Calculus 1 Using Definition of derivatives

Question

Assume that g(y) has a derivative at y = 3.5 with g′(−3.5) = −3. Using the definition of the derivative, explain why the function f (y) = g(y) − 10 has a derivative at y = 3.5 with

f′(3.5)=−3.

Assume that g(y) has a derivative at y = 3.5 with g′(−3.5) = −3. Use geometry to explain why the function f (y) = g(y) − 10 has a derivative at y = 3.5 with f ′(3.5) = −3.

Answer 1

From the definition of the limit you should note the limits

g'(x) = lim[(g(x+h)-g(x))/h] = lim[(f(x+h)-10 -f(x)+10)/h]= lim[(f(x+h)- f(x))/h] = f(x) for any x so apply to x = -3.5.

This is clearer to see if you write out the fractions involved and fill in the steps.

From geometry, note the curves are parallel since one is the other shifted up by 10 units. Parallel curves have parallel tangents at a given (fixed) value of x

Also make sure that you are not missing any signs.