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MIT 18.01 Problem Set 2A

Quotient, chain and product rules, higher derivatives and derivatives of inverse trigonometric functions.

1. MIT 18.01 Problem Set 2A — how this assessment works

This lesson reproduces the Problem Set “MIT 18.01 Problem Set 2A” from MIT OpenCourseWare 18.01 Single Variable Calculus (Fall 2006) (https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/), licensed under CC BY-NC-SA 4.0. Work through 6 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

2. MIT 18.01 PS2A Problem 1 — Qualitative graphs of the even/odd parts and their derivatives (4 pts)

(now; 4 pts) Graph the even and odd functions you found in Problem 1, Part II of PS1. Directly below, graph their derivatives. Do this qualitatively using your estimation of the slope. Do not use the formulas for the derivatives (except to check your work if you want). You can use a graphing calculator to check your answer, provided that you mention it in Problem 0. (Note, however, that you may not use books, notes or calculators during tests, so it is unwise to rely on a graphing calculator here.)

3. MIT 18.01 PS2A Problem 2 — Chain rule and product rule computations (5 pts (= 2 + 3))

(before Fri; 5 pts=2+3) Compute

a) (d/dx)tan⁡3(x4)(d/dx) \tan^3(x^4)

b) (d/dy)(sin⁡2ycos⁡2y)(d/dy)(\sin^2 y \cos^2 y)

(Do this two ways: first use the product rule, then write it as f(2y)f(2y). Show that the answers agree.)

4. MIT 18.01 PS2A Problem 3 — Higher derivatives of a product (3 pts (= 1 + 2))

(before Fri; 3pts=1+2)

a) If y=uvy=uv, show that y′′=u′′v+2u′v′+uv′′y''=u''v+2u'v'+uv''

b) Find y′′′y'''.

5. MIT 18.01 PS2A Problem 4 — Derivative of $\cos^{-1}x$ and its relation to $\sin^{-1}x$ (4 pts (= 3 + 1))

(Fri; 4pts 3+1)

a) The function cos⁡−1x\cos^{-1} x is the inverse of the cos⁡θ\cos \theta on 0≤θ≤π0 \leq \theta \leq \pi. Use implicit differentiation to derive the formula for (d/dx)cos⁡−1x(d/dx) \cos^{-1} x. Pay particular attention to the sign of the square root. (See the book or lecture for the case of the inverse of sine.)

b) Without calculation, explain why (d/dx)cos⁡−1x+(d/dx)sin⁡−1x=0(d/dx) \cos^{-1} x+(d/dx) \sin^{-1} x=0

6. MIT 18.01 PS2A Problem 5 — Textbook exercises: exponents and logarithms (10 pts (= 2+2+2+2+2))

(Tues+Thurs; 10pts=2+2+2+2+2) Do 8.2/8ac, 10, 11; 8.4/18,19a.

7. MIT 18.01 PS2A Problem 6 — Derivative of a product of $n$ functions (2 pts)

(Thurs; 2pts) Derive the formula for D(u1u2⋯un)D(u_1 u_2 \cdots u_n) from PS1, Part II, 7b, using logarithmic differentiation.

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