Question

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 to the graph of *f* at *P*(4,2).

d) Describe how to improve your approximation of the slope. (multiple choice)

-Choose secant lines that are nearly horizontal.

-Define the secant lines with points closer to
*P*.

-Define the secant lines with points farther away from
*P*

-Choose secant lines that are nearly vertical.

Answer #1

PROBLEM: Given f(x)=4x - x2 + k & (1,3+k) on the
graph of f.
(K=3)
a) Write your equation after substituting in the value of k.
(K=3)
b) Calculate the function values, showing your calculations,
then graph three secant lines
i. One thru P(1,f(1)) to P1(0.5,__)
ii. One thru P(1,f(1)) to P2(1.5,__)
iii. One thru P1 to
P2
c) Find the slope of each of these three secant lines showing
all
calculations.
d) NO DERIVATIVES ALLOWED HERE! Use the...

Find the derivative of f(x) - In(2x)
1. what point on the graph of f(x) at which the tangent line is
horizontal?
2. what point on the graph of f(x) at which the tangent is
vertical?
3. is there a point on the graph f(x) at which the tangent has a
slope 1?

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...

1.) P(2, 4) lies on the graph of y=x4-2x-4
Q is the point (x, x4-2x-4)
Write an equation for the slope of the secant line through P and
Q as a function of x.
2.) Using the formula above, find the slopes fo 1.9, 1.99,
1.999, 2.001, 2.01, and 2.1 (round 4 decimal places)
3.) What is the slope of the tangent line to the curve
y=x4-2x-4 at P(2, 4)
Can you please explain in detail how to get the...

Consider the graph of y=f(x)=1−x2 and a
typical point P on the graph in the first quadrant. The tangent
line to the graph at P will determine a right triangle in the first
quadrant, as pictured below.
a) Find the formula for a function A(x) that computes the area
of the triangle through the point P=(x,y)
b) Find the point P so that the area of the triangle is as small
as possible: P =()

Let f(x)=12x2. (a) If we wish to find the point Q=(x0,y0) on the
graph of f(x) that is closest to the point P=(4,1), what is the
objective function? (Hint: optimize the square of the distance from
P to Q.)
Objective function: O(x)=O(x)=____________________________
(b) Find the point Q=(x0,y0)Q=(x0,y0) as described in part (a).
Box your final answer.
(c) Verify that the line connecting PP to QQ is perpendicular to
the line tangent to f(x)f(x) at QQ. Hint: recall that the lines...

(1 point) The point P(1/5,10) lies on the curve y=2/x . Let Q be
the point (x,2/x). a.) Find the slope of the secant line PQ for the
following values of x.
If x=1.5/5, the slope of PQ is:
If x=1.05/5, the slope of PQ is:
If x=0.95/5, the slope of PQ is:
If x=0.5/5, the slope of PQ is:
Based on the above results, guess the slope of the tangent line
to the curve at P(1/5,10)

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))

find a point of the graph of the function f(x) = e^2x such that
the tangent line to the graph at that point passes through the
origin. Use a graphing utility to graph f and the tangent line is
the same viewing window

1. Use the x- and y- intercepts to graph each linear
equation.(Do not use graph paper)
a) 2x-5y=10
b) 3x=2y+6
2. Calculate the slope of the line passing through the given
points. If the slope is undefined, so state, Then indicate whether
the line rises, falls, is horizontal or is vertical.
a) (4,2) and (3,4)
b) (-1,3) and (2,4)

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