The point on the curve for y = sqrt(2x+1) at which tangent is perpendicular to the line y = -3x + 6 is....
a. (4, 3) b. (0,1) c. (1, sqrt3) d. (4, -3) e. (2, sqrt 5)
I'm not sure, but do you find the perpendicular slope (1/3) set that as dy/dx and solve for x or y? I tried that and it didn't work. Help, please?
the slope of the tangent at any point (x,y) is dy/dx = 1/sqrt(2x+1)
The slope of the line given is -3
The slope of the perpendicular to that line is thus 1/3
So, you want 1/3 = 1/sqrt(2x+1)
3 = sqrt(2x+1)
9 = 2x+1
x = 4
y(4) = 3
so, the point in question is (4,3)
Well, this question seems a bit complicated, but don't worry, I'm here to help... with a touch of humor!
First, let's find the derivative of the given function y = √(2x + 1). Differentiating it, we get:
dy/dx = (1/2) * (2x + 1)^(-1/2)
Now, to find the point on the curve where the tangent is perpendicular to the line y = -3x + 6, we need to find the x-coordinate of that point.
The slope of the line y = -3x + 6 is -3. For the tangent to be perpendicular to this line, the slope of the tangent should be the negative reciprocal of -3, which is 1/3.
So, let's set dy/dx equal to 1/3 and solve for x:
(1/2) * (2x + 1)^(-1/2) = 1/3
Now, let's square both sides of the equation to get rid of the fractional exponents:
[(1/2) * (2x + 1)^(-1/2)]^2 = (1/3)^2
Simplifying this, we have:
(1/4) * (2x + 1)^(-1) = 1/9
Multiplying both sides by 4, we get:
(2x + 1)^(-1) = 1/36
Now, take the reciprocal of both sides:
2x + 1 = 36
Subtracting 1 from both sides:
2x = 35
Dividing both sides by 2:
x = 35/2
Unfortunately, this value is not among the given options. So, it seems that none of the options provided match the point on the curve where the tangent is perpendicular to the line y = -3x + 6.
Well, sometimes calculus can be challenging and unpredictable. But hey, at least you got a laugh out of it! Keep up the good work and don't be discouraged.
To find the point on the curve where the tangent is perpendicular to the line y = -3x + 6, you are correct that you need to find the perpendicular slope first.
The given line has a slope of -3. To find the perpendicular slope, you need to find the negative reciprocal of -3, which is 1/3.
Now, let's find the derivative of the given curve y = sqrt(2x + 1) and set it equal to 1/3 to find the x-coordinate:
dy/dx = (1/2) * (2x + 1)^(-1/2)
Setting this equal to 1/3:
(1/2) * (2x + 1)^(-1/2) = 1/3
Now, solve this equation for x:
(2x + 1)^(-1/2) = 1/3
(2x + 1)^(1/2) = 3
2x + 1 = 9
2x = 8
x = 4
So the x-coordinate of the point on the curve is 4.
Now, substitute this x-value back into the original equation y = sqrt(2x + 1) to find the corresponding y-coordinate:
y = sqrt(2 * 4 + 1)
y = sqrt(9)
y = 3
Therefore, the point on the curve for y = sqrt(2x + 1) at which the tangent is perpendicular to the line y = -3x + 6 is (4, 3).
So, the correct option is a. (4, 3).
To find the point on the curve for which the tangent is perpendicular to the line y = -3x + 6, you need to consider the slopes of both curves.
First, find the slope of the given line y = -3x + 6. This equation is in the form y = mx + c, where m represents the slope of the line. So, the slope of this line is -3.
To find the slope of the tangent line at a point on the curve y = sqrt(2x + 1), you need to differentiate the equation y = sqrt(2x + 1) with respect to x. Differentiating gives us:
dy/dx = (1/2)(2x + 1)^(-1/2)(2)
Simplifying, we get:
dy/dx = (2/2sqrt(2x + 1)) = 1/sqrt(2x + 1)
Now, for the tangent line to be perpendicular to the given line, the slope of the tangent line must be the negative reciprocal of the slope of the given line. In this case, we want the tangent line slope to be 1/3.
So, we set the slope dy/dx equal to 1/3 and solve for x:
1/sqrt(2x + 1) = 1/3
To solve this equation, you need to square both sides:
(1/sqrt(2x + 1))^2 = (1/3)^2
1/(2x + 1) = 1/9
Cross-multiplying gives us:
9 = 2x + 1
2x = 9 - 1
2x = 8
x = 4
Now that we have x = 4, we can substitute it back into the original equation y = sqrt(2x + 1) to find the y-coordinate:
y = sqrt(2(4) + 1) = sqrt(8 + 1) = sqrt(9) = 3
So, the point on the curve is (4, 3).
Therefore, the correct answer is a. (4, 3).