Trig Review: General Shape of Functions Worksheet Solutions

Trig Review:General Shape of FunctionsWorksheet Solutions

#1) A critical point is any value that causes f ‘(x) = 0 or f ‘(x) to not exist f ‘(x) = 0 when sin x = 0 f ‘(x) = DNE when cos x = 0 Over the interval of 0 to 4π cos x = 0 at x = π/2 + nπ… π/2, 3π/2, 5π/2, 7π/2 Critical Points at nπ/2

#2) Increasing when f ‘(x) > 0 sin (x) = 0 at x = nπ sin (3x) = 0 at x = nπ/3 – + – + – + – + – sin starts at zero and goes up to 1 –sin and therefore –6sin will start at zero and go down

#3) Decreasing when f ‘(x) < 0 sin (x) = 0 at x = nπ sin (x/3) = 0 at x = nπ/(1/3) = 3nπ – + sin starts at zero and goes up to 1 –sin and also –(1/3)sin will start at zero and go down

#4) Max is when f ‘ goes from + to – cos (x) = 0 at x = π/2 + nπ cos (x + π/3) = 0 at x = π/2 + nπ – π/3 cos ( + π/3) = 0 at x = π/6 + nπ + – + – + x = 0 2cos(0 + π/3) = 2cos(π/3) = 2 * .5 = 1 = positive

#5) Min is when f ‘ goes from – to + cos (x) = 0at x = π/2 + nπ cos (3x) = 0 at x = (π/2 + nπ)/3 cos (3x) = 0 at x = π/6 + 2nπ/6 + – + – + – + – + x = 0 8cos(4*0) = 8 *1 = 8 = positive

#6) POI is when f “ = 0 and changes signs sin (x) = 0at x = nπ sin (πx) = 0 at x = nπ/π sin (πx) = 0 at x = n + – + – x = 0.5 -2π2 sin(π * .5) = -2π2 * 1 = negative Note that the range in the problem does not include 0 and 4 (uses < & >)

#7) Concave Up when f “ > 0 sin (x) = 0at x = nπ sin (x/2) = 0 at x = nπ/(1/2) sin (πx) = 0 at x = 2nπ – + x = π (-1/2)sin(π/2) = (-1/2) * 1 = -1/2 = Negative

#8) Concave Up when f “ < 0 sin (x) = 0at x = nπ sin (x – π/4) = 0 at x = nπ + π/4 + – + – + x = 0 -5sin(0 - π/4) = -5 * -1/√2 = Positive

#35) #32) A, only numbers inside the trig function affect the period of a trig function #37) f (x) = exsinx = 0 when either ex = 0 or sin x = 0 ex never equals zero sin x = 0 at x = nπ. This interval includes: 0, π, 2π Total of 3 zeros, D

Trig Review: General Shape of Functions Worksheet Solutions