Differentiation - Exercises
Below you will find the exercises we suggest you work on in connection with the Differentiation theme and the project problems which are to be done for the exercise classes in week 36 (2.-6. September) and week 37 (9.-13. September).
Suggested Exercises
Section 2.1: Tangent Lines and Their Slopes
2.1.1, 2.1.11, 2.1.16, 2.1.23.
Section 2.2: The Derivative
2.2.3, 2.2.19, 2.2.48, 2.2.55.
Section 2.3: Differentiation Rules
2.3.7, 2.3.29, 2.3.31, 2.3.46.
Section 2.4: The Chain Rule
2.4.5, 2.4.13, 2.4.24, 2.4.27.
Section 2.5: Derivatives of Trigonometric Functions
2.5.16, 2.5.35, 2.5.53, 2.5.58.
Section 2.6: Higher Order Derivatives
2.6.15.
Section 2.7: Using Differentials and Derivatives
2.7.12, 2.7.21, 2.7.23.
Section 2.8: The Mean Value Theorem
2.8.4, 2.8.15, 2.8.21.
Section 2.9: Implicit Differentiation
2.9.5, 2.9.16, 2.9.17.
Section 2.10: Antiderivatives and Initial-Value Problems
2.10.12, 2.10.26, 2.10.42.
Section 2.11: Velocity and Acceleration
2.11.7, 2.11.12.
Løsningsforslag
LF 2.7-2.11
Project Problems
Below are the problems to be presented in week 36 and week 37. See here for where you should meet, and here to sign up for presenting a problem.
Week 36
1. Oscillations and derivatives
Define the function \(g\) by \(g(x) = x^2\sin\frac 1x\) for \(x\neq 0\) and \(g(0) = 0\). Is \(g\) continuous in \(x=0\)? Differentiable in \(x=0\)? Continuously differentiable in \(x=0\)?
2. Using Maple to find derivatives
(Exercise 61 for chapter 2.5 in Adams.)
Let \(f(x) = \sqrt{2x^2+3\sin(x^2)} -\frac{(2x^2+3)^{3/2}\cos(x^2)}{x}\).
Use Maple to find \(f'(\sqrt \pi)\). Also use Maple to find the tangent at \(x= \sqrt \pi \).
3. Logarithmic derivation
In this exercise we will derive a useful method for calculating the differentials of complex expressions, called logarithmic differentiation.
a) Use the chain rule to show that if \(f\) is differentiable and \(f(x) \neq 0\), then
\(f'(x) = f(x)[\ln|f(x)|]' \).
b) Let \(f\) and \(g\) be two differentiable functions, and assume that \(f\) is positive. Show that \([f(x)^{g(x)}]' =f(x)^{g(x)}\left[g'(x)\ln f(x) +\frac{g(x)f'(x)}{f(x)}\right] \)
4. Tangents to polynomials
Let \(P(x)\) be a polynomial and let the straight line \(y=l(x)\) be the tangent to \(P(x)\) at \(x=0\). Show that \(P(x)-l(x)\) is a polynomial with no constant term and no first degree term.
Week 37
1. Lipschitz continuous functions
A function \(f\) is said to be Lipschitz continuous if there exists an \(L>0\) such that for all \(x\) and \(y\) we have \(|f(x)-f(y)|\leq L|x-y|\). The constant \(L\) is called a Lipschitz constant of \(f\).
a) Give an example of a Lipschitz continuous function.
b) Show that Lipschitz continuity implies continuity.
c) Show that if \(f\) is a differentiable function with bounded derivative (that is, there exists an \(L\) such that \(|f'(x)|\le L\) for all \(x\)), then \(f\) is Lipschitz continuous. (Hint: The Mean-Value Theorem)
2. Counterexamples
a) Find a continuous function \(f\) on \([-1,1]\) such that \(f\) has a maximal value at \(c\in (-1,1)\), but \(f'(c)\) does not exist.
b) Find a continuous function \(f\) on \([-1,1]\) such that \(\frac{f(1)-f(-1)}{1-(-1)} \neq f'(x)\) for any \(x\in(-1,1)\) where \(f'(x)\) is defined.
3. Driving a car
A car is driving at night along a level, curved road. It starts in the origin, the equation of the road is \(y = x^2 \), and the car's x-coordinate is an increasing function of time. There is a signpost located at \((2,3.75)\).
a) What is the position of the car when its headlight illuminates the signpost? Do you have any implicit physical assumptions in your solution?
b) What is the shortest distance between the signpost and the car? (You will need to solve a third degree equation in order to answer this question. Either use Maple to do that or explain what you would do if you had a solution to the third degree equation).
c) Let \( \frac{\mathrm{d}x}{\mathrm{d}t}= v_x\) and \(\frac{\mathrm{d}y}{\mathrm{d}t}=v_y.\) The car's velocity is then \([v_x, v_y]\). How are \(v_x\) and \(v_y\) related?
4. Ellipse and Parabola
Show that the ellipse \(\frac{1}{3}x^2 + y^2=1 \) and the hyperbola \(x^2 - y^2 =1 \) intersect at right angles. Use Maple to illustrate the situation.