The first derivative dy/dx gives us the rate of change of y with respect to x which is $(e^{(-t)} – t * e^{(-t)})$ / $e^t$ and the second derivative d²y/dx² is [(- $e^{(-t)} + t * e^{(-t)}) * e^t – (e^{(-t)} – t * e^{(-t)}) * e^t] / e^{(2t)}$ = -2. To find dy/dx and d²y/dx² at […]

# Category Archives: Calculus Q&A

Calculate the surface integral over a hemisphere defined by $x^2 + y^2 + z^2 = 9$ and z ≥ 0, with the integrand $x^2z + y^2z$. Using spherical coordinates, we parameterize the surface and compute the integral for this geometric region. Expert Answer To evaluate the surface integral over the hemisphere S defined by $x^2 […]

[ boldsymbol{ oint xy ds text{ where s is defined by } x = t^2 text{ and } y = 2t text{ over the interval } 0 leq t leq 4 } ] The aim of this question is to learn how to solve line integrals over some closed surfaces. To solve this question, we […]

What is the pressure difference? How much force is exerted on the roof? If the roof cannot withstand this much force, will it “blow in” or “blow out?” The main objective of this problem is to determine the air pressure, pressure difference, and the force exerted by hurricane wind on the roof. Bernoulli’s equation is being […]

The aim of this question is to deduce the equation of a tangent line of a curve at any point on the curve. For any given function $ y = f(x) $, the equation of its tangent line is defined by the following equation: [ boldsymbol{ y – y_1 = frac{ dy }{ dx } ( x – […]