Math 192 Final

Find f'(x) by using the definition of the derivative f(x)=1/7x²

-2/7x³

Find the limit if it exists. lim x→0 (x²-5)

0²-5=-5

Graph the rational function f(x)= (x²-9)/(x+2) a) Find the x-intercept b)Find the y-intercept c)Find the horizontal asymptotes d)Find the slat asymptotes e)Does your graph intersect the slant asymptote f)Find the first derivative h)From part (f) determine the critical points, and determine if f(x) has any min/max?

a)Find the x-intercept Let f(x)=0 0 x²-9 - = ----- 1 x+2 x²-9=0 x²=9 x=±3 (3,0) (-3,0) b)Find y-intercept Let x=0 f(0)= 0²-9 -9 ------ = --- 0+2 2 (0, -9/2) c)Find the horizontal asymptote DNE, degree of numerator is greater than degree of denominator. d)Find the slant asymptotes x+2√(x²-9) = x-2+ (5)/(x+2) y=x-2 e) Does your graph intersect the slant asymptote? x-2 x²-9 ---- = ---- 1 x+2 (x-2)(x+2) = x²-9 x²-4=x²-9 -4≠-9 No f)Find the first derivative

Determine where the given function is increasing and where it is decreasing

f'(x)=2x+9 f'(x)=0 2x+9=0 x=-9/2 inc. (-9/2, ∞) dec. (-∞, -9/2)

Find an equation for the line tangent to the graph of the given function at the indicated point. f(x)=x²-x at (3,6)

f'(x)=3x-1 f'(3)=2(3)-1 f'(3)=5 y-y₁=m(x-x₁) y-6=5(x-3) y-6=5x-15 y=5x-9

Find the point of inflection f(x)=x⁴-54x²

f'(x)=4x³-108x f''(x)=12x²-108 f''(x)=0 12x²-108=0 12(x²-9)=0 12(x+3)(x-3)=0 x=±3 inf. points f(-3)=(-3)⁴-54(-3)²=-405 (-3,-405) f(3)=(3)⁴-54(3)²=-405 (3,-405)