Question

AP CAL AB!!!! HELP!!

Answer 1

1)
`f'(x)= (3)/(2√(3x-5))`

2)
`y=(3)/(4)x-(1)/(4)`

Explanation:

So we have the function:

1)

And we want to find its derivative using the limit definition:

`\lim_(h \to 0) (f(x+h)-f(x))/(h)`

So, let's do so:

`\lim_(h \to 0) (√(3(x+h)-5)-√(3x-5))/(h)`

Simplify the first square root:

`\lim_(h \to 0) (√(3x+3h-5)-√(3x-5))/(h)`

Now, let's remove the square root by multiplying by its conjugate. So:

`\lim_(h \to 0) (√(3x+3h-5)-√(3x-5))/(h)\cdot (( √(3x+3h-5)+√(3x-5) )/(√(3x+3h-5)+√(3x-5) ))`

In the numerator, difference of two squares. In the denominator, multiply:

`\lim_(h \to 0) ((3x+3h-5)-(3x-5))/(h(√(3x+3h-5)+√(3x-5)))`

Distribute the numerator:

`\lim_(h \to 0) ((3x+3h-5)-3x+5)/(h(√(3x+3h-5)+√(3x-5)))`

Simplify:

`\lim_(h \to 0) (3h)/(h(√(3x+3h-5)+√(3x-5)))`

Cancel out the h:

`\lim_(h \to 0) (3)/(√(3x+3h-5)+√(3x-5))`

Direct substitution:

`= (3)/(√(3x+3(0)-5)+√(3x-5))`

Simplify:

`= (3)/(√(3x-5)+√(3x-5))`

Combine like terms:

`= (3)/(2√(3x-5))`

Therefore:

`f'(x)= (3)/(2√(3x-5))`

Your answer is indeed correct!

2)

Now, let's find the equation of the tangent line to the graph at x=3.

Remember what the derivative gives us. The derivative tells us the slope of the tangent line to a graph at a certain point. So, let's substitute 3 into f'(x) to find the slope of our tangent line:

`f'(3)= (3)/(2√(3(3)-5))`

Multiply and subtract:

`f'(3)= (3)/(2√(9-5))\\f'(3)= (3)/(2√(4))`

Simplify:

`f'(3)= (3)/(2(2))\\f'(3)= (3)/(4)`

Therefore, the slope of our tangent line is 3/4.

Now, let's find the equation of the line using the point-slope form. So, we need to point at x=3 of the original graph. So, substitute 3 into the original function:

`f(3)=√(3(3)-5)`

Multiply, subtract, and simplify:

`f(3)=√(9-5)\\f(3)=√(4)\\f(3)=2`

Therefore, our point is (3,2).

Now, we can use the point-slope form, where:

`y-y_1=m(x-x_1)`

Let (3,2) be (x₁, y₁) and substitute 3/4 for m. Therefore:

`y-2=(3)/(4)(x-3)`

Distribute:

`y-2=(3)/(4)x-(9)/(4)`

Add 2 to both sides:

`y=(3)/(4)x-(1)/(4)`

And we're done!