726Chapter 10 / Parametric and Polar Curves; Conic SectionsINTERSECTIONS OF POLAR GRAPHS
✔QUICK CHECK EXERCISES 10.3(See page 729 for answers.)1.(a) To obtaindy/dxdirectly from the polar equationr=f(θ), we can use the formuladydx=dy/dθdx/dθ=(b) Use the formula in part (a) to finddy/dxdirectly fromthe polar equationr=cscθ.2.(a) What conditions onf(θ0)andf (θ0)guarantee that thelineθ=θ0is tangent to the polar curver=f(θ)at theorigin?(b) What are the values ofθ0in[0,2π]at which the linesθ=θ0are tangent at the origin to the four-petal roser=cos 2θ?3.(a) To find the arc lengthLof the polar curver=f(θ)(α≤θ≤β), we can use the formulaL=.(b) The polar curver=secθ (0≤θ≤π/4) has arc lengthL=.4.The area of the region enclosed by a nonnegative polar curver=f(θ) (α≤θ≤β)and the linesθ=αandθ=βisgiven by the definite integral.5.Find the area of the circler=aby integration.EXERCISE SET 10.3Graphing UtilityCCAS1–6Find the slope of the tangent line to the polar curve for thegiven value ofθ.■1.r=2 sinθ;θ=π/62.r=1+cosθ;θ=π/23.r=1/θ;θ=24.r=asec 2θ;θ=π/65.r=sin 3θ;θ=π/46.r=4−3 sinθ;θ=π7–8Calculate the slopes of the tangent lines indicated in theaccompanying figures.■7.r=2+2 sinθ8.r=1−2 sinθ0c/2Figure Ex-70c/2Figure Ex-89–10Find polar coordinates of all points at which the polarcurve has a horizontal or a vertical tangent line.