Mathematics, 06.09.2020 01:01 brittanygibson2812

Find the slope of the secant line of the function f(x)=x2−5x+5 (a) On the interval [1,3]
(b) On the interval [1,2]
(c) On the interval [0,1]
(d) Use the slope of the secant lines computed in parts (b) and (c) above to estimate the slope of the tangent line of f(x) at =1

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Answer from: emily0259

a) The slope of the secant line of the function on the interval [1, 3] is -1.

b) The slope of the secant line of the function on the interval [1, 2] is -2.

c) The slope of the secant line of the function on the interval [0, 1] is -4.

d) The estimated slope of the tangent line of f(x) at x = 1 is -3.

Procedure - Determination of slopes of secant lines

Let f(x) be a differentiable curve, the slope of a line secant to that curve ( $m_{AB}$ ) is defined by the following expression:

$m_{AB} = \frac{f(b)-f(a)}{b-a}$ , $a \ne b$ (1)

a) On the interval [1,3]

If we know that $f(x) = x^{2}-5\cdot x + 5$ , a = 1 and b = 3 , then the slope of the secant line is:

$m_{AB} = \frac{-1-1}{3-1}$

$m_{AB} = -1$

The slope of the secant line of the function on the interval [1, 3] is -1. $\blacksquare$

b) On the interval [1, 2]

If we know that $f(x) = x^{2}-5\cdot x + 5$ , a = 1 and b = 2 , then the slope of the secant line is:

$m_{AB} = \frac{-1-1}{2-1}$

$m_{AB} = -2$

The slope of the secant line of the function on the interval [1, 2] is -2. $\blacksquare$

c) On the interval [0, 1]

If we know that $f(x) = x^{2}-5\cdot x + 5$ , a = 0 and b = 1 , then the slope of the secant line is:

$m_{AB} = \frac{1-5}{1-0}$

$m_{AB} = -4$

The slope of the secant line of the function on the interval [0, 1] is -4. $\blacksquare$

d) Estimation of the slope at x = 1

The slope of the tangent line of f(x) at x = 1 can be estimated by concept of weighted averages, whose application is shown below:

$m = \left(\frac{1-0}{2-0}\right)\cdot (-4) + \left(\frac{2-1}{2-0} \right)\cdot (-2)$

m = -3

The estimated slope of the tangent line of f(x) at x = 1 is -3. $\blacksquare$

To learn more on secant lines, we kindly invite to check this verified question: link

Answer from: hma9153

Step-by-step explanation:

Given function is f(x) = x² - 5x + 5

(a). Slope of the secant of the function in the interval [1, 3],

m = $\frac{\triangle y}{\triangle x}$

For x = 1

y_1 = (1)²- 5(1) + 5

y_1 = 1

For x = 3,

y_2 = (3)² - 5(3) + 5

y_2 = 9 - 15 + 5

y_2 = -1

Slope 'm' = $\frac{y_2-y_1}{x_2-x_1}$

m = $\frac{1+1}{1-3}$

m = -1

(b). Slope of the secant of the function in the interval [1, 2],

m = $\frac{\triangle y}{\triangle x}$

For x = 1,

y_1 = 1 [From part a]

For x = 2,

y_2= (2)² - 5(2) + 5

= 4 - 10 + 5

= -1

Slope = $\frac{-1-1}{2-1}$

= -2

(c). Slope of the secant of the function in the interval [0, 1]

For x = 0,

y_1=0-5(0)+5

= 5

For x = 1

y_2=-1 [From part (b)]

m = $\frac{5+1}{0-1}$

= -6

(d). Derivative of the function 'f',

f'(x) = 2x - 5

At x = 1,

f'(x) = 2(1) - 5

= -3

Slope of the tangent = -3

Answer from: Quest

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Answer from: Quest

a) one student is chosen from classroom a, then a student is chosen from classroom b.

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b. this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes. this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes. this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes. this makes them dependent events.

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Find the slope of the secant line of the function f(x)=x2−5x+5 (a) On the interval [1,3]
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