MAT1001 Differential Calculus: Lecture Notes
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Chapter 11 Tutorial Sheets
Here we collect all of the tutorial problems for the module. They are split into different weeks depending on the topic they relate to and when they were given out.
Many of these questions are taken from or adapted from the recommended books for the module or from some of the linked resources. These problems are to be attempted in the tutorial sessions and are there to help you familiarise yourself with the material that we have
covered in the lectures.
Problems marked with a star, \((\star )\) are particularly worth attempting. Problems marked with a dagger, \((\dagger )\), are more challenging and often go beyond what we directly discussed in the lectures.
The challenge problem sections contain extra problems. Some of them are just there for extra practice, but others are significantly more difficult than what you need to be able to solve to pass the module. If you are finding the content too easy then have a go at the
challenge problems. Sometime the challenge problems from one week will be quite similar to the ordinary problems of the next week. as the problems will become more accessible the more material that we cover.
11.1 Week 1
Functions
-
Problem 11.1.3 (\(\star \)). Consider the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
g(x)=x^2+2x+2,
\end{equation*}
produce a plot of the function by calculating its value at a selection of points. What do you notice as \(x\) gets very large? What happens for \(x=0\)?
-
Problem 11.1.4 (\(\star \)). Consider the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
g(x)=\frac {1}{x-4},
\end{equation*}
and plot the function.what happens as \(x\) gets large? What happens as \(x\) approaches \(4\)?
Plot the function and comment on its behaviour.
-
Problem 11.1.5. Draw a schematic of a one-to-one function between the sets
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\begin{split} &\left \{1,2,3,4,5\right \}\\ &\left \{a,b,c,d,e,f,g\right \}. \end {split}
\end{equation*}
Is there only one way to do this?
-
Problem 11.1.7. Consider the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
h(x)=\frac {x}{\sqrt {x^{2}-9}}.
\end{equation*}
Find the points where the denominator vanishes, then plot the function avoiding these points. What happens to the plot as the function approaches these points?
-
Problem 11.1.8. Given the functions
\(\seteqnumber{0}{11.}{0}\)
\begin{align*}
f(x)&=x^2-x+1,\\ g(x)&=2-x,
\end{align*}
find:
-
• \(\left (f\circ g\right )(2)\),
-
• \(\left (g\circ f\right )(2)\),
-
• \(\left (f\circ g\right )(x)\),
-
• \(\left (g\circ f\right )(x)\).
-
Problem 11.1.9 (\(\star \)). Given the functions
\(\seteqnumber{0}{11.}{0}\)
\begin{align*}
f(x)&=3x-2\\ g(x)&=\frac {x}{3}+\frac {2}{3},
\end{align*}
find:
-
• \(\left (f\circ g\right )(x)\),
-
• \(\left (g\circ f\right )(x)\),
-
• What is the relationship between \(f(x)\) and \(g(x)\)?
-
Problem 11.1.10 (\(\dagger \)). Given the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
h(x)=\frac {x+4}{2x-5},
\end{equation*}
identify when it has an inverse and calculate the inverse.
Challenge Problems
-
Problem 11.1.11. Consider the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
f(x)=\frac {x^{2}-x-12}{x-1}.
\end{equation*}
Identify the points where the numerator and denominator vanish. Plot the function and explain what happens to the function near these points.
-
Problem 11.1.12. Given the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
f(x)=2x-3,
\end{equation*}
find the inverse function \(f^{-1}(x)\).
-
Problem 11.1.13 (\(\dagger \dagger \)). Build a schematic of a bijection between the natural
numbers
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\N =\{1,2,3,4,5,\dots \},
\end{equation*}
and the integers
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\Z =\{0,1,-1,2,-2,3,-3,\dots \}.
\end{equation*}
Are there more integers than natural numbers?
Could you do the same for the real numbers \(\R \)? If you find this interesting you may want to explore the work of Cantor.
11.2 Week 2
Polynomials
-
Problem 11.2.1 (\(\star \)). For the polynomial equation
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
x^{3}-3x+2=0,
\end{equation*}
express it as a product of its factors.
Hint: this means write it as
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\left (x-a\right )\left (x-b\right )\left (x-c\right ),
\end{equation*}
where \(a,b,c\) are the roots of the polynomial.
Trig functions
Remember to work in radians for any problems related to trigonometry.
Exponentials and Logarithms
Limits and Asymptotes
-
Problem 11.2.8 (\(\star \)). For the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
f(x)=\frac {x^{2}+4x+12}{x^{2}-2x}
\end{equation*}
evaluate the limit
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\lim _{x\to 2}f(x).
\end{equation*}
-
Problem 11.2.9 (\(\star \)). For the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
g(x)=\frac {2x^{4}-x^{2}+8x}{7-5x^{4}}
\end{equation*}
evaluate the limits
\(\seteqnumber{0}{11.}{0}\)
\begin{align*}
&\lim _{x\to -\infty }g(x),\\ &\lim _{x\to \infty }g(x).
\end{align*}
-
Problem 11.2.10 (\(\dagger \)). For the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
h(x)=\frac {6e^{4x}-e^{-2x}}{8e^{4x}-2e^{2x}+3e^{-x}}
\end{equation*}
evaluate the limit
\(\seteqnumber{0}{11.}{0}\)
\begin{align*}
&\lim _{x\to \infty }h(x).
\end{align*}
-
Problem 11.2.11. Using an appropriate plot evaluate the limits
\(\seteqnumber{0}{11.}{0}\)
\begin{align*}
&\lim _{x\to \frac {\pi }{2}}\tan x,\\ &\lim _{x\to 0}\tan x.
\end{align*}
-
Problem 11.2.12 (\(\star \)). Determine where the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
g(x)=\frac {x^{3}+x^{2}-x-1}{x^{3}-2x^{2}-4x+8}
\end{equation*}
fails to be continuous. What do these points correspond to on a plot of \(g(x)\)?
Challenge Problems
-
Problem 11.2.13. By making a table of values of \(f(x), x\) estimate the limit
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
\lim _{x\to 2}f(x)
\end{equation*}
from above and below for the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
f(x)=\frac {x^{2}+4x-12}{x^{2}-2x}.
\end{equation*}
-
Problem 11.2.14. Use the definition of continuity to determine in the function
\(\seteqnumber{0}{11.}{0}\)
\begin{equation*}
g(x)=\frac {4x+10}{x^{2}-2x-15}
\end{equation*}
is continuous and if not find the points where it has discontinuities.
Hyperbolic Trig functions
11.3 Week 3
Differentiation
Challenge Problems