VIPSolutionsGive step-by-step solution with explanation end and final answer: Find an equation of the tangentline at x=4:y=sqrt[2x^2-7]€ Ng = ’ _—- & ~ Ny : aS fa De Sg Ba - Ma oy hah > e) F - SENT 1 EN SURE Eo BEAR x Na WN Ge HAY: * is : : se fy pati a x -
Question:
Give step-by-step solution with explanation end and final answer:
Find an equation of the tangentline at x=4:y=sqrt[2x^2-7]
€ Ng = ’
_—- & ~ Ny
: aS fa De Sg Ba - Ma oy hah > e) F -
SENT 1 EN SURE Eo BEAR x
Na WN Ge HAY: *
is : : se fy pati a x -
This is AI generated Response.
Asked by: pooja gargunde
Created at: 2025-10-07 11:00:30
Asked by: pooja gargunde
Created at: 2025-10-07 11:00:30
Thought Process:
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**Calculating tangent line equation**
I'm breaking down how to find an equation for the tangent line at x=4 for the function y = √(2x² - 7). First, I evaluate y at x=4, which gives me the point (4, 5). Next, I differentiate to find the slope, giving me dy/dx = 2x/√(2x² - 7). At x=4, the slope is 8/5. Thus, the tangent line is y - 5 = (8/5)(x - 4), which simplifies to y = (8/5)x - 7/5.**Preparing final message**
I want to ensure my final message includes all necessary derivative steps and uses the chain rule properly. I need to follow the instructions for a step-by-step solution with a clear explanation and the final answer. To keep it straightforward, I’ll avoid heavy formatting but can use numbered steps. It’s important to note the domain requirement of 2x² - 7 ≥ 0, and since x=4 is valid, everything checks out. Now I’m ready to finalize my response!
Final Answer:
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Step-by-step:
1) Function and point:
- y = sqrt(2x^2 − 7) = (2x^2 − 7)^(1/2)
- At x = 4: y = sqrt(2·4^2 − 7) = sqrt(32 − 7) = sqrt(25) = 5
- Point of tangency: (4, 5)
2) Slope of the tangent (derivative):
- dy/dx = (1/2)(2x^2 − 7)^(−1/2) · (4x) [chain rule]
- Simplify: dy/dx = 2x / sqrt(2x^2 − 7)
3) Evaluate slope at x = 4:
- m = 2·4 / sqrt(32 − 7) = 8/5
4) Equation of the tangent line through (4,5) with slope 8/5:
- Point-slope form: y − 5 = (8/5)(x − 4)
- Slope-intercept form: y = (8/5)x − 32/5 + 25/5 = (8/5)x − 7/5
Final answer:
y − 5 = (8/5)(x − 4), equivalently y = (8/5)x − 7/5.
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