Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x) = 7x ^ 2 + 2x + 3 at x = 1

Answer 1

16

Explanation:

Step 1: Write down the function
`f(x)=7x^2+2x+3.`

Step 2: Write down the limit definition of the derivative:

`f'(x)= lim_(h0) (f(x+h)=f(x))/(h) .`

Step 3: Substitute the function
`f(x)` into the limit definition:

`f'(x)=lim_(h0) ((7(x+h)^2+2(x+h)+3)-(7x^2+2x+3))/(h).`

Step 4: Simplify the expression inside the limit:

`f'(x)=lim_(h0)(7x^2+14xh+7h^2+2x+2h+3-7x^2-2x-3)/(h) .`

Step 5: Combine like terms:

`f'(x)=lim_(h0) (14xh+7h^2+2h)/(h) .`

Step 6: Factor out an
`h` from the numerator:

`f'(x)=lim_(h0) (h(14x+7h+2h)/(h) .`

Step 7: Cancel out the
`h` in the numerator and denominator:

`f'(x)=lim_(h0)(14x+7h+2).`

Step 8: Evaluate the limit as
`h` approaches 0:

`f'(x)=14x+2.`

Step 9: Substitute
`x=1` into the derivative:

`f'(1)=14(1)+2=14+2=16.`

The Slope of the tangent line to the curve
`f(x)=7x^2+2x+3` at
`x=1` would be
`16.`