More Applications of Differentiation - Exercises

Below you will find the exercises we suggest you work on in connection with the More Applications of Differentiation theme and the project problems which are to be done for the exercise classes 23-27 September.

Section 4.1: Related rates
4.1.1, 4.1.3, 4.1.6, 4.1.18.

Section 4.2: Finding roots of equations
4.2.1, 4.2.3, 4.2.8, 4.2.11.

Section 4.3: Indeterminate forms
4.3.1, 4.3.3, 4.3.12, 4.3.23, 4.3.26, 4.3.32.

Section 4.4: Extreme values
4.4.1, 4.4.3, 4.4.6, 4.4.9, 4.4.19, 4.4.25, 4.4.32.

Section 4.5: Concavity and inflections
4.5.1, 4.5.3, 4.5.4, 4.5.17, 4.5.28, 4.5.35.

Section 4.6: Sketching the graph of a function
4.6.7, 4.6.11, 4.6.16, 4.6.23.

Section 4.7: Graphing with computers
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Section 4.8: Extreme-value problems
4.8.1, 4.8.2, 4.8.13, 4.8.28, 4.8.31, 4.8.39, 4.8.49.

Section 4.9: Linear approximations
4.9.1, 4.9.3, 4.9.10, 4.9.29.

Section 4.10: Taylor polynomials
4.10.1, 4.10.6, 4.10.9, 4.10.10, 4.10.22, 4.10.27, 4.10.28.

Section 4.11: Roundoff error, truncation error, and computers
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These problems are to be presented 23-27 September. See here for where you should meet, and here to sign up for presenting a problem.

PDF-version of the project problems

Problem 1

Let the function \(f\) be defined by \[f(x) = \begin{cases} x\tanh(x) \ \text{for} \ x \geq 0 \\ ax^2+bx+c\ \text{for} \ x <0. \end{cases} \]

Find the constants \(a, b, c\) so that \(f\) is two times differentiable.

Problem 2

A water tank shaped like a cone pointing downwards is 10 metres high. 2 metres above the tip the radius is 1 metre. Water is pouring from the tank into a cylindrical barrel with vertical axis and a diameter of 8 metres. Assume that the height of the water in the tank is 4 metres, and is decreasing at a rate of 0.2 metres per second. How fast is the height of the water in the barrel changing?

Problem 3

The fuel consumption of a car, \(D\) measured in litres per hour, is related to the velocity \(v\) by the following formula\[D(v) = \frac{\mathrm{e}^v+\mathrm{e}^{-v}}{2}+\frac{1}{v},\] where \(v\) is measured in kilometres per hour.

a) Find an equation describing the velocity \(v_0\) that gives minimal fuel consumption.

b) Use Newton's Method to give an approximated solution the the equation found in a).

Problem 4

a) Prove that the polynomial \(p(x) = x^5+7x^3-20\) has exactly one root in the interval \((1, 2\)).

Hint: See the example Newton's Method.

b) Write a script in Maple to implement Newtons Method to find an approximated value for this root. For this and problem c) it may be useful to take a look at : Newton's Method.

c) Use Maples Student[Calculus]-package to approximate this root. How does the value compare to the one found in b)?

These problems are to be presented 30 September-4 October. See here for where you should meet, and here to sign up for presenting a problem.

PDF-version of the project problems}

Problem 1 (Problem 4.9.48 in Adams)

A sector is cut out of a disk of radius \(R\), as shown below. The remaining part of the disk is bent up so that the two edges join and a cone is formed. What is the largest possible volume for the cone?

Problem 2

The limit \[\lim_{x \to 0}\left(\frac{1}{\sin(px)}-\frac{1}{\mathrm{e}^{x/2}-1}\right)\] exists for exactly one real-valued \(p\). Find this value, and calculate the limit.

Problem 3

Adams defines a concave up, or convex, function on an interval \(I\) to be a differentiable function such that the derivative \(f'\) is an increasing function on \(I\). In this problem, we will show that for differentiable functions this is equivalent to the more general definition of a convex function. We will prove the following:

Assume \(f\) is differentiable on an interval \(I\). Then \(f'\) is an increasing function on \(I\) if and only if for all \(x, y \in I\) and all \(t \in [0,1]\) we have that \[f((1-t)x+ty)\leq (1-t)f(x)+tf(y).\]

a) First make the substitution \(x_1=x\), \(x_3 =y\) and \(x_2 =(1-t)x_1 +tx_3\) to prove that if the inequality above holds, then for any \(x_1, x_2, x_3\) in \(I\) such that \(x_1 < x_2 <x_3\), we have \[\frac{f(x_2)-f(x_1)}{x_2-x_1}\leq \frac{f(x_3)-f(x_2)}{x_3-x_2}.\] HINT: You will need to solve for \(t\) in the substitution.

b) Now assume that \[\frac{f(x_2)-f(x_1)}{x_2-x_1}\leq \frac{f(x_3)-f(x_2)}{x_3-x_2}.\] holds for \(x_1 < x_2 <x_3\). Show that this implies that the inequality \[f((1-t)x+ty)\leq (1-t)f(x)+tf(y).\] holds for \(x, y \in I\) and all \(t \in [0,1]\).

c) Now suppose \(f'\) increases on \(I\). Use the result from a) and b), and the Mean Value theorem to conclude that \[\frac{f(x_2)-f(x_1)}{x_2-x_1}\leq \frac{f(x_3)-f(x_2)}{x_3-x_2},\] and equivalently, \[f((1-t)x+ty)\leq (1-t)f(x)+tf(y).\]

d) For the other implication, suppose that for \(x_1 < x_2 <x_3\), we have \[\frac{f(x_2)-f(x_1)}{x_2-x_1}\leq \frac{f(x_3)-f(x_2)}{x_3-x_2}.\] Prove that \(f'\) is increasing on \(I\), and that the theorem holds.

Problem 4

a) Let \(f(x)= x^{10}\). Use Maple or other software to plot \(f(x)-\sqrt{f(x)^2}\) on the intervals \((-10, 10)\) and \((-\infty, \infty)\). Why does the plot indicate that the function is not identically equal to 0?

b) Assume that \(x>0\), and use Maple to calculate different values of \(f(x)-\sqrt{f(x)^2}\). Also use Maple to simplify the expression. Why does this answer differ from the one indicated in a)?

c) Use the function \(g(x)=2^x\ln(1+2^{-x})\) to determine the machine epsilon for your computer. What does the answer tell you? How does this relate to point a)?