Tangents and Normals to a Curve

Mathematics · Calculus & Analysis · ages 17-18

Name ______________________ Date ____________

For the curve y = x² - 3 at x = 2, what is the gradient of the normal?
- 4
- -1/4
- -4
- 1/4
For the curve y = x², what is the gradient of the tangent at x = 5?
- 25
- 5
- 10
- 2
For the curve y = x² + 5 at x = 3, the point of contact is (3, 6).
Circle one: True False
For y = x^2 - 3x, what is the gradient of the tangent at the point where x = 2?
Answer: ______________
The tangent at (2, 7) has gradient 5. At what value of x does it cross the x-axis?
Answer: ______________
Find the equation of the normal to y = x² at the point (3, 9).
- x + 6y - 57 = 0
- y = 6x - 9
- x - 6y + 51 = 0
- 6x + y - 27 = 0
The tangent and the normal at a point have the same gradient.
Circle one: True False
Sam finds a tangent gradient of 2, writes the normal gradient as -2, and says the lines are perpendicular because 2 times -2 is negative. Is Sam right?
Circle one: True False
The normal to y = x² at the point (1, 1) crosses the y-axis. At what value of y?
Answer: ______________
Priya works out a tangent gradient of 3 and writes the normal gradient as -3. Where is the slip?
- the normal gradient should have been 3 as well
- the tangent gradient was wrong, so the normal must be too
- a normal gradient is always negative, so -3 is fine
- flipping the fraction was skipped, so it should be -1/3

Answer key

For grown-ups. Fold this page away before handing over the rest.

-1/4 · dy/dx = 2x gives a tangent gradient of 4 at x = 2. The normal gradient is -1/m, so -1/4, and 4 times -1/4 is -1.
10 · dy/dx = 2x, and at x = 5 that is 10. The 25 would be the height of the curve, not its steepness.
False · The height comes from the curve: 9 + 5 is 14, so the point is (3, 14). The 6 is dy/dx at x = 3, which is the gradient, not the height.
1 · dy/dx = 2x - 3, so at x = 2 the gradient is 4 - 3 = 1.
0.6 · From y - 7 = 5(x - 2) you get y = 5x - 3. Setting y = 0 gives 5x = 3, so x = 0.6.
x + 6y - 57 = 0 · dy/dx = 2x gives a tangent gradient of 6, so the normal gradient is -1/6. From y - 9 = -(1/6)(x - 3), multiplying by 6 gives 6y - 54 = -x + 3, or x + 6y - 57 = 0.
False · They meet at a right angle, so their gradients multiply to -1. Equal gradients would make them the same line.
False · Perpendicular gradients multiply to -1, not just to any negative number. The normal gradient is the negative reciprocal, -1/2, and 2 times -1/2 is -1.
1.5 · dy/dx = 2x gives a tangent gradient of 2, so the normal gradient is -1/2. From y - 1 = -(1/2)(x - 1) you get y = -0.5x + 1.5, so the crossing is at 1.5.
flipping the fraction was skipped, so it should be -1/3 · She changed the sign but never flipped the fraction. The test is a product of -1, and 3 times -3 is -9, not -1.