MAT1001 Differential Calculus: Lecture Notes

Problem 11.1.1 (\(\star \)). Find the roots of the polynomial

\begin{equation*} g(x)=x^2-2x-12 \end{equation*}

and plot the function.

Problem 11.1.2. Find the roots of the polynomial

\begin{equation*} g(x)=x^3+x^2-x-1 \end{equation*}

and plot the function.

Problem 11.1.3 (\(\star \)). Consider the function

\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

\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

\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 11.1.6 (\(\star \)). Given the function

\begin{equation*} f(x)=x^3-2x^2-x+2 \end{equation*}

find:

• • \(f(0)\),

• • \(f(1)\),

• • \(f(-1)\),

• • \(f(2)\),

• • \(f(-2)\),

• • \(f(t)\),

• • \(f(x-1)\).

Problem 11.1.7. Consider the function

\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

\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

\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

\begin{equation*} h(x)=\frac {x+4}{2x-5}, \end{equation*}

identify when it has an inverse and calculate the inverse.

Problem 11.1.11. Consider the function

\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

\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

\begin{equation*} \N =\{1,2,3,4,5,\dots \}, \end{equation*}

and the integers

\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.

Problem 11.2.1 (\(\star \)). For the polynomial equation

\begin{equation*} x^{3}-3x+2=0, \end{equation*}

express it as a product of its factors.

Hint: this means write it as

\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.

Problem 11.2.2. Find the roots of the following polynomial equation

\begin{equation*} x^{4}-4x^{3}+6x^{2}-4x+1=0. \end{equation*}

Problem 11.2.3. Find the solutions to

\begin{equation*} \sqrt {2}\cos x =1. \end{equation*}

Problem 11.2.4. Identify any solutions to the equation

\begin{equation*} \sin (2x)=-2. \end{equation*}

Problem 11.2.5 (\(\star \)). Solve

\begin{equation*} 2x\sin x = x. \end{equation*}

Problem 11.2.6 (\(\star \)). Solve the equation

\begin{equation*} x=xe^{4x}. \end{equation*}

Problem 11.2.7 (\(\star \)). Solve

\begin{equation*} 2\ln x -\ln (x+2)=1. \end{equation*}

Problem 11.2.8 (\(\star \)). For the function

\begin{equation*} f(x)=\frac {x^{2}+4x+12}{x^{2}-2x} \end{equation*}

evaluate the limit

\begin{equation*} \lim _{x\to 2}f(x). \end{equation*}

Problem 11.2.9 (\(\star \)). For the function

\begin{equation*} g(x)=\frac {2x^{4}-x^{2}+8x}{7-5x^{4}} \end{equation*}

evaluate the limits

\begin{align*} &\lim _{x\to -\infty }g(x),\\ &\lim _{x\to \infty }g(x). \end{align*}

Problem 11.2.10 (\(\dagger \)). For the function

\begin{equation*} h(x)=\frac {6e^{4x}-e^{-2x}}{8e^{4x}-2e^{2x}+3e^{-x}} \end{equation*}

evaluate the limit

\begin{align*} &\lim _{x\to \infty }h(x). \end{align*}

Problem 11.2.11. Using an appropriate plot evaluate the limits

\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

\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 \(g(x)\)?

Problem 11.2.13. By making a table of values of \(f(x), x\) estimate the limit

\begin{equation*} \lim _{x\to 2}f(x) \end{equation*}

from above and below for the function

\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

\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.

Problem 11.2.15 (\(\dagger \)). For the function \(g(x)\) in the previous problem look at the definition of differentiability and work out where \(g(x)\) fails to be differentiable.

Problem 11.2.16 (\(\dagger \)). Show that

\begin{equation*} \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y. \end{equation*}

Problem 11.2.17 (\(\dagger \)). Following some of the examples in section 2.4 show that

\begin{equation*} \cosh ^{-1}x=\ln \left (x\pm \sqrt {x^{2}+1}\right ). \end{equation*}

Problem 11.3.1 (\(\star \)). Find the derivative of

\begin{equation*} f(x)=7x^{2} \end{equation*}

• a) using first principles,

• b) using the rule for differentiating monomials.

Problem 11.3.2. Find the derivative of

\begin{equation*} f(x)=3x^{3}-8x. \end{equation*}

Problem 11.3.3 (\(\star \)). Find the derivative of

\begin{equation*} f(x)=(x+4)(x+2). \end{equation*}

Problem 11.3.4 (\(\star \)). Find the derivative of

\begin{equation*} g(x)=6x^{4}-x^{3}. \end{equation*}

Problem 11.3.5. Find the derivative of

\begin{equation*} y=\frac {x^{9}}{3}+\frac {x^{4}}{4}. \end{equation*}

Problem 11.3.6 (\(\star \)). Calculate the derivative of

\begin{equation*} y=\left (x^{4}-3\right )^{2}. \end{equation*}

Problem 11.3.7 (\(\dagger \)). Consider the function

\begin{equation*} h(x)=\frac {x}{\sqrt {x^{2}-9}}, \end{equation*}

when can we calculate its derivative? When it exists calculate the derivative.

Problem 11.3.8. Identify when we can differentiate the function

\begin{equation*} y=\frac {3x^{3}\left (x^{2}-4x\right )}{x} \end{equation*}

and calculate its derivative.

Problem 11.3.9 (\(\star \)). Identify when we can differentiate the function

\begin{equation*} f(x)=9x^{4}-\frac {4}{x^{3}} \end{equation*}

and calculate its derivative.

Problem 11.3.10 (\(\star \)). Differentiate the function

\begin{equation*} f(x)=\sin (2x). \end{equation*}

Problem 11.3.11. Differentiate the function

\begin{equation*} f(x)=\cos (3x). \end{equation*}

Problem 11.3.12 (\(\star \)). Differentiate the following expressions

• a) \(\cos (x)-\sin (x)\)

• b) \(3\tan (x)-\cos (2x)\)

• c) \(4\sin (2x)+2\cos (5x)+5\)

• d) \(\ln (x+3)\)

• e) \(x\ln (x^{2})\)

Problem 11.3.13. Find the gradient of the following curves at the indicated points

• a) \(y=5\sin (2x)\) at the point \((x,y)=(\uppi /2,0)\)

• b) \(y=\tan (x/2)\) at the point \((x,y)=(\uppi /2,1)\)

Problem 11.3.14. Find the stationary points, the points where the derivative vanishes, for the following functions

• a) \(y=\sin (2x)+\cos (x)\), with \(0\leq x\leq \uppi \)

• b) \(f(x)=\log (\sin (x))\) for \(0\leq x\leq \uppi \)

Problem 11.3.15 (\(\dagger \)). Calculate the derivative of

\begin{equation*} y=x^{x}. \end{equation*}

Hint: You may want to take the logarithm of the function.

Problem 11.4.1 (\(\star \)). Solve the following, leaving your answer in terms of logs:

• a) \(2^{x+4}=6\)

• b) \(3^{2x-1}=17\)

• c) \(2^{1-4x}=5\)

• d) \(5^{3x+4}=31\)

Problem 11.4.2. Solve \(0.6=2^{-x}\).

Problem 11.4.3. The point \((K,5)\) lies on the curve \(y=2^{x}\). Find \(K\).

Problem 11.4.4 (\(\star \)). Simplify the expressions:

• a) \(\ln e^{\sin (x)}\)

• b) \(e^{2\ln (1+x)}\)

• c) \(e^{-\ln (5-x)}\)

Problem 11.4.5. Solve \(e^{2x}-5e^{x}+4=0\).

Problem 11.4.6 (\(\star \)). Evaluate the following limits:

• a) \(\lim _{x\to \uppi /2}\left (\frac {x}{1+\sin (x)}\right )\)

• b) \(\lim _{x\to 1}\left (\frac {\ln (x)}{1+\ln (x)}\right )\)

• c) \(\lim _{x\to \infty }\left (\frac {3+2x}{2+3x}\right )\)

• d) \(\lim _{x\to \uppi ^{-}/2}\left (\frac {x}{\tan {x}}\right )\)

• e) \(\lim _{x\to \uppi ^{+}/2}\left (\frac {x}{\tan {x}}\right )\)

Problem 11.5.1 (\(\star \)). Given the function

\begin{equation*} h(x)=\frac {x+4}{2x-5} \end{equation*}

identify when you can differentiate it and find its derivative.

Problem 11.5.2 (\(\star \)). Calculate the derivative of

\begin{equation*} y=x^{5}\sin (x) \end{equation*}

using the product rule.

Problem 11.5.3. Calculate the derivative of

\begin{equation*} f(x)=3e^{x}\cos (x) \end{equation*}

using the product rule.

Problem 11.5.4 (\(\star \)). Calculate the derivative of

\begin{equation*} f(x)=\left (10x-3\right )^{4} \end{equation*}

using the chain rule.

Problem 11.5.5. Use the quotient rule to calculate the derivative of

\begin{equation*} y=\tan (x)=\frac {\sin (x)}{\cos (x)}. \end{equation*}

Does it match your expectation?

Problem 11.5.6 (\(\star \)). Use the quotient rule to calculate the derivative of

\begin{equation*} f(x)=\frac {3x^{3}+8x^{2}+2}{2x+1}. \end{equation*}

Problem 11.5.7. Calculate the derivative of

\begin{equation*} f(x)=\ln (x^{2}+1). \end{equation*}

Problem 11.5.8 (\(\star \)). By identifying a function whose derivative is \(1/x\) solve the integral

\begin{equation*} I=\int \frac {1}{x}\ud x. \end{equation*}

Problem 11.5.9 (\(\star \)). Find the antiderivative of \(e^{x}\)

Problem 11.5.10 (\(\star \)). Find the antiderivatives of the following functions:

• a) \(\cos (x)\)

• b) \(x^{2}+2x\)

• c) \(\sqrt {X^{2}+2}\)

• d) \(x^{2}+1/x^{2}\)

Problem 11.5.11. Calculate the integral

\begin{equation*} I=\int \left (5x^{2}-8x+5\right )\ud x. \end{equation*}

Problem 11.5.12 (\(\star \)). Calculate the integrals:

• a) \(I=\int \left (x^{\frac {3}{2}}+2x+3\right )\ud x\)

• b) \(I=\int \left (\frac {8}{x}-\frac {5}{x^{2}}+\frac {6}{x^{3}}\right )\ud x\)

• c) \(I=\int \left (4e^{-7x}\right )\ud x\)

Problem 11.5.13 (\(\star \)). Calculate the integrals:

• a) \(I=\int \frac {x^{3}+4}{x^{2}}\ud x\)

• b) \(I=\int \left (12x^{\frac {3}{4}}-9x^{\frac {5}{3}}\right )\ud x\)

• c) \(I=\int 7\sin (x)\ud x\)

• d) \(I=\int 5\cos (x)\ud x\)

Problem 11.5.14 (\(\star \)). Calculate the following definite integrals:

• a) \(I=\int ^{4}_{1} \left (5x^{2}-8x+5\right )\ud x\)

• b) \(I=\int ^{9}_{1}\left (x^{\frac {3}{2}}+2x+3\right )\ud x\)

• c) \(I=\int ^{\frac {\uppi }{2}}_{0}\left (\frac {8}{x}-\frac {5}{x^{2}}+\frac {6}{x^{3}}\right )\ud x\)

• d) \(I=\int _{\frac {\uppi }{2}}^{\frac {3\uppi }{2}}\left (4e^{-7x}\right )\ud x\)

Problem 11.5.15. Evaluate the integral

\begin{equation*} I=\int \frac {3x}{4x-5}\ud x. \end{equation*}

Problem 11.5.16. Use integration by parts to evaluate

\begin{equation*} I=\int xe^{-2x}\ud x. \end{equation*}

Problem 11.5.17 (\(\dagger \)). Look at the definition of differentiation from first principles and prove the product rule from first principles.

Problem 11.5.18 (\(\star \)). Consider the function

\begin{equation*} f(x)=x^{2}+2x \end{equation*}

calculate its derivative. Is the derivative that you find a differentiable function? If it is, calculate its derivative, what do you get?

Do the same for \(f(x)=\sin (x)\), what do you notice here?

Problem 11.6.1. Calculate the integral

\begin{equation*} I=\int \left (5x^{2}-8x+5\right )\ud x. \end{equation*}

Problem 11.6.2. Calculate the integrals:

• a) \(I=\int \left (x^{\frac {3}{2}}+2x+3\right )\ud x\)

• b) \(I=\int \left (\frac {8}{x}-\frac {5}{x^{2}}+\frac {6}{x^{3}}\right )\ud x\)

• c) \(I=\int \left (4e^{-7x}\right )\ud x\)

Problem 11.6.3 (\(\star \)). Calculate the integrals:

• a) \(I=\int \frac {x^{3}+4}{x^{2}}\ud x\)

• b) \(I=\int \left (12x^{\frac {3}{4}}-9x^{\frac {5}{3}}\right )\ud x\)

• c) \(I=\int 7\sin (x)\ud x\)

• d) \(I=\int 5\cos (x)\ud x\)

Problem 11.6.4 (\(\star \)). Calculate the following definite integrals:

• a) \(I=\int ^{4}_{1} \left (5x^{2}-8x+5\right )\ud x\)

• b) \(I=\int ^{9}_{1}\left (x^{\frac {3}{2}}+2x+3\right )\ud x\)

• c) \(I=\int ^{\frac {\uppi }{2}}_{0}\left (\frac {8}{x}-\frac {5}{x^{2}}+\frac {6}{x^{3}}\right )\ud x\)

• d) \(I=\int _{\frac {\uppi }{2}}^{\frac {3\uppi }{2}}\left (4e^{-7x}\right )\ud x\)

Problem 11.6.5 (\(\star \)). Use an appropriate substitution to evaluate the following integrals:

• a) \(I=\int \left (x+5\right )^{6}\ud x\)

• b) \(I=\int \left (3-x\right )^{5}\ud x\)

• c) \(I=\int x\left (x^{2}-2\right )^{4}\ud x\)

• d) \(I=\frac {1}{2}\int x^{2}\left (3-x^{3}\right )^{5}\ud x\)

• e) \(I=\int 2x^{2}\left (x^{3}+1\right )^{3}\ud x\)

• f) \(I=\int \frac {2}{\left (x+7\right )^{3}}\ud x\)

• g) \(I=\int \left (\frac {x^{2}}{\sqrt {x^{3}+1}}\right )\ud x\)

• h) \(I=\int \left (\frac {3x^{2}}{(x^{3}-7)^{5}}\right )\ud x\)

• i) \(I=\int \left (\frac {x^{2}}{\sqrt {2x^{3}-3}}\right )\ud x\)

• j) \(I=\int \left (\frac {1}{\sqrt {x+4}}\right )\ud x\)

• k) \(I=\int \left (\frac {x}{(x^{2}-3)^{2}}\right )\ud x\)

• l) \(I=\int \left (\frac {x}{(3x^{2}+2)^{4}}\right )\ud x\)

Problem 11.6.6. Calculate the following integrals:

• a) \(I=\int \left (x-3\right )^{5}\) using \(u=x-3\)

• b) \(I=\int x\left (x+1\right )^{3}\ud x\) using \(u=x+1\)

• c) \(I=\int \frac {3x}{\left (4x-5\right )^{2}} \ud x\) using \(u=4x-5\)

• d) \(I=\int \frac {2x}{\sqrt {x+2}}\ud x\) using \(u=x+2\)

• e) \(I=\int \frac {8x^{2}}{(x^{3}-3)^{2}}\ud x\) using \(u=x^{3}-3\)

• f) \(I=\int \frac {x}{\sqrt {3x^{2}-1}}\ud x\) using \(u=3x^{2}-1\)

Hint: Some of these problems require you to use integration by parts as well as a substitution.

Problem 11.6.7 (\(\star \)). Calculate the following integrals:

• a) \(I=\int ^{1}_{0}\left (x-1\right )^{5}\ud x\)

• b) \(I=\int ^{1}_{0}\frac {x}{(x^{2}+8)^{3}}\ud x\) using \(u=x^{2}+8\)

• c) \(I=\int ^{0}_{-1}x\left (x^{2}-3\right )^{4}\ud x\) using \(u=x^{2}-3\)

• d) \(I=\int _{-1}^{1}\frac {3}{\sqrt {x+2}}\ud x\) using \(u=x+2\)

• e) \(I=\int _{0}^{2}x^{2}\left (7-x^{3}\right )^{3}\ud x\) using \(u=7-x^{3}\)

• f) \(I=\int ^{3}_{2}\frac {x}{\sqrt {x-1}}\ud x\) using integration by parts and \(u=x-1\)

Problem 11.6.8 (\(\star \)). Use integration by parts to calculate the following integrals

• a) \(I=\int x\cos (x)\ud x\)

• b) \(I=\int x\sin (3x)\ud x\)

• c) \(I=\int 2xe^{\frac {x}{3}}\ud x\)

• d) \(I=\int xe^{2x}\ud x\)

• e) \(I=\int 3xe^{5x}\ud x\)

• f) \(I=\int \frac {x}{e^{x}}\ud x\)

Problem 11.6.9. Evaluate the following integrals:

• a) \(I=\int e^{x}\cos (x)\ud x\)

• b) \(I=\int e^{2x}\sin (x)\ud x\)

• c) \(I=\int e^{x}\sin (4x)\ud x\)

Problem 11.6.10 (\(\star \)). Evaluate the following integrals:

• a) \(I=\int ^{\infty }_{1} x^{2}\ud x\)

• b) \(I=\int ^{\infty }_{0}4e^{-2x}\ud x\)

• c) \(I=\int ^{\infty }_{2}\frac {1}{x^{5}}\ud x\)

• d) \(I=\int _{0}^{\infty }xe^{-3x}\ud x\)

• e) \(I=\int _{4}^{\infty }\frac {1}{\sqrt {x}}\ud x\)

• f) \(I=\int ^{0}_{-\infty }e^{x}\ud x\)

• g) \(I=\int ^{0}_{-2}\frac {1}{\sqrt {x+2}}\ud x\)

• h) \(I=\int _{-2}^{2}\frac {1}{x^{\frac {2}{3}}}\ud x\)

Problem 11.6.11. Consider the following integrals and decide if they converge or not:

• a) \(I=\int ^{1}_{0} \frac {1}{x^{2}}\ud x\)

• b) \(I=\int ^{\frac {\uppi }{2}}_{0}\tan (x)\ud x\)

• c) \(I=\int ^{0}_{-\infty }\cos (x)\ud x\)

• d) \(I=\int _{1}^{\infty }\frac {1}{x^{2}}\ud x\)

Problem 11.6.12. Evaluate the integral

\begin{equation*} I=\int xe^{-ax^{2}}\ud x \end{equation*}

for an arbitrary constant \(a\).

Problem 11.6.13. Consider the function \(f(x)=x^{2}-4\) on the interval \([0,2]\). Use \(6\) subintervals and the midpoint approach to estimate the area under the curve. Then carry out the definite integral and see how close your got with the approximation.

Problem 11.6.14. Evaluate the integral

\begin{equation*} I=\int _{-\frac {\uppi }{2}}^{\frac {\uppi }{2}}\sin \left (\vert x\vert \right )\ud x. \end{equation*}

Hint: You want to think about the behaviour of \(\vert x\vert \) over the range of integration.

Problem 11.6.15 (\(\star \)). The average of a function over the interval \([a,b]\) is given by the integral

\begin{equation*} f_{\text {avg}}=\frac {1}{b-a}\int _{a}^{b}f(x)\ud x. \end{equation*}

Find the average of the following functions:

• a) \(\cos (2x)\) over the integral \([-\uppi /4,\uppi /4]\).

• b) \(\sin (x)\) over the interval \([0,2\uppi ]\).

• c) \(x^{2}\) over the interval \([-1,1]\).

Problem 11.7.1 (\(\dagger \)). Find \(f(2)\) for the data \(f(0)=1\), \(f(1)=3\), \(f(3)=55\) using Newton’s divided difference method.

Hint: Are the step sizes the same for all of the differences?

Problem 11.7.2 (\(\star \)). Find \(f(3)\) using Newton’s divided difference method for the data in the table 11.1

Problem 11.7.3. Find \(f(0.25)\) using Newton’s divided difference method for the data in the table 11.2

Problem 11.7.4 (\(\star \)). Find the interpolating polynomial for the data in table 11.3 and find \(f(4.3)\).

Problem 11.7.5 (\(\dagger \)). A sequence is defined by \(x_{n+1}=\sqrt {28-3x_{n}}\) with \(x_{0}=3\).

• • Find \(x_{1},x_{2},x_{3}\) and explain why they are all positive.

• • Given that there is a limit \(L\) to this sequence, show that the limit satisfies the equation \(L^{2}+3L-28=0\) and find the value of \(L\).

Problem 11.7.6 (\(\star \)). Show that the equation

\begin{equation*} x^{5}+7x^{3}-2=0 \end{equation*}

has a root between \(0.6\) and \(0.7\).

Problem 11.7.7 (\(\star \)). Show that the equation

\begin{equation*} 3x-4\cos (x)=0 \end{equation*}

has a root between \(0.8\) and \(0.9\).

Problem 11.7.8. Two students are attempting to use the Newton-Raphson method to solve \(x^{3}-3x+4=0\). Stduent A decides to use \(x_{0}=-1\) and student B decides to use \(x_{0}=-3\) as a first approximation. Explain why one of the studnets will be successful in finding a root while the other will not.

Problem 11.7.9 (\(\star \)). The equation

\begin{equation*} 4x^{3}-5x^{2}+2=0 \end{equation*}

has a single real root \(\alpha \). Use the Newton-Raphson method with first approximation \(x_{0}=-0.5\) to find \(x_{2} \) and \(x_{3}\) to four decimal places.

Problem 11.7.10 (\(\star \)). Evaluate the following integrals:

• a) \(I=\int ^{\infty }_{1} x^{2}\ud x\)

• b) \(I=\int ^{\infty }_{0}4e^{-2x}\ud x\)

• c) \(I=\int ^{\infty }_{2}\frac {1}{x^{5}}\ud x\)

• d) \(I=\int _{0}^{\infty }xe^{-3x}\ud x\)

• e) \(I=\int _{4}^{\infty }\frac {1}{\sqrt {x}}\ud x\)

• f) \(I=\int ^{0}_{-\infty }e^{x}\ud x\)

• g) \(I=\int ^{0}_{-2}\frac {1}{\sqrt {x+2}}\ud x\)

• h) \(I=\int _{-2}^{2}\frac {1}{x^{\frac {2}{3}}}\ud x\)

Problem 11.7.11. Consider the following integrals and decide if they converge or not:

• a) \(I=\int ^{1}_{0} \frac {1}{x^{2}}\ud x\)

• b) \(I=\int ^{\frac {\uppi }{2}}_{0}\tan (x)\ud x\)

• c) \(I=\int ^{0}_{-\infty }\cos (x)\ud x\)

• d) \(I=\int _{1}^{\infty }\frac {1}{x^{2}}\ud x\)

Problem 11.7.12. Starting from \(x_{0}=1\) apply Newton’s method to find the solution to \(f(x)=\sqrt [3]{x}=0\).

Problem 11.7.13. Starting from \(x_{0}=1\) apply Newton’s method to find the solution to

\begin{equation*} 40x=e^{x}, \end{equation*}

to four decimal places.

Problem 11.7.14. Write a computer program to implement Newton’s method and test it out on the examples from this tutorial sheet.

Problem 11.8.1 (\(\star \)). Consider the integral

\begin{equation*} I=\int _{0}^{2}3^{x}\ud x \end{equation*}

Evaluate this using:

• a) The midpoint rule with \(4\) strips.

• b) The Trapezium rule with \(4\) strips.

• c) Simpson’s rule with \(4\) strips.

Problem 11.8.2 (\(\star \)). Consider the integral

\begin{equation*} I=\int _{-1}^{1}\sqrt {x^{3}+1}\ud x \end{equation*}

Evaluate this for \(3\) strips using:

• a) The midpoint rule.

• b) The Trapezium rule.

• c) Simpson’s rule.

Problem 11.8.3. Consider the integral

\begin{equation*} I=\int _{0}^{1}\frac {1}{x^{2}+1}\ud x \end{equation*}

Evaluate this for \(9\) strips using:

• a) The midpoint rule.

• b) The Trapezium rule.

• c) Simpson’s rule.

Problem 11.8.4 (\(\star \)). Consider the integral

\begin{equation*} I=\int _{0}^{1}\sqrt {x(2x-1)}\ud x \end{equation*}

Evaluate this for \(4\) strips using:

• a) The midpoint rule.

• b) The Trapezium rule.

• c) Simpson’s rule.

Problem 11.8.5. Consider the integral

\begin{equation*} I=\int _{1}^{2}\ln \left (1+\sqrt {x}\right )\ud x \end{equation*}

Evaluate this for \(5\) strips using:

• a) The midpoint rule.

• b) The Trapezium rule.

• c) Simpson’s rule.

Problem 11.8.6. Use Simpson’s rule with \(6\) strips to calculate

\begin{equation*} I=\int _{0}^{2}\sqrt {1+\sin (x)+\cos (x)}\ud x \end{equation*}

to \(6\) decimal places.

Problem 11.8.7. Write a computer program to implement Simpson’s rule and use it to check the problems on this sheet.

Problem 11.9.1. Given \(f(x)=\cos (x)\),

• a) find \(f'\left (\frac {\uppi }{3}\right )\) using the forward difference method with \(h=0.1,0.01, 0.001, 0.0001\).

• b) Now find \(f'\left (\frac {\uppi }{3}\right )\) using the backward difference method with \(h=0.1,0.01, 0.001, 0.0001\).

• c) Calculate the exact value of the derivative at \(x=\frac {\uppi }{3}\) and compare it to the approximations.

Problem 11.9.2 (\(\star \)). Use the forward difference method with \(h=0.05\) to approximate the derivative of \(f(x)=4e^{2x}\) at \(x=1\) and compare this to the exact result.

Problem 11.9.3 (\(\star \)). Given \(f(x)=\cos (x)\) find the value of the derivative at \(x=\frac {\uppi }{4}\) with step sizes \(h=0.1\) and \(h=0.05\) using:

• a) The forward difference method.

• b) The backwards difference method.

• c) The central difference method.

Problem 11.9.4. Consider the function \(f(x)=\ln (x)\) for step size \(h=0.1\), use the central difference method to find the value of the derivative of \(f(x)\) at \(x=\frac {1}{2}\).

Problem 11.9.5. Find the second derivative of \(f(x)=x^{2}\).

Problem 11.9.6 (\(\star \)). Find the first and second derivative of \(f(x)=\cos (x)\) and evaluate these at \(x=0\). Then compare the values of \(f(x)\) near zero to the values of \(f(0)+f'(0)x+\frac {1}{2}f''(0)x^{2}\) near \(x=0\).

Problem 11.9.7. Using the definition of the derivative and the forward difference method find a numerical expression for the second order derivative.

Problem 11.9.8. Write a computer programme to implement the three different numerical differentiation methods and use it to check the problems on this sheet.

Problem 11.10.1 (\(\star \)). For the function \(f(x)=x^{3}-3x\) find and classify the critical points.

Problem 11.10.2. Find the critical points of the function

\begin{equation*} f(x)=x^{4}-3x^{2}+2. \end{equation*}

Problem 11.10.3 (\(\star \)). By considering the first and second derivatives of \(f(x)=\sin (x)\) find the maxima and minima.

Problem 11.10.4. Suppose that the population of a certain type of insect after \(t\) months is given by the formula

\begin{equation*} P(t)=3t+\sin (4t)+100 \end{equation*}

determine the minimum and maximum population in the first four months.

Problem 11.10.5. Suppose we have a model \(\hat {y}=wx\) with one parameter \(w\), that takes an input value \(x\), gives an output of \(\hat {y}\), and has a target output of \(y\) with loss function

\begin{equation*} \lambda (w)=\left (wx-y\right )^{2}. \end{equation*}

If the input value is \(x=2\) and the target output is \(y=5\), find the value of the parameter \(w\) which minimises the loss function. Then find the optimal output.

Problem 11.10.6. Consider the one parameter model

\begin{equation*} \hat {y}=e^{kx} \end{equation*}

with input \(x\), output \(\hat {y}\), target output \(y\), and parameter \(k\). If we take the loss function to be

\begin{equation*} \lambda (w)=\left (e^{kx}-y\right )^{2} \end{equation*}

with input value \(x=1\) and target output \(y=3\), minimise the loss function and find the output that corresponds to the minimum value of \(k\).

Problem 11.10.7 (\(\dagger \)). In the previous two problems does it matter if we change the loss function? What do you think would happen if our model had more than one parameter in it?