MAT265: Calculus for Engineers I Classwork and Derivative Reference - - PDF document

MAT265: Calculus for Engineers I Classwork and Derivative Reference Sheet 11 September, 2015 Name: Instructions: Complete the following problems with a partner, referring to the reference sheet on the attached page. Please use scratch paper; there is not enough room on this page to thoroughly show your work. If there is enough time at the end of the class, partners will present some of the solutions to these problems on the chalkboard. You do NOT need to turn this assignment in for a grade.

• 1. Let f : R → R be defined by f(x) = x3 + x2 + x + 1.

(a) Using the definition of the derivative, find f ′(a) at an arbitrary point a ∈ R. (b) Write the formula for f ′(x). (c) Using the definition of the derivative, find f ′′(a) at an arbitrary point a ∈ R. (d) Write the formula for f ′′(x). (e) Does f ′′′ exist? What about f (n) for n > 3? (f) If your answer for part (e) was yes, make a conjecture as to what f ′′′ might be.

• 2. Let f : R → R be defined by f(x) = xm for some fixed number m. Recall the binomial

theorem: (x+h)m =

• m
• xmh0+
• m

1

• xm−1h1+
• m

2

• xm−2h2+· · ·+
• m

m − 1

• x1hm−1+
• m

m

• x0hm,

where

• m

k

• =

m! k!(m − k)!. Use the binomial theorem to help you calculate f ′(x), using the definition of the deriva- tive.

• 3. The following is a proof that the derivative of the sine function is the cosine function

(in other words, if f(x) = sin x, then f ′(x) = cos x). After carefully reading this proof, prove that if g(x) = cos x, then g′(x) = − sin x. You will need the following trigonometric identities: sin(x + h) = sin x cos h + cos x sin h cos(x + h) = cos x cos h − sin x sin h.

• Proof. Let f : R → R be defined by f(x) = sin x. Using the definition of the derivative,

we calculate that f ′(x) = lim

h→0

f(x + h) − f(x) h = lim

h→0

sin(x + h) − sin x h = lim

h→0

[sin x cos h + cos x sin h] − sin x h = lim

h→0

cos x sin h − sin x(1 − cos h) h = cos x

• lim

h→0

sin h h

• − sin x
• lim

h→0

1 − cos h h

• = cos x · 1 − sin x · 0

= cos x. The facts that limh→0 sin h h = 1 and limh→0 1 − cos h h = 0 can be verified graphically (we will learn a way to calculate these limits algebraically later in the semester). Note: You may be interested in watching the YouTube channel Mathispower4u, which has a few videos that explains how to use the definition of the derivative to find many different common derivatives. The link for this channel is www.youtube.com/user/bullcleo1. You will see the tabs “home, videos, playlists, channels, discussion, about,” and then a magnifying glass to perform a search. Click on the search option and type “derivative of sin, derivative

• f cos, power rule,” or whatever else you might be looking for.

Derivative Reference Sheet Definition 1. Let f : D → R be a function. If a ∈ D and if h is sufficiently small so that a + h ∈ D as well, then we define the derivative of f at a to be: f ′(a) = lim

h→0

f(a + h) − f(a) h , provided this limit exists. Definition 2. Let f : D → R be a function. If f ′(a) exists for all a ∈ D, we say that f is

• differentiable. In this case, we may define a derivative function f ′ : D → R, pronounced

“f prime.” Remark 1. Typically, if we write f as a function of x, then f ′ is written as a function of x as well. In other words, if we write f(x), then we write f ′(x). Plugging in a number a would yield f ′(a), but the notation f ′(a) implies that we are looking for the derivative at a specific point, whereas the notation f ′(x) implies that we have a formula for the derivative at any point in the domain. Remark 2. Given the function f, we sometimes write f ′(x) as d f dx instead. This notation will make more sense later in the semester. Definition 3. Let f : D → R be a function, and suppose f is differentiable. If the function f ′ has a derivative at all points in its domain, we say that f has a second derivative,

• r that f is twice differentiable, and we write the second derivative of f at a as f ′′(a)

(pronounced “f double prime of a”). Similarly, if f ′′ has a derivative at all points in its domain, we denote the third derivative by f ′′′. If f has higher derivatives, we denote these by f (n).