Chapter 1. Calculus Tutorial 2.8.005.

1.1 Problem Statement

{3,5,7,9}

eval round(-1,2)

eval round($b - cos($a),2)

Show that cos($a x) = $b x has a solution on the interval [0,1].

Hint: Show that f(x) = $b x - cos($a x) has a zero in [0,1].

1.2 Step 1

To show that cos($a x) = $b x has a solution in the interval [0,1], we could try to directly solve for some value of x in the interval [0,1] that satisfies the equation.

However, this is not easily done. We can make this problem easier by noting that finding an x satisfying cos($a x) = $b x is equivalent to finding an x that satisfies $b x - cos($a x) = 0.

We can therefore set f(x) = $b x − cos($a x) and look for zeroes of f(x) in the interval [0, 1]. To help prove the existence of a zero of f(x) in [0, 1], we will use the Intermediate Value Theorem.

Question Sequence

Question 1.1

The Intermediate Value Theorem states that if f(x) is continuous on a closed interval [a,b] and f(a)2jjYt7TAnKuJZuyPe892aUvjxRc=f(b), then for every value M between f(a) and f(b), there exists at least one value c (a,b) such that f(c) = FoCpM7rR/d9iB27bsAsiW7e7QcYkYEUJwpETSjpZBPUYEanp1qOUKB+byLs=.

Incorrect.

Correct.

Question 1.2

As the difference of a polynomial and this trigonometric function, both of which are continuous for all real numbers, f(x) = $b x − cos($a x) PMOdEMjAlCTr7j/lZi0uQA== continuous on the interval [0,1] and therefore we 1x1GEHSmQB7Xf9RkNYh/Nw== apply the Intermediate Value Theorem.

Incorrect.

Correct.

1.3 Step 2

Question Sequence

Question 1.3

To apply the Intermediate Value Theorem to f(x) = $b x − cos($a x) on [0,1], evaluate f(0) and f(1). (Round your answer to two decimal places.)

f(0) = ynBgSgbe83A=

f(1) = wXYIfDe/F4g=

Incorrect.

Correct.

1.4 Step 3

Question 1.4

Because the graph of f(x) = $b x − cos($a x) is continuous in the interval [0, 1], the Intermediate Value Theorem tells us that f(x) will take on all values between f(0) = $f0 and f(1) ≈ $f1. Since f(0) < 0 < f(1) and f(x) is continuous, the Intermediate Value Theorem asserts that
f(x) = $b x − cos($a x) 4jrRan3ReFFExJmp2KLqbjXHMYrQdHV8VrX5HvUTXLmvbMnZafVW3g== have a zero in the interval [0, 1].

sqjMdnL2Von541GtW9tH28CnuN9FBwM5TDhZihLxnx3l3s+37HnQYX+uV30R1sDAN5Q7ybXXTMKf9OInz/LBK4qEXAwa0gQbQqwg/Qpe8AFJ0KsepNOshG5Y9MMXV+fTN75klqyVJtkEgmxSSshszrUvP7hXse2yh2OFnRVrl77LPeIm74dV01f4A+ZOno6pV2rDUGw7Dblu0MC7frzju2Kwn/FNoFNl

Incorrect.

Correct.