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
f(x) = −∞ we mean that for every > 0 there exists a δ > 0 such that
whenever x > 1
(b) At a height of 2km the barometric pressure is changing at a rate of 1 kiloPascal per
(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
x2+ 1 =1
3. (a) f0(x) = ln x+ 1.
(b) g0(x) = −cos xsin x
(c) h0(x) = sec(x2+x) tan(x2+x)(2x+ 1)(3e2x+ 1) −6e2x(sec(x2+x))
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 +