Expert: Paul Klarreich - 4/21/2006

Question
Hello Paul,
This  following problem is probably the most difficult problem I have ever faced in Calculus as I know how to do it, but I don't get the right answer choice:Here is the question that I'm hoping you can help me with:
A curve has slope 2x+3 at each point (x,y) on the curve. Which of the following  is an equation for this curve if it passes through the point (1,2)?
The answer from the answer key is y=x^2+3x-2;Hopefully, you'll get that!
I tried 3 different ways already and I'm out of ideas.
1rst attempt:y-2=2x+3(x-1)=2x+3x-3 so y=5x-1[not even an answer choice!]
2nd attempt:I plugged 1 from(1,2) into 2x+3 to get 5. Then y-2=5(x-1) getting y=5x-3
3rd attempt:y-2=(2x+3)(x-1)=2x^2-2x+3x-3 so y=2x^2+x-1[not an answer choice;they don't even let me guess!]

Answer
Hi, Jeff,

Subject:  tangent line
Question:  Hello Paul,
This following problem is probably the most difficult problem I have ever faced in Calculus as I know how to do it, but I don't get the right answer choice:Here is the question that I'm hoping you can help me with:

A curve has slope 2x+3 at each point (x,y) on the curve. Which of the following is an equation for this curve if it passes through the point (1,2)?
The answer from the answer key is y=x^2+3x-2;Hopefully, you'll get that!

I tried 3 different ways already and I'm out of ideas.

First attempt:  y-2=2x+3(x-1)=2x+3x-3 so y=5x-1[not even an answer choice!]

>>> NO WAY.


2nd attempt:I plugged 1 from(1,2) into 2x+3 to get 5. Then y-2=5(x-1) getting y=5x-3

>> NO GOOD, EITHER.

3rd attempt:y-2=(2x+3)(x-1)=2x^2-2x+3x-3 so y=2x^2+x-1[not an answer choice;they don't even let me guess!]
--------------------------------------------

I see it is getting close to crunch time and you are panicking.  No need -- just keep your wits about you and you'll do fine.

If the slope is given, that means dy/dx is given.
m = 2x + 3 means
dy/dx = 2x + 3  
which means, after you integrate,
y = x^2 + 3x + C.

Now substitute: (1,2)

2 = 1^2 + 3(1) + C
2 = 1 + 3 + C
C = -2

The equation is  y = x^2 + 3x - 2

Is that one of the choices?