MATH 100 Study Guide - Midterm Guide: Scilab, Hypotenuse
SOLUTIONS TO MID TERM #1, MATH 100
1. [6 marks] Using only the definition of the derivative, and not the rules, find f0(x) for the
function f(x)=√x2+1.
Solution:
f0(x) = lim
h→0
f(x+h)−f(x)
h= lim
h→0p(x+h)2+1−√x2+1
h
= lim
h→0 p(x+h)2+1−√x2+1
h×p(x+h)2+1+√x2+1
p(x+h)2+1+√x2+1
!
= lim
h→0
(x+h)2+1−(x2+1)
h(p(x+h)2+1+√x2+1) = lim
h→0
2hx +h2
h(p(x+h)2+1+√x2+1)
= lim
h→0
2x+h
p(x+h)2+1+√x2+1 =x
√x2+1
2. [12 marks] Find the derivatives of the following functions.
(a) f(x)=sin3x+cos
3x2.
(b) f(x)=q1+√x+x2.
(c) f(x)=x2+1
x2−1.
(d) f(x)=(x2+x+1)(x3+1).
Solution:
(a) f0(x)=2sin3x+cos
3x(3 sin2x×cos x+3cos
2x×(−sin x)).
(b) f0(x)=1
21+√x+x2−1/2
×1
2(x+x2)−1/2(1 + 2x).
(c) f0(x)=(x2−1)2x−(x2+1)2x
(x2−1)2.
(d) f0(x)=(2x+1)(x3+1)+(x2+x+1)3x2.
3. [8 marks]
(a) Determine lim
θ→0
tan 2θ
θ.
(b) Find the absolute maximum and minimum of the function f(x)=x2+1
x2on the interval
1
2≤x≤3.
1
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MATH 100 Full Course Notes
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Document Summary
[6 marks] using only the de(cid:12)nition of the derivative, and not the rules, (cid:12)nd f. 1. function f (x) = p x2 + 1: p. = lim h!0 f (x + h) f (x) p (x + h)2 + 1 p p (x + h)2 + 1 (x2 + 1) p (x + h)2 + 1 + p. 2hx + h2 (x + h)2 + 1 + x2 + 1) x2 + 1) h( = lim h!0 xp x2 + 1: [12 marks] find the derivatives of the following functions. (a) f (x) = sin3 x + cos3 x. 2 : (cid:1) (cid:0) q p (b) f (x) = 1 + x + x2: (c) f (x) = x2 + 1 x2 1 (d) f (x) = (x2 + x + 1)(x3 + 1): 0 (a) f (x) = 2 (cid:0) (cid:16) p.
