how do you find the horizontal and vertical assymptotes of the curve

y= x/ x+3 and also the intercepts.

so that you can approximate the area under the curve from x=1 to x=5 using 5 function values?

THANKYOU,

There is a vertical asymptote at x = -3 and a horizontal asymptote of y = 1, as x approaches + or = infinity.

The x-intercept is where x = 0, which would be y = (0,0). The y intercept would be where y = 0, or (also) (0,0). There is ony one intercept.

To approximate the area uner the curve, use the trapezoidal or Simpson's rule. and computed values of f(x) at x = 1,2,3,4 and 5. These values are 1/4, 2/5, 1/2, 4/7 and 5/8.

What happens to the function as x>>-3 ?

What happens to the function as x>>inf
What happens to the function as x>>-inf

appx area. I am not certain of the technique you have been taught.

To find the horizontal asymptote of a curve, you need to analyze the behavior of the function as x approaches positive or negative infinity. For the equation y = x / (x + 3), the horizontal asymptote can be determined by looking at the highest power of x in the numerator and denominator.

In this case, the numerator has an x to the power of 1, while the denominator has an x to the power of 1 as well. Since the powers are the same, we divide the leading coefficients (which are both 1) to obtain the horizontal asymptote.

Therefore, the equation has a horizontal asymptote at y = 1.

To find the vertical asymptote of a curve, you need to identify the values of x that make the denominator equal to zero, because division by zero is undefined.

In this case, the denominator is x + 3. Setting it equal to zero, we get x = -3. Thus, there is a vertical asymptote at x = -3.

Now, to find the intercepts of the curve, we set y equal to zero and solve for x to find the x-intercept, and set x equal to zero and solve for y to find the y-intercept.

For the x-intercept, we set y = 0:
0 = x / (x + 3)
Since the numerator is zero, we have x = 0.

For the y-intercept, we set x = 0:
y = 0 / (0 + 3)
Simplifying, y = 0.

Therefore, the x-intercept is 0 and the y-intercept is 0.

Finally, to approximate the area under the curve from x = 1 to x = 5 using 5 function values, you can use numerical integration methods like the trapezoidal rule or Simpson's rule. These methods involve dividing the interval from x = 1 to x = 5 into smaller subintervals and evaluating the function at specific points within those subintervals.

For example, using the trapezoidal rule, you divide the interval into 4 subintervals (since you have 5 function values and 4 intervals between them).

The subintervals would be:
[1, 2], [2, 3], [3, 4], [4, 5].

Then, you evaluate the function at the endpoints and midpoint of each subinterval. The sum of the areas of the trapezoids formed by each subinterval will approximate the area under the curve.

Note: This is a general explanation of the steps involved. The specific calculations and values depend on the function and the method of numerical integration you choose to use.