Class 2 - Anti-derivatives / 2nd Fundamental Theorem (Thusday, 9/7/2017)
Before we talk about mathematics, here is some important logistics:
Last time, we studied the fundamental theorem of calculus:
\[\int_{a}^{b} f'(x) dx = f(x)|_{x=a}^{x=b} := f(b) - f(a),\]
which holds for any continuously differentiable functions $f$ (meaning that derivative is continuous). The left-hand side of the above identity is often referred to definite integral of $f'(x)$.
For example, take $f(x) = x^{3}/3$. Then $f'(x) = x^{2}$, so the above identity reads
\[\int_{a}^{b}x^{2}dx = x^{3}/3|_{x=a}^{x=b} = \dfrac{b^{3} - a^{3}}{3}.\]
Exercise. Compute the area between the graph $y = e^{x}$ and the $x$-axis from $x = 0$ to $x = 1$. (Hint: this is equal to $\int_{0}^{1}e^{x}dx$. Take $f'(x) = e^{x}$ in the fundamental theorem above. Then you may choose $f(x) = e^{x}$.)
Exercise. Compute the area between the graph $y = 1/x$ and the $x$-axis from $x = 1$ to $x = e$. (Hint: this is equal to $\int_{1}^{e}(1/x)dx$. Take $f'(x) = 1/x$ in the fundamental theorem above. Then you may choose $f(x) = \ln(x)$.)
The following are class notes for Thursday, Sept 7, 2017. They roughly cover Section 5.4, 6.1, 6.2, and 6.4 of your book.
Example. If $f'(x) = 0$, then we can choose $f(x) = c$ for any constant real number $c$.
Exercise. Is there any other function whose derivative is constantly $0$? (Hint: use the mean-value theorem to realize that for any distinct real numbers $a$ and $b$, you get $f(a) = f(b)$.)
Given a function $f$, we denote $\int f(x) dx$ to mean the set of all anti-derivatives of $f$. In other words, this means that
\[\int f(x)dx = \{F(x) : F'(x) = f(x)\}.\]
Examples. The previous exercise shows that
\[\int 0 dx = \{c : c \text{ is a constant}\}.\]
This implies that for any $n \geq 0$, we have
\[\int x^{n} dx = \{x^{n+1}/(n+1) + c : c \text{ is a constant}\},\]
and
\[\int 3^{x} dx = \{3^{x}/\ln(3) + c : c \text{ is a constant}\},\]
among many other similar examples.
Exercise*. Solve Parts (a) of Number 4 in Exam 1, Winter 2017. (Hint: to see whether $F(x)$ is an anti-derivative of $f(x)$, you just need to take the derivative $F'(x)$ of $f(x)$ and check $F'(x) = f(x)$.)
Exercise*. Compute the anti-derivatives of the following functions (Warning: don't forget the arbitrary constant $c$, and the last one is quite tricky to think about at first):
Suppose that $f(t)$ is a function in time variable $t$. Let $g(t)$ be the rate of the change of $f$ in at $t$. This just means that $g'(t) = f(t)$.
Again, Fundamental Theorem of Calculus. If we know $g(t)$ for all $t$ and $f(a)$ for one $a$, we can restore $f(t)$ for all $t$. With more explicit notation, we have
\[f(t) = f(a) + \int_{a}^{t}g(x)dx.\]
Case study: Part (a) of Number 2 of Exam 1, Winter 2017. Our discussion above implies that $y(t) = B(t) - B(3) = \int_{3}^{t}Z(x)dx$. Now, you can compute
Exercise*. Finish Part (a) of Number 2 we are discussing. (Hint: $B'(t) = Z(t)$, so the given graph of $Z(t)$ must present how the slope of $B(t)$ change at every point. This also means that when $Z(t)$ strictly increases $B(t)$ is concave up, and when $Z(t)$ strictly decreases $B(t)$ is concave-down.)
Second Fundamental Theorem of Calculus. If $f$ is a continuous function, then
\[\frac{d}{dx}\int_{a}^{x}f(t)dt = f(x),\]
for any constant $a$.
Exercise*. Solve Parts (c) of Number 4 in Exam 1, Winter 2017. (Hint: to see whether $F(x)$ is an anti-derivative of $f(x)$, you just need to take the derivative $F'(x)$ of $f(x)$ and check $F'(x) = f(x)$. If you have $F(x) = \int_{a}^{g(x)}f(t)dt$, then you may apply chain rule and 2nd fundamental theorem together to have $F'(x) = f(g(x))g'(x)$.)
Remark (only for people who care about proofs). The first fundamental theorem and the second fundamental theorem have very different proofs. The first one uses mean-value theorem, while the second one uses extreme-value theorem.
Your book only proves the second one, most likely because to prove the first one, you need to understand the definition of integral quite rigorously. (Our "area" definition is intuitive but rather ad-hoc, which makes it somewhat difficult if you want to be micro-analytic about math.) In case you like proofs, you can see the proof of the first fundamental theorem in Theorem 6.21 of this book.
Nevertheless, you don't need to understand any of these proofs for this class!
1st Gateway (Computer based Pass/Fail test)
Where? B069 EH
When? Sept 11 ~ Sept 25 (Hours are the same as Math Lab hours.)
You have two trials a day, and if you don't pass during the above period, then you will loose 1/3 of your letter grade (i.e., B- to C). The contents will be about derivatives, which is prerequisite for this course.
Early pass event. If you pass by 9/15, then I will give you 10% extra credit on your next quiz.
Web Homework
Here are your web homework dues:
- 5.1, 5.2, 5.3 : 9/11 Mon
- 5.4, 6.1, 6.2 : 9/13 Wed
Fundamental Theorem of Calculus
Last time, we studied the fundamental theorem of calculus:
\[\int_{a}^{b} f'(x) dx = f(x)|_{x=a}^{x=b} := f(b) - f(a),\]
which holds for any continuously differentiable functions $f$ (meaning that derivative is continuous). The left-hand side of the above identity is often referred to definite integral of $f'(x)$.
For example, take $f(x) = x^{3}/3$. Then $f'(x) = x^{2}$, so the above identity reads
\[\int_{a}^{b}x^{2}dx = x^{3}/3|_{x=a}^{x=b} = \dfrac{b^{3} - a^{3}}{3}.\]
Exercise. Compute the area between the graph $y = e^{x}$ and the $x$-axis from $x = 0$ to $x = 1$. (Hint: this is equal to $\int_{0}^{1}e^{x}dx$. Take $f'(x) = e^{x}$ in the fundamental theorem above. Then you may choose $f(x) = e^{x}$.)
Exercise. Compute the area between the graph $y = 1/x$ and the $x$-axis from $x = 1$ to $x = e$. (Hint: this is equal to $\int_{1}^{e}(1/x)dx$. Take $f'(x) = 1/x$ in the fundamental theorem above. Then you may choose $f(x) = \ln(x)$.)
The following are class notes for Thursday, Sept 7, 2017. They roughly cover Section 5.4, 6.1, 6.2, and 6.4 of your book.
Computing anti-derivatives: reverse process of taking derivative
Note that when we use the fundamental theorem stated in the beginning, we are given $f'(x)$ and our task is to find $f(x)$. Such an $f(x)$ is called an anti-derivative or an indefinite integral of $f'(x)$. Notice we said "an" instead of "the". It is because there can be many functions $f(x)$ that gives the same derivative $f'(x)$.Example. If $f'(x) = 0$, then we can choose $f(x) = c$ for any constant real number $c$.
Exercise. Is there any other function whose derivative is constantly $0$? (Hint: use the mean-value theorem to realize that for any distinct real numbers $a$ and $b$, you get $f(a) = f(b)$.)
Given a function $f$, we denote $\int f(x) dx$ to mean the set of all anti-derivatives of $f$. In other words, this means that
\[\int f(x)dx = \{F(x) : F'(x) = f(x)\}.\]
Examples. The previous exercise shows that
\[\int 0 dx = \{c : c \text{ is a constant}\}.\]
This implies that for any $n \geq 0$, we have
\[\int x^{n} dx = \{x^{n+1}/(n+1) + c : c \text{ is a constant}\},\]
and
\[\int 3^{x} dx = \{3^{x}/\ln(3) + c : c \text{ is a constant}\},\]
among many other similar examples.
Exercise*. Solve Parts (a) of Number 4 in Exam 1, Winter 2017. (Hint: to see whether $F(x)$ is an anti-derivative of $f(x)$, you just need to take the derivative $F'(x)$ of $f(x)$ and check $F'(x) = f(x)$.)
Exercise*. Compute the anti-derivatives of the following functions (Warning: don't forget the arbitrary constant $c$, and the last one is quite tricky to think about at first):
- $x \mapsto 3x^{3} + 4x + 10$;
- $x \mapsto \cos(x)$;
- $x \mapsto \sin(x)$;
- $x \mapsto e^{x}$;
- $x \mapsto a^{x}$ where $a$ is a positive constant;
- $x \mapsto 1/x$ where $a$ is a positive constant.
The exercise above is not too difficult, but the last one is somewhat tricky. That is, you are supposed to get
\[\int \frac{1}{x} dx = \ln|x| + c,\]
where $c$ denotes the arbitrary constant. The reason we need the absolute value is that when $x < 0$, we have $|x| = -x$ so that $(d/dx)\ln(|x|) = (d/dx)\ln(-x) = -(-1/x) = 1/x$. Notice that we got extra minus sign due to chain rule.
Restoring graph of a function by from the rate of it
Suppose that $f(t)$ is a function in time variable $t$. Let $g(t)$ be the rate of the change of $f$ in at $t$. This just means that $g'(t) = f(t)$.Again, Fundamental Theorem of Calculus. If we know $g(t)$ for all $t$ and $f(a)$ for one $a$, we can restore $f(t)$ for all $t$. With more explicit notation, we have
\[f(t) = f(a) + \int_{a}^{t}g(x)dx.\]
Case study: Part (a) of Number 2 of Exam 1, Winter 2017. Our discussion above implies that $y(t) = B(t) - B(3) = \int_{3}^{t}Z(x)dx$. Now, you can compute
- $y(-4) = \int_{3}^{-4}Z(x)dx = -\int_{-4}^{3}Z(x)dx = -(4 - 1) = -3$;
- $y(-1) = \int_{3}^{-1}Z(x)dx = -\int_{-1}^{3}Z(x)dx = -(1 - 1) = -0 = 0$;
- $y(3) = \int_{3}^{3}Z(x)dx = 0$;
- $y(6) = \int_{3}^{6}Z(x)dx = 0 + 2 + \pi/2 = 2 + \pi/2$.
Exercise*. Finish Part (a) of Number 2 we are discussing. (Hint: $B'(t) = Z(t)$, so the given graph of $Z(t)$ must present how the slope of $B(t)$ change at every point. This also means that when $Z(t)$ strictly increases $B(t)$ is concave up, and when $Z(t)$ strictly decreases $B(t)$ is concave-down.)
Second Fundamental Theorem of Calculus
The second fundamental theorem looks a lot like the first one we have discussed in the beginning.Second Fundamental Theorem of Calculus. If $f$ is a continuous function, then
\[\frac{d}{dx}\int_{a}^{x}f(t)dt = f(x),\]
for any constant $a$.
Exercise*. Solve Parts (c) of Number 4 in Exam 1, Winter 2017. (Hint: to see whether $F(x)$ is an anti-derivative of $f(x)$, you just need to take the derivative $F'(x)$ of $f(x)$ and check $F'(x) = f(x)$. If you have $F(x) = \int_{a}^{g(x)}f(t)dt$, then you may apply chain rule and 2nd fundamental theorem together to have $F'(x) = f(g(x))g'(x)$.)
Remark (only for people who care about proofs). The first fundamental theorem and the second fundamental theorem have very different proofs. The first one uses mean-value theorem, while the second one uses extreme-value theorem.
Your book only proves the second one, most likely because to prove the first one, you need to understand the definition of integral quite rigorously. (Our "area" definition is intuitive but rather ad-hoc, which makes it somewhat difficult if you want to be micro-analytic about math.) In case you like proofs, you can see the proof of the first fundamental theorem in Theorem 6.21 of this book.
Nevertheless, you don't need to understand any of these proofs for this class!
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