Unit 3 and 4: Derivatives and Transcendental functions - Problem Set FAQ

This problem set is based on Unit 3 and 4: Derivatives and Transcendental functions. Please read the Problem Set FAQ for details on submission policies, collaboration rules, and general instructions. Remember

1. Submissions are only accepted by Gradescope. Do not send anything by email. Late submissionsare not accepted under any circumstance. Remember you can resubmit anytime before the deadline.

2. Submit your polished solutions using only this template PDF. You will submit a single PDFwith your full written solutions. If your solution is not written using this template PDF (scanned print or digital) then you will receive zero. Do not submit rough work. Organize your work neatly in thespace provided.

3. Show your work and justify your steps on every question, unless otherwise indicated. Put your final answer in the box provided, if necessary.
We recommend you write draft solutions on separate pages and afterwards write your polished solutions here. You must fill out and sign the academic integrity statement below; otherwise, you will receive zero.

1. Let a, b, c ∈ R and a < c < b.Suppose that f : (a, b) → R is differentiable at c and f 0 (c) > 0.

(a) Prove that there exists δ > 0 such that f(x) > f(c) for all x in (c, c + δ) and f(x) < f(c) for all x in (c − δ, c).

2. Does f have to be increasing in some neighborhood of c? If so, prove it. If not, find a counterexample and justify.
Remark: We say a function F is increasing on an inverval I

3. Assume h is continuous everywhere and h(0) = 0. Here is the graph of its derivative:

Use the space below to sketch the graph of h. Remember h(0) = 0 and h is continuous everywhere.

3. Let a ∈ R. We want to study the curve with equation

Notice that for each value of a we get a different curve.

4. You can see the graph on desmos.com/calculator/fuxts9nua4. You will find one slider that allows you to change the value of a. Note that the curve is differentiable at every point for every value of a, except at the origin, which is always a singular point.You can use this information
without justification for this problem.

a. Fix a = 4. Find out how many points on the curve have a horizontal tangent line. Find all of their coordinates.
Hint: Use implicit differentiation. Think of y as a function of x.

(b) Fix a = 4. Find out how many points on the curve have a vertical tangent line. Find all of their coordinates.
Hint: Use implicit differentiation. Think of x as a function of y.

(c) Now play with the slider and try different values of a. You will notice that sometimes the curve has exactly 2 points with a vertical tangent line, sometimes it has no points with a vertical tangent line and sometimes it has exactly 10 points with a vertical tangent line. For which values of a does it have 10, for which values does it have 2, and for which values does it have 0? Prove it.