Find dydx and d2ydx2 and find the slope and concavity (if possible) at the given value of the parameter. Parametric Equations x=2+sec⁡θ, y=1+2tan⁡θ Parameter θ=−π3

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Find dydx and d2ydx2 and find the slope and concavity (if possible) at the given value of the parameter. Parametric Equations  x=2+sec⁡θ, y=1+2tan⁡θ Parameter θ=−π3

Beulah Bryan · Accepted Answer

Step 1  To find dydx and d2ydx2  and to find the slope and concavity at the given value of the parameter.  Parametric equations are x=2+sec⁡θ, y=1+2tan⁡θ, parameter θ=−π3  Step 2  Given Parametric equations are x=2+sec⁡θ, y=1+2tan⁡θ parameter θ=−π3  dxdθ=sec⁡θtan⁡θ  dydθ=2sec2θ  dydx=dydθ×dθdx=2sec2θsec⁡θtan⁡θ=2sec⁡θtan⁡θ  Now to find d2ydx2  d2ydx2=ddθ(dydx)×dθdx  ddθ(dydx)=ddθ(2sec⁡θtan⁡θ)=−2cos⁡θcos⁡sec⁡θ  d2ydx2=−2cot⁡θcos⁡sec⁡θsec⁡θtan⁡θ=−2cot3θ  Step 3  For the slope at θ=−π3  dydxθ=−π3=2sec⁡(π3)−tan⁡(π3)=−43  For the concavity at

Find \frac{dy}{dx} and \frac{d^{2}y}{dx^{2}} and find the slope and concavity

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