Application Of Derivatives

Step 1: Concept
The slope of the tangent at a point is the value of the first derivative at that point.
Step 2: Analysis
. At , slope .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Concept
The slope of the tangent at a point is the value of the first derivative at that point.
Step 2: Analysis
. At , slope .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Understand the given curves.
We are given two curves: 1.
2.
We need to determine whether these curves cut at right angles, touch each other, or cut at some other angle.

Step 2: Calculate the gradients of the curves.
The angle between two curves at their point of intersection can be determined by calculating the gradients (slopes) of the tangents to the curves at the intersection point. - For the first curve, , the gradient is given by the partial derivatives: - For the second curve, , the gradient is:

Step 3: Calculate the angle between the curves.
The angle between the two curves is given by the formula: where and are the gradients of the tangents to the curves at the point of intersection. After calculating the gradients and substituting into the formula, we find that , meaning the curves cut at right angles.

Step 4: Conclusion.
Thus, the two curves cut at right angles, corresponding to option (A).

Step 5: Verification.
By verifying the gradients and calculating the angle, we confirm that the curves indeed intersect at right angles.

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}


Step 3: Final Answer

Concept: Use tangent form and area of triangle.

Step 1: Tangent equation.
Intercept form gives triangle area.

Step 2: Area condition.


Step 3: Solve differential equation.
Leads to:

Concept: Let . In , .

Step 1: Find minimum value of for .

Step 2: Minimum value.

Step 3: Conclusion. For the equation to have at least one solution, must be at least the minimum value of the function:

Concept: Use Intermediate Value Theorem (IVT): If is continuous and changes sign in an interval, then it has a root in that interval.

Step 1: Given condition.


Step 2: Form function.


Step 3: Evaluate at two points.
Now use: Multiply by 2: Subtract from given: Thus:

Step 4: Substitute back.


Step 5: Function becomes.


Step 6: Solve quadratic.


Step 7: Check interval.
Conclusion:

Concept: Newton-Raphson iteration: , where and .

Step 1: Choose initial guess. (since , and ).

Step 2: First iteration.

Step 3: Second iteration (for verification).