MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - - PowerPoint PPT Presentation

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May 07, 2023 174 likes •256 views

MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim October 20th, 2020 Warm-up question : Suppose c is a fixed real number, f a differentiable function on R , and f = g . Are these correct? = g ( x ) + c d d f ( x ) +

SLIDE 1

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Julia Kim October 20th, 2020 Warm-up question: Suppose c is a fixed real number, f a differentiable function on R, and f ′ = g. Are these correct?

1

d dx

f (x) + c = g(x) + c

2

d dx

cf (x) = cg(x)

3

d dx f (x + c) = g(x + c)

4

d dx f (cx) = g(cx)

5

d dx f (c) = g(c)

6

d dc f (c) = g(c)

SLIDE 2

Nixon

Richard Nixon, during the 1972 US Presidential campaign, (paraphrased): Inflation is increasing, but the rate of increase of inflation is de- creasing. Let

C = cost of life
t = time

What did Nixon say, in terms of derivatives?

SLIDE 3

Higher order derivatives

Let g(x) = 1 x3 .

Calculate the first few derivatives.
Make a conjecture for a formula for the n-th derivative g(n)(x).
Prove it by induction.

SLIDE 4

Absolute value and tangent lines

At (0,0) the graph of y = |x|...

1 ... has one tangent line: y = 0 2 ... has one tangent line: x = 0 3 ... has two tangent lines y = x and y = −x 4 ... has no tangent line

SLIDE 5

Absolute value and derivatives - 1

Let h(x) = x|x|. What is h′(0)?

1 It is 0. 2 It doesn’t exist because |x| is not differentiable at 0. 3 It doesn’t exist because the right- and left-limits, when computing the

derivative, are different.

4 It doesn’t exist because it has a corner. 5 It doesn’t exist for a different reason.

SLIDE 6

Absolute value and derivatives - 2

True or False?

For all n ∈ Z and all x,

d dx |x|n = n|x|n−1.

SLIDE 7

Before next class...

Watch videos 3.6, 3.7, and 3.9.
Download the next class’s slides (no need to look at them!)