# MATH137 Study Guide - Midterm Guide: Binary Logarithm, Intermediate Value Theorem, Liquid Oxygen

Math 137 Sample Midterm 1 Answers

NOTE: Only answers (and some proofs) are contained here. On the test you

are required to put full solutions and show all steps/all of your work.

1. (a) By lim

x→∞

f(x) = −∞ we mean that for every > 0 there exists a δ > 0 such that

f(x)<−1

whenever x > 1

δ.

(b) At a height of 2km the barometric pressure is changing at a rate of 1 kiloPascal per

km.

(c) tanh x=ex−e−x

ex+e−x. Its domain is Rand its codomain is (−1,1).

(d) log2(2) −log2(5) + log2(20) = log28 = 3

(e) Observe that f(0) = 2, f(1) = 1 −2−3 + 2 = −2, and f(x) is continuous on the

interval [0,1]. Thus, by the Intermediate Value Theorem there exists a root of fbetween

0 and 1.

2. (a) lim

x→2

tan π

2x

x2+ 1 =1

5

(b) lim

x→0

sin(3x)

x= 3

(c) lim

x→03x2sin 1

3x2= 0

3. (a) f0(x) = ln x+ 1.

(b) g0(x) = −cos xsin x

√cos2x

(c) h0(x) = sec(x2+x) tan(x2+x)(2x+ 1)(3e2x+ 1) −6e2x(sec(x2+x))

(3e2x+ 1)2

1

## Document Summary

Note: only answers (and some proofs) are contained here. whenever x > 1 (b) at a height of 2km the barometric pressure is changing at a rate of 1 kilopascal per km. (c) tanh x = ex e x ex + e x . Thus, by the intermediate value theorem there exists a root of f between. 3x2 sin (cid:20) (b) lim x 0 (c) lim x 0. Cos x sin x cos2 x sec(x2 + x) tan(x2 + x)(2x + 1)(3e2x + 1) 6e2x(sec(x2 + x)) (3e2x + 1)2. 2 (cid:18: x = k , cos sin 1. 3: f (x) is continuous on ( , 1) (1, ). (a) show that sinh 1 x = ln(x + Then, x2 + 1) ey e y x = sinh y = Then, by the quadratic formula we get ey = Since, ey 0 we must have and so y = ln(x +