1
Suppose a particular caribou has a top speed of 70 kph, and in one year it migrates 5000 km. What do you know about the amount of time the caribou spent travelling during its migration?
Subsection 2.13.5 Exercises
Exercises — Stage 1
2
Suppose a migrating sandhill crane flew 240 kilometres in one day. What does the mean value theorem tell you about its speed during that day?
3
Below is the graph of \(y=f(x)\text{,}\) where \(x\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) Mark on the graph the approximate location of a value \(c\) guaranteed by the mean value theorem.
4
Give a function \(f(x)\) with the properties that:
- \(f(x)\) is differentiable on the open interval \(0 \lt x \lt 10\)
- \(f(0)=0\text{,}\) \(f(10)=10\)
but for all \(c \in (0,10)\text{,}\) \(f'(c)=0\text{.}\)
5
For each of the parts below, sketch a function \(f(x)\) (different in each part) that is continuous and differentiable over all real numbers, with \(f(1)=f(2)=0\text{,}\) and with the listed property, or explain why no such function exists.
- \(f'(c)=0\) for no point \(c \in (1,2)\)
- \(f'(c)=0\) for exactly one point \(c \in (1,2)\)
- \(f'(c)=0\) for exactly five points \(c \in (1,2)\)
- \(f'(c)=0\) for infinitely many points \(c \in (1,2)\)
6
Suppose you want to show that a point exists where the function \(f(x)=\sqrt{|x|}\) has a tangent line with slope \(\frac{1}{13}\text{.}\) Find the mistake(s) in the following work, and give a correct proof.
The function \(f(x)\) is continuous and differentiable over all real numbers, so the mean value theorem applies. \(f(-4)=2\) and \(f(9)=3\text{,}\) so by the mean value theorem, there exists some \(c \in (-4,9)\) such that \(f'(x) = \dfrac{3-2}{9-(-4)}=\dfrac{1}{13}\text{.}\)
Exercises — Stage 2
7 (✳)
Let \(f(x)=x^2-2\pi x+ \cos(x)-1\text{.}\) Show that there exists a real number \(c\) such that \(f'(c)=0\text{.}\)
8 (✳)
Let \(f(x)=e^x + (1-e)x^2 - 1\text{.}\) Show that there exists a real number \(c\) such that \(f'(c)=0\text{.}\)
9 (✳)
Let \(f(x)=\sqrt{3 + \sin(x)} + (x - \pi)^2\text{.}\) Show that there exists a real number \(c\) such that \(f'(c)=0\text{.}\)
10 (✳)
Let \(f(x)=x\cos(x) - x\sin(x)\text{.}\) Show that there exists a real number \(c\) such that \(f'(c)=0\text{.}\)
11
How many roots does the function \(f(x)=3x-\sin x\) have?
12
How many roots does the function \(f(x)=\dfrac{(4x+1)^4}{16}+x\) have?
13
How many roots does the function \(f(x)=x^3+\sin\left(x^5\right)\) have?
14
How many positive-valued solutions does the equation \(e^x=4\cos(2x)\) have?
15 (✳)
Let \(f(x)=3x^5-10x^3+15x+a\text{,}\) where \(a\) is some constant.
- Prove that, regardless of the value \(a\text{,}\) \(f'(x) \gt 0\) for all \(x\) in \((-1,1)\text{.}\)
- Prove that, regardless of the value \(a\text{,}\) \(f(x)=3x^5-10x^3+15x+a\) has at most one root in \([-1,1]\text{.}\)
16 (✳)
Find the point promised by the Mean Value Theorem for the function \(e^x\) on the interval \([0, T]\text{.}\)
17
Use Corollary 2.13.12 and Theorem 2.12.8 to show that \(\arcsec x=C-\arccsc x\) for some constant \(C\text{;}\) then find \(C\text{.}\)
Exercises — Stage 3
18 (✳)
Suppose \(f(0) = 0\) and \(f'(x) = \dfrac{1}{1 + e^{-f(x)}}\) . Prove that \(f(100) \lt 100\text{.}\)
Remark: an equation relating a function to its own derivative is called a differential equation. We'll see some very basic differential equations in Section 3.3.
19
Let \(f(x)=2x+\sin x\text{.}\) What is the largest interval containing \(x=0\) over which \(f(x)\) is one--to--one? What are the domain and range of \(f^{-1}(x)\text{?}\)
20
Let \(f(x)=\dfrac{x}{2}+\sin x\text{.}\) What is the largest interval containing \(x=0\) over which \(f(x)\) is one--to--one? What are the domain and range of \(f^{-1}(x)\text{,}\) if we restrict \(f\) to this interval?
21
Suppose \(f(x)\) and \(g(x)\) are functions that are continuous over the interval \([a,b]\) and differentiable over the interval \((a,b)\text{.}\) Suppose further that \(f(a) \lt g(a)\) and \(g(b) \lt f(b)\text{.}\) Show that there exists some \(c \in [a,b]\) with \(f'(c) \gt g'(c)\text{.}\)
22
Suppose \(f(x)\) is a function that is differentiable over all real numbers, and \(f'(x)\) has precisely two roots. What is the maximum number of distinct roots that \(f(x)\) may have?
23
How many roots does \(f(x)=\sin x + x^2 + 5x +1\) have?