Finding the Slope of an Implicit Curve at a Specific Point
Determine the slope of the tangent line to the curve \(x^2 + xy + y^2 = 7\) at the point \( (1,2) \).
Minimizing Travel Distance with Constraints
A courier company is planning a new hub location along a straight road. A warehouse is located 10 km north of the road. To minimize travel time, the company wants to find the point on the road where the distance from the warehouse is minimized under the constraint that the road has a curved section […]
Concavity and Second Derivative Analysis
Given the circle defined by the implicit equation \(x^2 + y^2 = 25\), use implicit differentiation to find the second derivative \(\frac{d^2y}{dx^2}\). Then, determine the concavity of the graph in the first quadrant.
Drainage Rate and Optimization of a Non-uniform Tank
An industrial storage tank with a complex shape is draining. The volume \(V\) of liquid in the tank is related to the liquid height \(h\) by the equation \(V = \frac{1}{3}\pi h^3 + 2h\). If the liquid height decreases at a rate of \(0.5 \text{ m/min}\) when \(h = 4 \text{ m}\), determine the rate […]
Tangent Line to an Implicitly Defined Curve at an Intersection Point
Consider the curve defined implicitly by \(2x^2 + 3xy + y^2 = 20\). The curve intersects the horizontal line \(y = 2\) at two points. Identify the intersection point with a positive \(x\)-coordinate and use implicit differentiation to determine the slope of the tangent to the curve at that point. Then, write the equation of […]
Optimizing Profit Function using Derivatives
A company is analyzing its profit function based on its pricing strategy. Given the price-demand function \(P(x)=100-2x\) and a cost per unit of \(c=20\), perform the following tasks:\n\n(a) Write the profit function \(\pi(x)\) in terms of \(x\).\n\n(b) Differentiate \(\pi(x)\) with respect to \(x\).\n\n(c) Determine the value of \(x\) that maximizes the profit and compute the […]
Implicit Differentiation and Horizontal Tangents
Consider the implicit curve defined by \(x^2 + xy + y^2 = 7\). Using implicit differentiation, find an expression for \(\frac{dy}{dx}\). Then determine the coordinates of the points on the curve where the tangent is horizontal.
Ladder Sliding: Implicit Differentiation and Related Rates
A 10-meter ladder leans against a vertical wall. As the ladder slides, the base moves away from the wall along the horizontal ground, and the top slides downward along the wall, maintaining a right triangle at all times.
Optimizing the Surface Area of a Closed Cylinder
A manufacturer wishes to design a closed cylindrical can (with both a top and a bottom) having a fixed volume \(V\) (in cubic units). The surface area of the can is given by \(S = 2\pi r^2 + 2\pi r h\), where \(r\) is the radius and \(h\) is the height. Using the volume constraint […]
Determining the Rate of Increase of a Balloon’s Radius
A spherical balloon is being inflated such that its volume increases at a constant rate of 100 cm\(^3\)/min. Given the volume of a sphere is \(V = \frac{4}{3}\pi r^3\), find the rate of change of the radius \(r\) when \(r = 5\) cm.