Math 1000 Tutorial
Quiz 9
Week 10
The solutions to Quiz 9 are below. I might be showing more steps than necessary, this is to aid your
understanding.
2
1. (4.9 Q39) Find f given f 00 (x) = −2 + 12x − 12x2 , f (0) = 4, f 0 (0) = 12.
Solution: f 0 (x) = −2x+6x2 −4x3 +C is the most general antiderivative of f 00 (x). Since f 0 (0) = 12,
we have f 0 (0) = C = 12, so f 0 (x) = −2x + 6x2 − 4x3 + 12.
f (x) = −x2 + 2x3 − x4 + 12x + D is the most general antiderivative of f 0 (x). Since f (0) = 4, we
have f (0) = D = 4, so
f (x) = −x2 + 2x3 − x4 + 12x + 4
2
2. (Parts of 5.1 Q5a) Estimate the area under the graph of f (x) = 1 + x2 from x = −1 to x = 2 using six
rectangles and right endpoints.
Solution: a = −1 and b = 2 since these are the endpoints of the interval; n = 6 is the number of
2+1
1
rectangles; thus ∆x = b−a
n = 6 = 2.
xi = a + i∆x = −1 +
R6 = ∆x
i
2
so x1 = −1 +
X
1
2
= −0.5, x2 = −1 +
2
2
= 0, x3 = 0.5, x4 = 1, x5 = 1.5, x6 = 2
6f (xi ) = ∆x[f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ) + f (x5 ) + f (x6 )]
i=1
1
[f (−0.5) + f (0) + f (0.5) + f (1) + f (1.5) + f (2)]
2
1
1
= [1.25 + 1 + 1.25 + 2 + 3.25 + 5] = [5/4 + 1 + 5/4 + 2 + 13/4 + 5]
2
2
1
1 55
55
= (13.75) = ·
= 6.875 =
2
2 4
8
=
(Either fractions or decimals is fine, you don’t have to show both.)
1/
2
(bonus)
(a) Sketch the curve and the approximating rectangles.