3.4 (Graded) Cubic splines 2. Determine the clamped cubic spline s that interpolates the data f (0) 0, f(1) 1, f(2) 2 and satisfies s' (0) = s'(2) = 1 Note: this can be done effectively by hand. 4c. Construct the natural cubic spline for the following data. f (x) х -0.29004996 0.1 -0.56079734 0.2 -0.81401972 0.3 Note: this can be done effectively with the aid of software - avoid ugly numbers by hand. 8c. Construct the clamped cubic spline using the data of Exercise 4 and the fact that f'(0.1) =-2.801998 and f'(0.3) = -2.453395. Note: Book has f(0.1) listed incorrectly. Also this can be done effectively with the aid of software avoid ugly numbers by hand. 12. A clamped cubic splines for a function f is defined on [1,3] by Jso()3(-1)+2( -1)2 - (x - 1)3, if 1 < 2 S1()a+b(x - 2)+ c( - 2)2 + d(x - 2)3, s(x) = if 2
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
#4c from this handout
From Numerical Analysis 8th edition by Burden, Section 3.4
![3.4 (Graded) Cubic splines
2. Determine the clamped cubic spline s that interpolates the data
f (0) 0, f(1) 1, f(2) 2
and satisfies s' (0) = s'(2) = 1
Note: this can be done effectively by hand.
4c. Construct the natural cubic spline for the following data.
f (x)
х
-0.29004996
0.1
-0.56079734
0.2
-0.81401972
0.3
Note: this can be done effectively with the aid of software - avoid ugly numbers by hand.
8c. Construct the clamped cubic spline using the data of Exercise 4 and the fact that
f'(0.1) =-2.801998 and f'(0.3) = -2.453395.
Note: Book has f(0.1) listed incorrectly. Also this can be done effectively with the aid
of software avoid ugly numbers by hand.
12. A clamped cubic splines for a function f is defined on [1,3] by
Jso()3(-1)+2( -1)2 - (x - 1)3, if 1 < 2
S1()a+b(x - 2)+ c( - 2)2 + d(x - 2)3,
s(x) =
if 2 <I< 3.
Given f'(1) f'(3), find a, b, c, and d.
Note: this can be done effectively by hand.
16. Construct a natural cubic spline to approximate f(x) = e by using the values given by
f(x) at x 0, 0.25, 0.75, and 1.0.
1
1
e dx 1-
the result to
(a) Integrate the spline over [0, 1], and
compare
е
(b) Use the derivatives of the spline to approximate f (0.5) and f"(0.5). Compare the
approximations to the actual values.
Note: this can be done effectively with the aid of software.
19. Suppose that f(x) is a polynomial of degree 3. Show that f(x) is its own clamped cubic
spline, but that it cannot be its own natural cubic spline.
29. (Bonus/graduate) A car traveling along a straight road is clocked at a number of points.
The data from the observations are given in the following table, where the time is in
seconds, the distance is in feet, and the speed is in feet per second.
13
8.
0
3
5
Time
Distance 0
Speed
225 383 623 993
72
80
74
75
77
(a) Use a clamped cubic spline to predict the position of the car and its speed when t = 10 s.
(b) Use the derivative of the spline to determine whether the car ever exceeds a 55
speed limit on the road; if so, what is the first time the car exceeds this speed?
(c) What is the predicted maximum speed for the car?
Note: this can be done effectively with the aid of software.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4001e78-f573-4ab7-b337-97f71cf09379%2Fbe470bcc-ff7b-447c-ad17-50ff70043109%2Fjyhn2hc.jpeg&w=3840&q=75)
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