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. 91. f ( x ) = 2 x 2 - 3 x + 1
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. 91. f ( x ) = 2 x 2 - 3 x + 1
Solution Summary: The author explains how the function f (x) = 2 x 2 + 3 + 1 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.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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