RevisionDojo

Question 1

Specification

Use implicit differentiation to find the equation of the tangent line to the curve $x^2y - y^2 = 0$ at the point $(1,1)$ . Markscheme:

Differentiate with respect to $x$ : $2xy + x^2\frac{dy}{dx} - 2y\frac{dy}{dx} = 0$ M1 A1
Substitute $x=1$ and $y=1$ into the derivative equation (or isolate $\frac{dy}{dx}$ first): $2(1)(1) + (1)^2\frac{dy}{dx} - 2(1)\frac{dy}{dx} = 0$ M1
Calculate the gradient: $2 - \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = 2$ A1
Substitute the point $(1,1)$ and gradient $m=2$ into the equation of a line: $y - 1 = 2(x - 1)$ M1
State the final equation in the form $y = mx + c$ or $ax + by + d = 0$ : $y = 2x - 1$ A1

6 marks total

Use implicit differentiation to find the equation of the tangent line to the curve $x^2y - y^2 = 0$ at the point $(1,1)$ .

[6]

Question 2

Find the equation of the tangent line to the curve $e^x + e^y = 2e$ at the point $(1,1)$ .

[6]

Question 3

Find the equation of the tangent line to the curve $\ln(x) + \ln(y) = 0$ at the point $(1,1)$ .

[5]

Question 4

Find the equation of the tangent line to the curve $e^x + xy = e + 1$ at the point $(1,1)$ .

[6]

Question 5

Find the equation of the tangent line to the curve $x^3 + y^3 = 2$ at the point $(1,1)$ .

[5]

Question 6

Find the equation of the tangent line to the curve $\ln(xy) = 0$ at the point $(1,1)$ .

[6]

Question 7

Find the equation of the tangent line to the curve $xy + y^2 = 2$ at the point $(1,1)$ .

[5]

Question 8

Find the equation of the tangent line to the curve $x^2 + y^2 + xy = 3$ at the point $(1,1)$ .

[6]

Question 9

Find the equation of the tangent line to the curve $\arctan\left(\frac{y}{x}\right) = \frac{\pi}{4}$ at the point $(1,1)$ .

[4]

Question 10

Find the equation of the tangent line to the curve $x^2 + y^2 = 2$ at the point $(1,1)$ .

[6]

Question 11

Find the equation of the tangent line to the curve $\sin(xy) = \sin(1)$ at the point $(1,1)$ .

[6]

Question 12

Find the equation of the tangent line to the curve $y^2 = x + \sin(x) - \sin(1)$ at the point $(1,1)$ .

[6]