Question

Let f(x) = x^3 - x

a) Find the equation of the secant line through (0,f(0)) and
(2,f(2))

b) State the Mean-Value Theorem and show that there is only one
number c in the interval that satisfies the conclusion of the
Mean-Value Theorem for the secant line in part a

c) Find the equation of the tangent line to the graph of f at point
(c,f(c)).

d) Graph the secant line in part (a) and the tangent line in part c
in the same coordinate system and confirm that the lines are
parallel

Answer #1

refer to the graph of y=f(x)=x^2+x shown
a. Find the slope of the secant line joining(-3,f(-3)) and
(0,f(0))
b. Find the slope of the secant line joining (-3,f(-3))
and(-3+h,f(-3+h))
c . Find the slope of the graph at (-3,f(-3))
d. Find the equation of the tangent line to the graph at
(-3,f(-3))

1) Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem. (Enter your answers as a
comma-separated list.)
f(x) = 1 − 12x + 2x^2, [2, 4]
c =
2) If f(2) = 7 and f '(x) ≥ 1 for 2 ≤
x ≤ 4, how small can f(4) possibly be?
3) Does the function satisfy the hypotheses of the Mean Value
Theorem...

Consider the function f(x) = √x and
the point P(4,2) on the graph f.
a)Graph f and the secant lines passing through the
point P(4, 2) and Q(x,
f(x)) for x-values of 3, 5, and 8.
b) Find the slope of each secant line. (Round your answers to
three decimal places.)
(line passing through Q(3, f(x)))
(line passing through Q(5, f(x)))
(line passing through Q(8, f(x)))
c)Use the results of part (b) to estimate the slope of the
tangent line...

Suppose f(x) = (2/x) + 5 .
a. *Graph this function.
b. *Find the equation of the secant line to f(x) on the interval
[1, 3]. Call this line g(x). Add g(x) to your graph
c. Find the equation of the tangent line to f(x) at the point
(2,6). Call this line h(x). Add h(x) to your graph.
Please neatly show your work.

Find the slope of the secant lines over the specified intervals for
the functions f(x)= -1/2x^2+2x .
a) [2,3]
b) [2,2.1]
c) Estimate the slope of the tangent line at x=2
d) Use your estimate to find the equation of the tangent line
at x=2

To illustrate the Mean Value Theorem with a specific function,
let's consider f(x) = x^3 − x, a = 0, b = 5. Since f is a
polynomial, it is continuous and differentiable for all x, so it is
certainly continuous on [0, 5] and differentiable on (0, 5).
Therefore, by the Mean Value Theorem, there is a number c in (0, 5)
such that
f(5) − f(0) = f '(c)(5 − 0).
Now f(5) = ______ , f(0) =...

Find an equation of the line that is tangent to the graph of
f and parallel to the given line.
Function
Line
f(x) = x2
6x − y + 9 = 0

Let
fx=x2+12x-3
Find the equation of the line tangent to the graph of
f(x) at x=3
Find the value(s) of x where the tangent line is
horizontal.
2.The total sales of a video game months after being introduced
is given by the function
St=5ex2+ex
Find
S(10) and S'(10). What do these values represent
in terms of sales?
Use these
results to estimate the total sales at t=11 months after
the games release.

let f(x) = sqrt x^4+4x+4.find the equation of the tangent line
to the graph of f −1(a) when a = 3

1. Let f(x)=(x^2+1)(2x-3)
Find the equation of the line tangent to the graph of f(x) at
x=3.
Find the value(s) of x where the tangent line is horizontal.
2. The total sales S of a video game t months after being
introduced is given by the function
S(t)=(5e^x)/(2+e^x )
Find S(10) and S'(10). What do these values represent in terms
of sales?
Use these results to estimate the total sales at t=11 months
after the games release.

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