewhereville in the year 2002?
b) What was the population of Somewhereville in the year 2014?
c) What was the instantaneous rate of change of the population of Somewhereville in the year 2008?
d) What was the instantaneous rate of change of the population of Somewhereville in the year 2014?
e) Use your answers from parts a-d to ESTIMATE the population of Somewhereville in 2015.
5) Find the equation of the line tangent to 𝑔(𝑥) = 3 + 5𝑥 − 12√𝑥 at 4=x . {10 pts} 6) Evaluate the following limits: {5 pts each}
a) lim 𝑥 → 2
𝑥3 − 10𝑥2 + 16𝑥 𝑥3 − 5𝑥2 + 6𝑥
b) lim 𝑥 → −3
6𝑥2 + 24𝑥 + 18 𝑥2 − 9𝑥 + 18
