A-Level Edexcel Maths Pure Differentiation: 2. (a) Sketch the curve with equ

2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 2

Question 4

2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes. (b) Solve 4^x = 100 giving your answer to 2 decima... show full transcript

Worked Solution & Example Answer:2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 2

Step 1

Sketch the curve with equation y = 4^x stating any points of intersection with the coordinate axes.

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Answer

To sketch the curve defined by the equation $y = 4^x$ , we observe that:

Behavior of the Curve:
- The curve is an exponential function which rises steeply as $x$ increases. It is defined for all real $x$ and passes through the point (0, 1) since $4^0 = 1$ .
- As $x$ approaches negative infinity, $y$ approaches 0, meaning the curve never quite touches the x-axis (horizontal asymptote).
- The curve lies entirely in quadrants I and II since $y$ is always positive.
Points of Intersection:
- X-axis Intersection: There is no intersection with the x-axis as $y = 0$ has no solution in this context.
- Y-axis Intersection: The curve intersects the y-axis at the point (0, 1).

Step 2

Solve 4^x = 100 giving your answer to 2 decimal places.

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Answer

To solve the equation $4^x = 100$ , we can proceed as follows:

Taking Logarithms:
- Apply logarithm on both sides to make the exponent manageable. Using natural logarithm or logarithm base 10, the equation can be transformed:
x imes ext{log}(4) &= ext{log}(100)\ x &= \frac{ ext{log}(100)}{ ext{log}(4)}\ ext{Since, log}(100) = 2\ x &= \frac{2}{ ext{log}(4)} \ herefore x & \approx 3.32\ \ ext{Final Answer:} & x \approx 3.32\ \ ext{Thus, rounding to two decimal places, the solution is: }& 3.32. \end{align*}$$

2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 2

Worked Solution & Example Answer:2. (a) Sketch the curve with equation y = 4^ x stating any points of intersection with the coordinate axes - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 2

Sketch the curve with equation y = 4^x stating any points of intersection with the coordinate axes.

Solve 4^x = 100 giving your answer to 2 decimal places.

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