Applications Of Derivatives

Step 1: Use the derivative. The function is strictly increasing if its derivative is positive. The derivative of is .
Step 2: Solve for . For the function to be increasing on , must be positive on this interval. Solving for , we get the least value .
Step 3: Conclusion. Thus, the least value of is -2.

Step 1: Finding the critical points.
Differentiate the function and set the derivative equal to zero to find the critical points. After solving, evaluate the maximum value at the critical point.

Step 2: Conclusion.
Thus, the maximum value of is . Hence, the correct answer is option (C).

Final Answer:

Step 1: Equation of tangents.
Use the parametric equations to find the derivatives of the curve to obtain the slopes of the tangents at the points where the curve intersects the x-axis. Then calculate the angle between the tangents.

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer:

Step 1: Write volume of sphere.

Step 2: Differentiate w.r.t. time .

Step 3: Substitute given values.


Final Answer:

Step 1: Find point of intersection.
Then:
Intersection point is .
Step 2: Find slopes of tangents at intersection.
For :
For :
Step 3: Angle between curves formula.
Substitute:
Using base-10 logs ( ), constant cancels, so:
Final Answer:

We are asked to find the maximum value of the function: Step 1: Substituting .
Let , where because takes values between -1 and 1. Now, the function becomes: This is a quadratic function in , and we need to find its maximum value for .
Step 2: Finding the vertex of the quadratic function.
The general form of a quadratic function is , and the vertex occurs at: For the function , we have: So, the vertex occurs at:
Step 3: Checking if the vertex is within the domain.
Since the vertex occurs at , which is outside the domain , we need to check the function values at the endpoints of the domain.
Step 4: Evaluating the function at the endpoints.
First, evaluate at : Next, evaluate at :
Step 5: Conclusion.
The maximum value of the function occurs at , and the value is . Therefore, the correct answer is (d) None of these.

Step 1: Understanding the surface area change.
The surface area of a sphere is given by . The rate of change of surface area is proportional to . Since the radius is increasing at a rate of , the rate of change of surface area is proportional to .

Step 2: Conclusion.
The rate of change of the surface area is proportional to , corresponding to option (2).

Step 1: Understanding the problem.
We are given the curves and , and we need to find the value of such that the area enclosed by the curves is 5. This can be done by setting up the area integral and solving for .

Step 2: Conclusion.
The value of is , corresponding to option (2).

Step 1: Applying the Mean Value Theorem.
By the Mean Value Theorem, since is differentiable on the interval , there exists a point such that: We know that and for all .
Step 2: Substituting known values.
Substituting the known values into the Mean Value Theorem equation, we get: Since , we have: Multiplying both sides by 5: Thus:
Conclusion.
The correct answer is (2) , as we have shown that must be at least 8.

The value of is derived from the derivative of the given function and applying the mean value theorem. We find that the value at satisfies the condition.

The minimum value of the function can be found by differentiating the function and finding the critical point where the derivative is zero. The minimum occurs at .

The equation of the tangent to the curve is given by the derivative at the point of tangency. The area of the triangle formed by the tangent line and the coordinate axes can be calculated, and the value of is found to be 6.

The normal equation at a point on a curve can be derived by finding the slope of the curve at that point and using the point-slope form of the line. The equation at is calculated as .

After simplifying the derivative, we find that the maximum of occurs at .

Step 1: Find the Velocity and Acceleration.
To find the velocity, differentiate the displacement equation with respect to time. The velocity is the first derivative, and the acceleration is the second derivative. Set acceleration to zero to find the instant when acceleration is zero. Then use the velocity equation to find the velocity.
Step 2: Conclusion.
The correct answer is (C), 28.

Step 1: Find the Velocity and Acceleration.
The velocity is the first derivative of the displacement function, and the acceleration is the second derivative. Set the acceleration equal to zero and calculate the velocity at that instant.
Step 2: Conclusion.
The correct answer is (C), 28.

Step 1: Find the maximum and minimum values of the function.
To find the maximum and minimum values, use the derivative of to determine the critical points.
Step 2: Conclusion.
Thus, the difference between the maximum and minimum values of is .
Final Answer:

Step 1: Differentiate the function.
The derivative of is . For the function to decrease, we need , which holds when .
Step 2: Conclusion.
Thus, the function decreases when .
Final Answer:

To find the rate of change of the depth of water, use the chain rule to differentiate the volume equation with respect to time.
Final Answer:

Step 1: To find the value of , differentiate the function and find the critical points.
Step 2: By substituting into the function and ensuring that it is a maximum, we solve for , which gives .

Final Answer:

Step 1: Rolle's Theorem guarantees that there exists at least one point in the interval where the derivative of is zero.
Step 2: To find , first compute the derivative of . We get: Set , which gives . Solving this gives .

Final Answer:

Step 1: The volume of a sphere is given by: Step 2: Differentiating with respect to time, we get: Step 3: Given and , substitute to find the rate of change of volume:
Final Answer:

Using the mean value theorem, calculate the derivative of and solve for in the interval .
Final Answer:

Given the condition , the derivative of will indicate that it is a strictly increasing function.
Final Answer:

The maximum area for a quadrilateral with right angles and a fixed perimeter occurs when it is a square. The area is calculated using the side length obtained from the perimeter.
Final Answer:

The volume of a cylinder is . Differentiate the equation with respect to time and substitute the given rates of change to get the rate of change of volume.
Final Answer:

Downstream speed = 10 km/h, upstream = 6.67 km/h

Equation: