I have a few problems from my homework that I am having trouble with.

#1 Find the difference quotient of f foreach function

f(x)=1/(x+3)

I got -2x+6

#2 Express the area of an isosceles right triangle as a function of the length x of one of the two eual sides

Thanks

The difference quotient,

∆f/∆x = [f(x+h)-f(x)]/h
= [1/((x+h)+3) - 1/(x+3)]/h
= 1/h [(x+3) - (x+h+3)] / (x+h+3)(x+3)
= 1/h (-h)/(x+h+3)(x+3)
= -1/[(x+h+3)(x+3)]

The legs are x and x. One leg is the base, the other leg is the height, so the area is 1/2 bh = 1/2 x^2

Why can't the hypotenuse be one of the equal sides? Because the hypotenuse is longer than either leg.

for the second problem what is th height??

ohh wait ok that was me just having a dumb moment haha. THANKS sooo much:)

Sure! I can help you with both of these problems.

#1: To find the difference quotient of a function, we need to find the slope of the secant line between two points on the graph of the function. In this case, we can use the formula for the difference quotient:

Difference quotient = [f(x + h) - f(x)] / h

Let's apply this formula to the given function f(x) = 1/(x + 3):

Step 1: Replace f(x) with the given function: f(x) = 1/(x + 3)
Step 2: Replace f(x + h) with 1/(x + h + 3)
Step 3: Simplify the numerator: 1/(x + h + 3) - 1/(x + 3)
Step 4: Combine the fractions over a common denominator: [(x + 3) - (x + h + 3)] / [(x + h + 3)(x + 3)]
Step 5: Simplify the numerator: (x + 3 - x - h - 3) / [(x + h + 3)(x + 3)]
Step 6: Simplify further: -h / [(x + h + 3)(x + 3)]

Therefore, the difference quotient of f(x) = 1/(x + 3) is -h / [(x + h + 3)(x + 3)].

#2: An isosceles right triangle has two equal sides, so let's call the length of each equal side x. The area of a triangle is given by the formula A = (1/2) * base * height. In this case, the base is x and the height is x.

Step 1: Replace the base with x and the height with x in the formula: A = (1/2) * x * x
Step 2: Simplify the formula: A = (1/2) * x^2

Therefore, the area of an isosceles right triangle in terms of the length x of one of the equal sides is (1/2) * x^2.

I hope this helps you with your homework problems! Let me know if you have any other questions.