MAT 140 Chapter 5 notes - Transcendental Functions and Calculus
5.1 Here, we took a first look at the Natural Logarithmic Function, and some of it's properties.
ln properties:
(i)
(ii)
(iii)
Ex: Type I problem. Expand the logarithmic statement into, sum, differences and multiples of logarithmic expressions.
(a)
Soln: We use a combination of the principles above:
(ii) (i) (iii)
(b) Type II problem. Compress the statement into a single logarithm.
(iii)
Soln:
(i) (ii)
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-> Then we looked at the differentiation rules for natural log functions:
rules:
The thing that we're 'sticking in' to the function ln is called it's argument. If the argument is more complicated than just a single variable (like x), then we have to use the second version of the rule. Also, we may have to use these rules in conjunction with others, like the product rule.
Ex: Find the derivative of each function:
(c)
Soln: This is going to require the use of the product rule, since we have the product of a polynomial (x) and a transcendental function (ln(x)).
(d)
Soln: Ok, here we have to use the 2nd version of the derivative rule, since the argument is 2x + 3, which is more complicated than just x. We take the derivative of the argument, and put it over the argument in a fraction:
(e)
Soln: again, we have a more complicated argument than just x, so we use the second version of the rule. note:
(f)
Soln: I'm going to use ln properties to expand the expression, then take the derivative:
Homework Problems:
(1) Use ln properties to expand the expression:
(2) Use ln properties to compress into a single ln:
(3) Find the derivative of the function:
(a) (b) (c)
5.2 Now we look at the integral rules for using ln(x). They are:
rules:
The presence of the u is predictable - the rules look the same, but the second one is valid when we use u-substitution.
Ex:
(g) Find the area of the region under between a = 1 and b = 6.
Soln: Of course, we set up area as a definite integral.
Area =
(h) Evaluate the integral:
Soln: Here, we have two polynomials present, and they differ in degree by 1, so I'm going to revert to my old strategy here, and select u to be the bigger-degreed of them.
\
Seems to work.
(i) Derive (find) the rule for the integral of tan(x):
Soln: I'm going to rewrite tan as a quotient, and then use u-substitution:
And this is the rule.
Homework: Evaluate the integral:
(4) (a) (b) (c)
(5) Use u-substitution to find the rule for:
5.3 Here, the topic is the so-called inverse function.
->Two functions f and g are inverses if they meet the following property:
That is, if we plug one function into the other, either way, we get the identity function
y = x.
Ex: Show that are inverses.
Soln:
So, plugging in one function to the other produces x either way.
->Not every function has an inverse. Only one-to-one (1-1) functions do. A function is
1-1 if it passes the horizontal line test (HLT) - meaning that you can not hit the graph of the function twice with the same horizontal line.
Ex: Are the following functions 1-1?
(k) (l)
Soln: Just graph the functions, and apply the HLT:
(k)
Not 1-1
(l)
1-1
-> Granted that a function is 1-1, we can find it's inverse by the following procedure:
(1) Replace f(x) with y, if necessary,
(2) Solve the equation for x,
(3) Switch x and y, then set
Ex: Find the inverse of f(x), and graph both f and on the same set of axes:
(m)
Soln:
(replace f with y)
(solve for x)
(switch x, y = )
Graphing both functions, we see a telltale feature of inverse functions: they are symmetric about the identity function x. This is depicted in the second picture - y = x is the line 'running up the middle'.
(n) (in class)
Homework: (6) For the function f(x) = 3x + 2, find the inverse of f, verify that f and are inverses (plug into each other) and graph both f and on the same set of axes.
5.4 Wherein we discuss the second transcendental function, the natural exponential function .
->The constant e was discovered as the solution to an old accounting problem. It is the horizontal asymptote of the function , i.e. . It is used as the base of the natural exponential function.
->and g(x) = ln x are inverse functions. That means that following properties are true:
In essence, the e and the ln 'cancel' each other out. Moreover, this property applies when we have a more complicated argument than just x to either function:
These properties are important, as they help us to solve two types of equations:
Exponential equations involve exponentials (duh). We solve them by taking the natural log of both sides.
Ex: (o) Solve the equation:
Soln: Take the ln of both sides, and cancel the e and ln:
(p) Solve the equation:
Soln: First, we have to isolate the exponential:
Then we take the ln of both sides:
-> More rarely, we may have to solve a logarithmic equation. Again, we use the inverse properties of e and ln - we 'take e to both sides' and cancel.
Ex: (q) Solve the equation:
Soln: Take e to both sides, and isolate x:
-> Now, we look at the application of Calculus to f(x) = ex, starting with derivative rules. It has a unique property - exis it's own derivative.
rules:
Ex: (r) Find the derivative of
Soln: Here, we have to use the product rule:
(s) Find the derivative of .
Soln: Because we have something more complicated than x in the exponent, we have to use the second version of the rule:
derivative of exponent:
(t) Find the derivative of
Soln: We have to use the quotient rule:
-> Next, we need to look at the integration rules for the exponential function.
rules:
The second one, as you might guess, is used with u-substitution.
Ex: Evaluate the integral:
(u)
Soln: As the exponent is more complicated than just x, we use u-substitution. Again, we set u to be the higher-degree of the polynomials which appear.
(v) .
Soln: Seems easy, but we have to use u-substitution again, with u equal to the exponent. We also have to 'rig the constant'.
(w)
Soln: Same strategy - I'm going to set u equal to the exponent on e:
Homework: (7) Solve the exponential equation for t:
(8) Solve the exponential equation for k:
(9) Give the derivative of the function: (a) (b)
(10) Evaluate the integral:
(11) Evaluate the integral if a is any real number: