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Given the functions defined by $f(x) = ext{cos}(x - 45^ ext{o})$ and $g(x) = -2 ext{sin}(x)$, where $x ext{ is in } [0^ ext{o}, 360^ ext{o}]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2024 - Paper 2
Question 5
![Given-the-functions-defined-by---$f(x)-=--ext{cos}(x---45^-ext{o})$-and-$g(x)-=--2--ext{sin}(x)$,-where-$x--ext{-is-in-}-[0^-ext{o},-360^-ext{o}]$----5.1-Draw-sketch-graphs-of-$f$-and-$g$-on-the-same-set-of-axes-provided-in-the-ANSWER-BOOK-NSC Technical Mathematics-Question 5-2024-Paper 2.png](https://cdn.simplestudy.io/assets/backend/uploads/question/subject_1520_paper_20926_q5_679a5deb.jpg)
Given the functions defined by $f(x) = ext{cos}(x - 45^ ext{o})$ and $g(x) = -2 ext{sin}(x)$, where $x ext{ is in } [0^ ext{o}, 360^ ext{o}]$ 5.1 Draw sketch... show full transcript
Worked Solution & Example Answer:Given the functions defined by $f(x) = ext{cos}(x - 45^ ext{o})$ and $g(x) = -2 ext{sin}(x)$, where $x ext{ is in } [0^ ext{o}, 360^ ext{o}]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2024 - Paper 2
Step 1
5.1 Draw graph of $f$ and $g$
Answer
To sketch the graphs of ( f(x) ) and ( g(x) ), we need to determine their characteristics:
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Graph of ( f(x) = \text{cos}(x - 45^\circ) ):
- Amplitude: 1
- Period: 360°
- X-intercepts: Occurs when ( f(x) = 0 ), which is at ( x = 45^\circ + 90n ) for integers ( n )
- Turning Points: Maximum at ( x = 0^\circ ) and Minimum at ( x = 180^\circ )
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Graph of ( g(x) = -2\text{sin}(x) ):
- Amplitude: 2
- Period: 360°
- X-intercepts: Occurs when ( g(x) = 0 ), which happens at ( x = 0^\circ, 180^\circ, \text{ and } 360^\circ )
- Minimum: At ( 90^\circ )
Both graphs should be plotted on the same set of axes, indicating turning points, intercepts and any end-points clearly.
Step 2
Step 3
Step 4
5.4 Use the letters A and B to indicate on the graphs where $ rac{1}{2} ext{cos}(x - 45^ ext{o}) = ext{sin}(x)$.
Answer
To solve for where ( \frac{1}{2} \text{cos}(x - 45^\circ) = \text{sin}(x) ), mark the intersection points on the graph as:
- Point A: Intersection graphically at ( x > 45^\circ )
- Point B: Intersection graphically at ( x < 225^\circ )
Step 5
5.5 Use the graphs drawn in QUESTION 5.1 to determine the values of $x$ for which $f''(x) < 0$.
Answer
From the graph of ( f(x) ), ( f''(x) < 0 ) in the intervals where the graph is concave down, which occurs:
- For ( 45^\circ < x < 225^\circ )
- In other words, between the critical values of ( 45^\circ ) and ( 225^\circ ).