A-Level Edexcel Maths Pure Equation of a Straight Line: A curve C has equation y = e^{2

A curve C has equation y = e^{2x} an x, x ≠ (2n + 1) rac{ au}{2} - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 6

Question 2

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A curve C has equation y = e^{2x} an x, x ≠ (2n + 1) rac{ au}{2}. (a) Show that the turning points on C occur where tan x = -1. (b) Find an equation of the ta... show full transcript

Worked Solution & Example Answer:A curve C has equation y = e^{2x} an x, x ≠ (2n + 1) rac{ au}{2} - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 6

Step 1

Show that the turning points on C occur where tan x = -1.

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Answer

To find the turning points of the curve C, we first differentiate the given equation.

Differentiate the function: The derivative of y with respect to x is computed using the product rule: $\frac{dy}{dx} = e^{2x} \tan x + e^{2x} \sec^2 x \cdot \frac{dx}{dx}$ Therefore, we get: $\frac{dy}{dx} = e^{2x} \tan x + 2e^{2x} \sec^2 x.$
Set the derivative to zero: To find the turning points, we set the derivative to zero: $\frac{dy}{dx} = 0 \Rightarrow 2e^{2x} \tan x + e^{2x} \sec^2 x = 0.$ We can simplify this expression: $2 \tan x + 1 \sec^2 x = 0.$
Solve the equation: Rearranging gives: $2 \tan x + 1 = 0 \Rightarrow \tan x = -1.$
Hence, turning points occur when (\tan x = -1).

Step 2

Find an equation of the tangent to C at the point where x = 0.

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Answer

To find the equation of the tangent to the curve C at the point where x = 0, follow these steps:

Evaluate the derivative at x = 0: Calculate (\frac{dy}{dx}) when (x = 0): $\frac{dy}{dx} \bigg|_{x=0} = e^{0} \cdot \tan(0) + 2e^{0} \cdot \sec^2(0) = 0 + 2 \cdot 1 = 2.$
Thus, the slope of the tangent line at (x = 0) is (2).
Determine the point on the curve: Calculate y when (x = 0): $y = e^{0} \cdot \tan(0) = 1 \cdot 0 = 0.$
So, the point is (0, 0).
Use the point-slope form to write the tangent equation: With point (0, 0) and slope 2, the equation is: $y - 0 = 2(x - 0) \Rightarrow y = 2x.$
Hence, the equation of the tangent line is (y = 2x).

A curve C has equation y = e^{2x} an x, x ≠ (2n + 1) rac{ au}{2} - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 6

Worked Solution & Example Answer:A curve C has equation y = e^{2x} an x, x ≠ (2n + 1) rac{ au}{2} - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 6

Show that the turning points on C occur where tan x = -1.

Find an equation of the tangent to C at the point where x = 0.

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