Differential Calculus

The curve passing through the point (1, 2) given that the slope of the tangent at any point (x, y) is represents which type of curve?

The slope of the tangent at any point is .

We can separate variables to solve this differential equation:

Integrate both sides:

Since the curve passes through (1, 2), we can substitute x = 1 and y = 2 to find C:

So the equation of the curve is . Rearranging, we get , or .

This is the equation of a hyperbola.

Therefore, the correct option is (D) Hyperbola.

We need to find the maximum value of for x > 0.

Let . To find the maximum value, we need to find the critical points by finding the first derivative and setting it to zero.

Using the quotient rule:

Set :

Now we need to verify that x = e corresponds to a maximum. We can use the second derivative test.

Using the quotient rule:

Evaluate :

Since , the critical point at x = e corresponds to a maximum.

Now, we find the maximum value of f(x) at x = e:

Therefore, the maximum value of is .

Thus, the correct option is (C) .

If , then we need to find .

Rewrite y as:

First, find the first derivative :

Next, find the second derivative :

Now, multiply by :

Since , we have:

Therefore, the correct option is (B) .

If , then we need to find .

Differentiate both sides with respect to x using implicit differentiation:

Using the chain rule and the derivative of is :

Divide by :

Isolate :

Since , substitute in the above expression:

Therefore, the correct option is (B) .

The angular displacement is given by the equation: To find the angular velocity , we need to take the first derivative of the angular displacement with respect to time : Differentiating with respect to : The derivative of is: The derivative of is: Therefore, the angular velocity is: Now, at , the angular velocity is: We are told that the angular velocity at is , so . Therefore, the value of is: Thus, the correct answer is , which corresponds to option (D).

We are asked to find the value of the derivative at for the function:

This function is a product of powers, so logarithmic differentiation is a convenient method.

Step 1: Take the natural logarithm of both sides.

Using logarithm properties and :

Step 2: Differentiate both sides with respect to .

Using implicit differentiation for and the chain rule for the terms on the right:

Step 3: Solve for .

Substitute the original expression for :

Step 4: Evaluate the derivative at .

First, evaluate at :

Next, evaluate the expression in the brackets at :

Now, substitute these values back into the expression for :

The value of the derivative at is 516.

So, the correct answer ia (D): 516

We are tasked with differentiating the expression: We will differentiate each term separately. 1. Differentiating : The derivative of can be computed using logarithmic differentiation. Take the natural logarithm of : Now differentiate both sides with respect to : Using the product rule: So: 2. Differentiating : The derivative of (where is constant) is: 3. Differentiating : The derivative of (where is constant) is: 4. **Differentiating : Since is a constant, its derivative is 0. Now, summing all the derivatives:

Therefore, the correct answer is (B).

We are given the function:

We need to find the derivative of this function.
Step 1: Differentiate
Using the power rule, the derivative of is:

Thus, the first term gives: .
Step 2: Differentiate
Using the power rule, the derivative of is:

Thus, the second term gives: .
Step 3: Differentiate
The derivative of is:

Thus, the third term gives: .
Step 4: Combine the results
Thus, the derivative of the given function is:

Therefore, the correct answer is(B): , ,

The correct answer is: (C) units.

An enemy fighter jet is flying along the curve given by:

A soldier is placed at the point and wants to shoot down the jet when it is nearest to him. We are tasked with finding the nearest distance from the soldier to the jet.
Step 1: Find the distance between the soldier and the jet

The distance between a point and the point is given by the distance formula: Substituting into this formula: Simplifying:
Step 2: Minimize the distance function

To find the point where the distance is minimum, we differentiate with respect to and set it equal to 0. Minimizing is equivalent to minimizing , since is always positive. First, compute : Now, differentiate with respect to : Set the derivative equal to 0 to find the critical points: Divide through by 2:
Step 3: Solve for

Using trial and error or numerical methods, we find that satisfies the equation.
Step 4: Calculate the distance when

Now, substitute into the distance formula: Therefore, the nearest distance is units.
Thus, the correct answer is (C) units.