Problem Set 6 — Exponential Derivatives and Implicit Differentiation

Purpose

This problem set reinforces several things we have covered recently, centering on derivatives of exponential and logarithmic functions and implicit differentiation, but also including a little more practice with antiderivatives and a simple application of derivatives. It addresses the following learning outcomes:

• Compute limits and derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 1)
• Compute definite and indefinite integrals of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 2)
• Solve problems in a range of mathematical applications using the derivative or the integral (outcome 5).

Background

This problem set draws mainly on sections 3.8 and 3.9 of our textbook. Antiderivatives of the functions in those sections will be covered in class on October 19, but haven’t been discussed in the book yet. We discussed section 3.9 in class on October 12, and section 3.8 (implicit differentiation) and derivatives of inverse functions between October 14 and 19.

Activity

Solve the following problems:

Question 1

Exercise 302 in section 3.8 of our textbook: Use implicit differentiation to find \(\frac{dy}{dx}\) given that \(x^2y = y - 7\).

Question 2

Find \(\frac{dy}{dx}\) given that \(\ln x + \ln y = 1\).

Question 3

(Based on exercise 320 in section 3.8 of our textbook.)

Consider the equation

Part A

Find the slope of the line tangent to the graph of Equation 1 at point \((0,\tfrac{1}{2})\).

Part B

(This part exceeds the mastery I expect for applying derivatives because it uses ideas that we haven’t talked about in class — although they are in various examples in the textbook.)

Find the equation for the line tangent to the graph of Equation 1 at point \((0,\tfrac{1}{2})\).

Question 4

Part A

Use antiderivative rules we have discovered, possibly including the antiderivative sum (\(\int f(x) + g(x)\,dx = \int f(x)\,dx + \int g(x)\,dx\)) and constant multiple (\(\int cf(x)\,dx = c\int f(x)\,dx\)) rules, to evaluate

Part B

(This part goes beyond the mastery I expect for integration in this course because it connects what we’re doing to an integration technique from calculus 2.)

If you put the terms in the integrand from Equation 2 (i.e., \(\frac{1}{2x} + \frac{2}{1+x^2}\)) over a common denominator and combine them, you get a function that doesn’t look like the original, but must have the same antiderivative, because it’s the same function. What is that function?

(Doing what you just did in reverse, i.e., starting with a fraction whose denominator is a product, and asking what sum of simpler fractions would have that product as a common denominator, sometimes turns an apparently intractable antiderivative problem into one that can be solved using antiderivative rules we know. This strategy is called the method of partial fractions, and is one of the topics taught in calculus 2.)

Follow-Up

I will grade this exercise during one of your weekly individual meetings with me. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.