If f(x)=3x^2-5x, find the f'(2) & use it to find an equation of tangent line to the parabola
y=3x^2-5x at the point (2,2).
My ans is f'(2)=7 & y=7x-12.
What is parabola exactly? & is the any possibility if the eqtn of f(x) is not the same with parabola eqtn? If yes, how to calculate the tangent equation ? Can u give any clue or tips or keywords for answering this kind of question....
excuse me? You're in calculus, and don't know what a parabola is?
Regardless, just work with the math.
y'(x) = 6x-5
y'(2) = 7
y(2) = 2
So, you want the equation of the line passing through (2,2) with slope 7:
(y-2) = 7(x-2)
Your answer is correct.
Any quadratic equation represents a parabola. Go to wolfram and type in some quadratic functions. You will see that they are all parabolas.
A parabola is a U-shaped curve that is symmetric about a certain line called the axis of symmetry. The equation of a parabola in general form is y = ax^2 + bx + c, where a, b, and c are constants.
In this case, the equation of the parabola is y = 3x^2 - 5x. The given function, f(x), is the same as the equation of the parabola, so you're correct on that.
To find the derivative of f(x) and evaluate it at x = 2 (which will give you f'(2)), you can use the power rule of differentiation. For y = ax^n, the derivative is dy/dx = anx^(n-1). Applying this rule to f(x) = 3x^2 - 5x, you would differentiate each term separately:
f'(x) = d/dx (3x^2) - d/dx (5x)
= 6x - 5
Now, to find f'(2), you substitute x = 2 into the derived function:
f'(2) = 6(2) - 5
= 7
So, your answer of f'(2) = 7 is correct.
Next, to find the equation of the tangent line to the parabola at the point (2,2), you can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope of the line.
The slope of the tangent line is the value of f'(2), which is 7. The point on the line is (2, 2). Substituting these values into the point-slope form, we get:
y - 2 = 7(x - 2)
To simplify this equation, distribute 7 into (x - 2):
y - 2 = 7x - 14
Now, simplifying further, move 2 to the other side:
y = 7x - 12
So, your answer for the equation of the tangent line, y = 7x - 12, is correct.
In general, when finding the tangent line to a curve at a given point, you need to find the derivative of the function and evaluate it at the point to find the slope. Then, you can use the point-slope form to find the equation of the tangent line. Some key concepts and keywords to keep in mind are derivatives, tangent, slope, and point-slope form.