Textbook Question
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
y² – 2x = 1 – 2y
4. Applications of Derivatives
Implicit Differentiation
6:10 minutesProblem 3.7.39
Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
Verified step by step guidance1
First, verify that the point (1, 0) is on the curve by substituting x = 1 and y = 0 into the equation y = 2 sin(πx - y). Check if the left-hand side equals the right-hand side.
To find the tangent line, we need the derivative of the function. Differentiate the equation y = 2 sin(πx - y) implicitly with respect to x. Remember to apply the chain rule and implicit differentiation.
Solve the resulting equation for dy/dx, which represents the slope of the tangent line at any point (x, y) on the curve.
Substitute the point (1, 0) into the expression for dy/dx to find the slope of the tangent line at this specific point.
The normal line is perpendicular to the tangent line. Use the negative reciprocal of the tangent slope to find the slope of the normal line. Then, use the point-slope form of a line equation to write the equations of both the tangent and normal lines at the point (1, 0).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Point Verification on a Curve
To verify that a given point is on a curve, substitute the point's coordinates into the curve's equation. If the equation holds true, the point lies on the curve. For the equation y = 2 sin(πx - y), substitute x = 1 and y = 0 to check if the left side equals the right side.
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Summary of Curve Sketching
Tangent Line to a Curve
The tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. To find the tangent line, calculate the derivative of the curve's equation to determine the slope at the given point, then use the point-slope form of a line to write the equation of the tangent line.
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Normal Line to a Curve
The normal line to a curve at a given point is perpendicular to the tangent line at that point. To find the normal line, first determine the slope of the tangent line, then use the negative reciprocal of this slope for the normal line. Apply the point-slope form using the given point to write the equation of the normal line.
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