Transcendental Functions - Exercises
Below you will find the exercises we suggest you work on in connection with the Transcendental Functions theme and the project problems which are to be done for the exercise classes 16-20 September.
Suggested Exercises
Section 3.1: Inverse Functions
3.1.3, 3.1.7, 3.1.11, 3.1.21, 3.1.25, 3.1.28, 3.1.34.
Section 3.2: Exponential and Logarithmic Functions
3.2.3, 3.2.5, 3.2.15, 3.2.26, 3.2.31.
Section 3.3: The Natural Logarithm and Exponential
3.3.3, 3.3.7, 3.3.15, 3.3.17, 3.3.39, 3.3.45, 3.3.54, 3.3.60.
Section 3.4: Growth and Decay
3.4.1, 3.4.3, 3.4.7, 3.4.11, 3.4.12.
Section 3.5: The Inverse Trigonometric Functions
3.5.11, 3.5.17, 3.5.19, 3.5.29, 3.5.33, 3.5.43.
Section 3.6: Hyperbolic Functions
3.6.2, 3.6.5, 3.6.7, 3.6.9.
Project Problems
These problems are to be presented 16-20 September. See here for where you should meet, and here to sign up for presenting a problem.
Problem 1
a) Let \(f(x) =\mathrm{e}^x\) with domain \((-\infty,\infty)\), and \(g(x)=x^2\) with domain \([0,\infty)\). Show that \((f\circ g)^{-1}(x) =\sqrt{\ln x}\), and show that \(g^{-1}(f^{-1}(x)) =\sqrt{\ln x}\).
b) In general, let \(f\) and \(g\) be functions with inverses \(f^{-1}\) and \(g^{-1}\) respectively. Show that \((f\circ g)^{-1} = g^{-1}\circ f^{-1}\).
Hint: Solution to Exercise 3.1.34.
Problem 2 (Problem 75, p.183 in Adams)
We will show that \(2<\mathrm{e} <3\). Let \(f(t) = 1/t \) for \(t>0\).
a) Show that the area bounded by \(y =f(t)\), \(y =0\) and \(t=1\), \(t=2\) is less than \(1\). Deduce from this that \(\mathrm{e} >2\).
b) Show that all tangent lines to the graph of \(f\) lies below the graph. (Hint: Show that \(f''(t) >0\) for all \(t\).)
c) Find the lines \(T_2\) and \(T_3\) that are tangent to \(y =f(t)\) at \(t=2\) and \(t=3\) respectively.
d) Find the area \(A_2\) enclosed by \(y=T_2\) , \(y=0\) and \(t=1\), \(t=2\). Also find the area \(A_3\) enclosed by \(y=T_3\) , \(y=0\) and \(t=2\), \(t=3\).
e) Show that \(A_2+A_3 >1\), and deduce that \(\mathrm{e}<3\).
Problem 3
The number of people infected by a virus at time \(t\) is given by \[ y(t) =\frac{L}{1+M\mathrm{e}^{-kt}}\] for \(t \geq 0\), where \(t\) is measured in months.
a) Assuming that we at the start of the outbreak had 200 patients, and after 1 month there were 1000 people infected. Eventually, the number of patients stabilizes at 10 000. Use this information to find the constants \(L, M\) and \(k\).
b) How many people were infected after 3 months? How fast was he infection spreading at this time?
c) At what point was the number of patients growing fastest?