高中數(shù)學(xué)必修4配北師版-課后習(xí)題Word版習(xí)題課-函數(shù)y=Asin(ωx+φ)的綜合應(yīng)用

1、習(xí)題課函數(shù)y=Asin(x+)的綜合應(yīng)用課后篇鞏固探究1.下列函數(shù)中,在4,2上是減少的,且周期為的是()A.y=sin2x+2B.y=cos2x+2C.y=sinx+2D.y=cosx+2解析C,D中函數(shù)周期為2,所以錯(cuò)誤.當(dāng)x4,2時(shí),2x+2,32,函數(shù)y=sin2x+2為減少的,而函數(shù)y=cos2x+2為增加的.答案A2.已知函數(shù)f(x)=2sin x(0)在區(qū)間-3,4上的最小值是-2,則的最小值等于()A.23B.32C.2D.3解析0,-3x4,-3x4.由已知條件知-3-2,32.答案B3.將函數(shù)y=2sin2x+6的圖像向右平移14個(gè)周期后,所得圖像對(duì)應(yīng)的函數(shù)為f(x),則函

2、數(shù)f(x)的單調(diào)遞增區(qū)間是()A.k-12,k+512(kZ)B.k+512,k+1112(kZ)C.k-524,k+724(kZ)D.k+724,k+1924(kZ)解析函數(shù)y=2sin2x+6的周期T=22=,將函數(shù)y=2sin2x+6的圖像向右平移14個(gè)周期后,所得圖像對(duì)應(yīng)的函數(shù)為f(x)=2sin2x-4+6=2sin2x-3,令2k-22x-32k+2,kZ,可得k-12xk+512,kZ,函數(shù)f(x)的單調(diào)遞增區(qū)間是k-12,k+512,kZ.故選A.答案A4.函數(shù)f(x)=sin(2x+)|2的圖像向左平移6個(gè)單位長(zhǎng)度后關(guān)于原點(diǎn)對(duì)稱,則函數(shù)f(x)在0,2上的最小值為()A.-3

3、2B.-12C.12D.32解析函數(shù)f(x)=sin(2x+)的圖像向左平移6個(gè)單位長(zhǎng)度得y=sin2x+6+=sin2x+3+的圖像.又其為奇函數(shù),則3+=k,kZ,解得=k-3.又|0)取得最小值,則函數(shù)y=f34-x()A.是奇函數(shù)且圖像關(guān)于點(diǎn)2,0對(duì)稱B.是偶函數(shù)且圖像關(guān)于點(diǎn)(,0)對(duì)稱C.是奇函數(shù)且圖像關(guān)于直線x=2對(duì)稱D.是偶函數(shù)且圖像關(guān)于直線x=對(duì)稱解析當(dāng)x=4時(shí),函數(shù)f(x)取得最小值,函數(shù)f(x)的圖像關(guān)于直線x=4對(duì)稱,由f(0)=f2得=4+k,kZ,f(x)=Asinx+4+k,kZ,f34-x=Asin34-x+4+k=Asin(-x+k)=Asinx,k為偶數(shù),-A

4、sinx,k為奇數(shù).y=f34-x是奇函數(shù),且圖像關(guān)于直線x=2對(duì)稱.答案C6.已知關(guān)于x的方程2sin2x+4=k在區(qū)間0,2上有兩個(gè)不同的實(shí)數(shù)解,則k的取值范圍為.解析設(shè)f(x)=sin2x+4.x0,2,42x+454.易知函數(shù)f(x)=sin2x+4在0,8上是增加的,在8,2上是減少的,當(dāng)方程sin2x+4=k2時(shí),有f(0)k2f8,即1k0)和g(x)=2cos(2x+)+1的圖像的對(duì)稱軸完全相同.若x0,2,則f(x)的取值范圍是.解析由題意知=2,所以f(x)=3sin2x-6.因?yàn)閤0,2,所以2x-6-6,56,所以f(x)-32,3.答案-32,38.函數(shù)y=Asin(

5、x+)其中A0,0,|0)的圖像向右平移4個(gè)單位長(zhǎng)度,所得圖像經(jīng)過點(diǎn)34,0,則的最小值是.解析將函數(shù)y=sin x(其中0)的圖像向右平移4個(gè)單位長(zhǎng)度,所得圖像對(duì)應(yīng)的函數(shù)為y=sin x-4.再由所得圖像經(jīng)過點(diǎn)34,0,可得sin 34-4=sin2=0,2=k,kZ.故的最小值是2.答案210.已知函數(shù)f(x)=2sin2x+3+1.(1)當(dāng)x=43時(shí),求f(x)的值;(2)若存在區(qū)間a,b(a,bR且a0,xR)的最小正周期為10.(1)求的值;(2)設(shè),0,2,f5+53=-65,f5-56=1617,求cos ,sin 的值.解(1)由已知得2=10,=15.(2)f(x)=2cos

6、15x+6,f5+53=2cos155+53+6=-2sin ,f5-56=2cos155-56+6=2cos .又f5+53=-65,f5-56=1617,sin =35,cos =817.又,0,2,cos =45,sin =1517.12.導(dǎo)學(xué)號(hào)93774035已知f(x)=Asin2x+6(A0)的最大值為6.(1)求A;(2)將函數(shù)y=f(x)的圖像先向左平移12個(gè)單位長(zhǎng)度,再將所得圖像上各點(diǎn)的橫坐標(biāo)縮短為原來(lái)的12倍,縱坐標(biāo)不變,得到函數(shù)y=g(x)的圖像.求g(x)在0,524上的值域.解(1)因?yàn)锳0,所以由題意知A=6.(2)由(1)得f(x)=6sin2x+6.將函數(shù)y=f(x)的圖像先向左平移12個(gè)單位長(zhǎng)度后得到y(tǒng)=6sin2x+12+6=6sin2x+3的圖像,再將得到的圖