Thank you Dauhla Dnshl blem 4 Section 35 Remember for Calcu

Dauhla Dnshl blem 4: (Section 3.5) - Remember for Calculus! ob a. Let f(x) sin(r). Show that the difference quotient for f simplifies to + h)-f(x) = cos(x) , T sin(h) +sinCx) f(x + h)-f(x) = cos(x) + sin(x) b. Let g(x) = cos(x). Show that the difference quotient for g simplifies to sin(h)+cos(x) cos(h)1 (x + h) - g(x) -sin(x

Solution

f(x) = sin x

f(x+h) = sin (x+h)

plugging the values in difference quotient formula

(sin (x+h) - sin x ) / h

sin (x+h) = sin x cos h + cos x sin h

= sin x cos h + cos x sin h - sin x / h

taking the gcf out of sin x cos h - sin x

= sin x ( cos h -1 ) / h + cos x sin h / h

= cos x . sin h / h + sin x . ( cos h -1 ) / h

which is right side

proved !

2) g(x) = cos x

g (x+h) = cos ( x+h)

plugging the value in difference quotient formula

cos ( x+h) = cos x cos h - sin x sin h

so we have

(cos x cos h - sin x sin h - cos x )/ h

( cos x cos h - cos x - sin x sin h ) / h

taking the gcf out

{cos x ( cos h -1 ) - sin x sin h } / h

= - sin x . sin h / h + cos x . ( cos h -1 ) / h

which is right side

proved !