Maths Questions - HELPA parabola

Find the equation of the curve $\displaystyle y=x^2 - 7x^2 + 5$ at the point (1, -1)


Also

One card is selected at random from a pack of 20 numbered 1 to 20. What is the probability that it is either a multiple of 5 or 3. I had 1/2, but apparently its wrong. Please tell where I went wrong.


Thanks

Quote:

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Originally Posted by mibamars

Find the equation of the curve $\displaystyle y=x^2 - 7x^2 + 5$ at the point (1, -1)

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Hello,

1. I assume that there is a typo and your function reads:
$\displaystyle y = x^2-7x+5$

2. do you want to find the equation of the tangent line at the graph of your function with the tangent point (1, -1)? If so:

The slope of the tangent has the same value as the gradient of the function in the tangent point.
Calculate the 1st derivation:

EDIT: As angel.white pointed out I've made one of my usual silly mistakes. The following calculations are corrected - by me! Therefore check my calculations!

$\displaystyle y'=2x-7$ . Now plug in x = 1. You'll get

$\displaystyle y'= -5$ That means the tangent has a slope of m = -5.

Now use point-slope-formula of a straight line:

$\displaystyle (y-(-1))=-5(x-1)~\iff~\boxed{y=-5x+4}$

Quote:

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Originally Posted by mibamars

One card is selected at random from a pack of 20 numbered 1 to 20. What is the probability that it is either a multiple of 5 or 3. I had 1/2, but apparently its wrong. Please tell where I went wrong....

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Hi,

obviously you counted th 15 twice because 15 is a multiple of 5 and a multiple of 3. But you can count the 15 only once!

Quote:

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Originally Posted by earboth

Hello,

1. I assume that there is a typo and your function reads:
$\displaystyle y = x^2-7x+5$

2. do you want to find the equation of the tangent line at the graph of your function with the tangent point (1, -1)? If so:

The slope of the tangent has the same value as the gradient of the function in the tangent point.
Calculate the 1st derivation:

$\displaystyle y'=2x+7$ . Now plug in x = 1. You'll get

$\displaystyle y'= 9$ That means the tangent has a slope of m = 9.

Now use point-slope-formula of a straight line:

$\displaystyle (y-(-1))=9(x-1)~\iff~\boxed{y=9x-10}$

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Looks like a negative sign got misplaced in y'

Hello, mibamars!

Quote:

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One card is selected at random from a pack of 20 numbered 1 to 20.
What is the probability that it is either a multiple of 5 or 3.

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As earboth pointed out, you overcounted.

The cards with a multiple of 3 or 5 are: .$\displaystyle \{3,\,5,\,6,\,9,\,10,\,12,\,15,\,18,\,20\}$
. . There are only nine of them.


They expected you count carefully ... or use this formula:

. . $\displaystyle P(\text{mult.of 3 } \vee \text{ mult. of 5}) \;\;=\;\;P(\text{mult.of 3}) \;+\; P(\text{mult.of 5})$ $\displaystyle \;-\; P(\text{mult.of 3 } \wedge \text{ mult.of 5}) $