Find the equation of the tangent line and use Desmos to draw the graph of (x, y) and the tangent line (save it to file and print). The equation is y = x^2 at x = 2 and y = 5 at x = 8. Also, find the equation y = zx^2 + vi + # at x = [
The Correct Answer and Explanation is:
To find the equation of the tangent line to the curve y=x2y = x^2y=x2 at the point where x=2x = 2x=2, let’s go through the steps in detail.
Step 1: Find the slope of the tangent line
The slope of the tangent line is given by the derivative of the function y=x2y = x^2y=x2.
The derivative of y=x2y = x^2y=x2 is: dydx=2x\frac{dy}{dx} = 2xdxdy=2x
At x=2x = 2x=2, substitute into the derivative: dydx=2(2)=4\frac{dy}{dx} = 2(2) = 4dxdy=2(2)=4
So, the slope of the tangent line at x=2x = 2x=2 is 444.
Step 2: Find the point on the curve at x=2x = 2x=2
The point on the curve is given by substituting x=2x = 2x=2 into the original function: y=22=4y = 2^2 = 4y=22=4
Thus, the point of tangency is (2,4)(2, 4)(2,4).
Step 3: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is: y−y1=m(x−x1)y – y_1 = m(x – x_1)y−y1=m(x−x1)
Where:
- mmm is the slope of the tangent line (444),
- (x1,y1)(x_1, y_1)(x1,y1) is the point on the curve (2,42, 42,4).
Substitute these values into the equation: y−4=4(x−2)y – 4 = 4(x – 2)y−4=4(x−2)
Simplify: y−4=4x−8y – 4 = 4x – 8y−4=4x−8 y=4x−4y = 4x – 4y=4x−4
Step 4: Plot the curve and tangent line using Desmos
- Open Desmos.
- Enter the function y=x2y = x^2y=x2 in the input box.
- Enter the tangent line equation y=4x−4y = 4x – 4y=4x−4 in another input box.
- Add the points (2,4)(2, 4)(2,4) and (8,64)(8, 64)(8,64) to show the specific points on the curve.
Once you’ve done this, you can export the graph as an image by clicking the “Share” button in the top right corner and selecting “Export Image”.
For the second equation, y = zx^2 + vi + #, it’s not clear what you mean by “vi” and “#”. Could you clarify these terms so I can proceed with the correct interpretation?
