Let f(x) be the function 1/x+5. Then the quotient
f(6+h)−f(6)/h can be simplified to −1/ah+b for:
a= 11
b=
I did all the work to get a but for some reason the number that I keep getting for "B" is wrong
prove d/dx (csc x) = −csc x cot x
First of all , I think you meant f(x) = 1/(x+5)
the difference quotient is
(f(l+h) - f(6))/h
= (1/(11+h) - 1/11 )/h
= (11 - 11 - h)/(11(11+h) (1/h)
= -1 / (11(11+h))
I notice another error is your typing
you probably meant to type:
..... can be simplified to -1/(a(h+b) )
compare -1/(11(11+h)) to -1/(a(h+b))
since we are told a=11, clearly b has to be 11 for them to match
To simplify the quotient f(6+h)−f(6)/h, we need to substitute the values into the function f(x) = 1/x + 5.
Let's first find f(6+h):
f(6+h) = 1/(6+h) + 5
Next, let's find f(6):
f(6) = 1/6 + 5
Now, we can substitute these expressions into the quotient:
(f(6+h)−f(6))/h = (1/(6+h) + 5 - (1/6 + 5))/h
To simplify this expression, we need to find a common denominator.
The common denominator for (6+h) and 6 would be 6(6+h). Let's rewrite the expression:
= (6(1/(6+h)) + 6(5) - (6(1/6) + 6(5)))/h
= ((6/(6+h)) + 30 - (6/6) - 30)/h
Now, let's simplify further:
= ((6/(6+h)) - (6/6))/h
Since 6 divided by 6 is 1, we can rewrite the expression as:
= (6/(6+h) - 1)/h
To simplify this further, we need to find a common denominator again.
The common denominator for (6+h) and 1 is (6+h). Let's rewrite the expression:
= (6 - (6+h))/(h(6+h))
= (6 - 6 - h)/(h(6+h))
= (-h)/(h(6+h))
Now, we can cancel out h:
= -1/(6+h)
Since we have a = 11, let's substitute that into the expression:
= -1/(6+11)
= -1/17
Hence, the simplified quotient is -1/17.
Regarding the value of b, please check your calculations again. If you provide the steps you took to calculate it, I can help you identify any errors.