Math 1P97. Assignment 5 (4%). Due date: Monday, June 26, 6:45 pm Student’s name: Box #: 1 Math 1P97. Assignment 5 (4%). Due date: Monday, June 26, 6:45 pm Student’s name:
ID:
Total mark:
NOTE: Students are expected to complete all questions on the assignment. However, only
a subset of questions will be considered for marking. Marks will be deducted for incomplete
assignments.
1. Solve the differential equation y 0 = 4e2x + 6x2 with the initial condition y(0) = 4. 2. Find the indefinite integrals
Z
2x − 8
dx =
a)
2
x − 8x + 16 Z
b) (x2 2x
dx =
+ 3)3 2 Z
c) Z
d) Z
e) Z
f) 2xex √
5 2 +5 dx = 7x + 12 dx = e8x−20 dx = (ln x)2
dx =
x 3 3. According to a model for predicting the height of preschool children, the rate of growth of
a typical child is


1
0
−t
R (t) = 26e + 6.3,
6t66
4
centimeters per year, where t is measured in years. The height of a typical 3-month old
preschool child is 60 cm.
a) Find a function R(t) for predicting the height of a typical preschool child at age t. b) Estimate the height of a typical 3-yr-old child. 4. Evaluate the definite integral.
Z 1


√
a)
4x3 − 6x2 + 3 x + 1 dx =
0 Z 1 b) x2 + 1
2 dx = −1 4 Z 1 6x x2 + 9 c)
2 dx = 0 Z 0 e2x+4 dx = d)
−2 Z e e)
1 Z (3 ln x)2
dx =
x −1 f)
−2 4x3 − 3x2 + 1
dx =
x2 5 5. The rate at which the risk of Down syndrome is changing is approximated by the function
r0 (x) = 0.004641x2 − 0.3012x + 4.9, (20 6 x 6 45) where r0 (x) is measured in percentage of all births/year and x is the maternal age at
delivery.
a) Find a function f giving the risk as a percentage of all births when the maternal age
at delivery is x years, given that the risk of Down syndrome at age 30 is 0.14% of all
births b) Based on this model, what is the risk of Down syndrome when the maternal age at
delivery is 25 years? 6 c) What is the estimated average of the risk of Down syndrome when the maternal age
at
 delivery is between 25 and 35 years?

Z b
1
Hint: The average value (AV) of f over [a, b] is AV = b−a
f (x) dx
a 6. (Maple Question) Find the area of the region enclosed by the following graphs.
To find the area between a function f and g, use the command
[> int(f-g,x=a..b);
a) y = x2 − 3x − 10, y = −x2 + x + 20
b) xy = 9, x + y = −10
Submit a printout of your Maple worksheet.
2 x and
7. (Maple Question) Plot the graphs of the functions of f = ex −3x−4 and g = −x2 + 15
4
find the area of the region totally enclosed by the graphs of these functions using Maple.
a) Write your name, restart Maple and load the plot package
[> #Q7. <name> <student number>
[> restart:with(plots):
b) Define the function f (x):
[> f:=exp(x∧ 2-3*x-4);
c) Define the function g(x);
[> g:=-x∧ 2+15*x/4);
d) Find the intersections of these graphs:
[> a:=fsolve(f=g,x=0..1);
[> b:=fsolve(f=g,x=3..4);
e) plot both functions f and g on the same graph for x:
[> plot([f,g],x=a..b,color=[blue,green],thickness=2);
f) Find the exact area between f (x) and g(x) on the interval [a, b]:
[> S:=int(g-f,x=a..b);
g) Submit a printout of your Maple worksheet 7

 

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