Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points ( x , f ( x ) ) and ( x + h , f ( x + h ) ) on the graph of a function y = f ( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find m sec for h = 0.5 , 0.1, and 0.01 at x = 1 . What value does m sec approach as h approaches 0? (c) Find an equation for the secant line at x = 1 with h = 0 .01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 89. f ( x ) = x 2 + 2 x
Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points ( x , f ( x ) ) and ( x + h , f ( x + h ) ) on the graph of a function y = f ( x ) may be given as In calculus, this expression is called the difference quotient of f (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find m sec for h = 0.5 , 0.1, and 0.01 at x = 1 . What value does m sec approach as h approaches 0? (c) Find an equation for the secant line at x = 1 with h = 0 .01 . (d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window. 89. f ( x ) = x 2 + 2 x
Solution Summary: The author explains the function f (x) = x 2 + 2. Calculation requires the discussion of a secant line.
Problems 87-94 require the following discussion of a secant line. The slope of the secant line containing the two points
and
on the graph of a function
may be given as
In calculus, this expression is called thedifference quotient of f
(a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer.
(b) Find msec for
, 0.1, and 0.01 at
. What value does msec approach as h approaches 0?
(c) Find an equation for the secant line at
with
.
(d) Use a graphing utility to graph f and the secant line found in part (c) in the same viewing window.
1. A bicyclist is riding their bike along the Chicago Lakefront Trail. The velocity (in
feet per second) of the bicyclist is recorded below. Use (a) Simpson's Rule, and (b)
the Trapezoidal Rule to estimate the total distance the bicyclist traveled during the
8-second period.
t
0 2
4 6 8
V
10 15
12 10 16
2. Find the midpoint rule approximation for
(a) n = 4
+5
x²dx using n subintervals.
1° 2
(b) n = 8
36
32
28
36
32
28
24
24
20
20
16
16
12
8-
4
1
2
3
4
5
6
12
8
4
1
2
3
4
5
6
=
5 37
A 4 8 0.5
06
9
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
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