# Calculus

There is a diagram with the curve y=(2x-5)^4.The point P has coordinates (4,81) and the tangent to the curve at P meets the x-axis at Q.Find the area of the region enclosed between curve ,PQ and the x-axis.

1. 👍 0
2. 👎 0
3. 👁 340
1. y = (2x-5)^4
dy/dx = 4(2x-5)^3 (2) = 8(2x-5)^3
P(4,81) lies on the curve, and the slope of the tangent at that point is
8(3^3) = 216
and the equation of the tangent is
y-81 = 216(x-4)
y = 216x - 783
At Q, the intercept with the x-axis, x = 783/216 = 29/8

the curve makes contact with the x-axis at 5/2

Let a vertical from P cut the x-axis at R

So find the area of the shape between the x-axis the vertical PR and the curve, then subtract the area of triangle PRQ.

you will need the integral of (2x-5)^4 which would be (1/10)(2x-5)^5

take over

1. 👍 0
2. 👎 0
👨‍🏫
Reiny
2. How come At Q, the intercept with the x-axis, x = 783/216 = 29/8

1. 👍 0
2. 👎 0
3. How does the vertical cut x-axis?

1. 👍 0
2. 👎 0
4. From the tangent equation, y = 216x - 783
at the x-axis, y = 0
so 0 = 216x - 783
216x = 783 ----> x = 783/216 = 29/8

your 2nd post: the x-axis is horizontal, any vertical line would cut it, wouldn't it?

1. 👍 0
2. 👎 0
👨‍🏫
Reiny
5. I found area of triangle but couldn't get area of the shape between the x-axis the vertical PR and the curve

1. 👍 0
2. 👎 0
6. I got it.Thanks

1. 👍 1
2. 👎 0

## Similar Questions

Given the curve x^2-xy+y^2=9 A) write a general expression for the slope of the curve. B) find the coordinates of the points on the curve where the tangents are vertical C) at the point (0,3) find the rate of change in the slope

2. ### calculus

1. Given the curve a. Find an expression for the slope of the curve at any point (x, y) on the curve. b. Write an equation for the line tangent to the curve at the point (2, 1) c. Find the coordinates of all other points on this

3. ### calculus

Notice that the curve given by the parametric equations x=25−t^2 y=t^3−16t is symmetric about the x-axis. (If t gives us the point (x,y),then −t will give (x,−y)). At which x value is the tangent to this curve horizontal?

4. ### math

Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, −25). Find the coordinates of the points where these tangent lines intersect the parabola.

1. ### calculus

Consider the curve defined by 2y^3+6X^2(y)- 12x^2 +6y=1 . a. Show that dy/dx= (4x-2xy)/(x^2+y^2+1) b. Write an equation of each horizontal tangent line to the curve. c. The line through the origin with slope -1 is tangent to the

2. ### calculus

Consider the curve given by the equation y^3+3x^2y+13=0 a.find dy/dx b. Write an equation for the line tangent to the curve at the point (2,-1) c. Find the minimum y-coordinate of any point on the curve. the work for these would

3. ### Calculus 1

The curve y = |x|/(sqrt(5- x^2)) is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (2, 2)

4. ### math

A curve has implicit equation x^2-2xy+4y^2=12 a)find the expression for dy/dx in terms of y and x. hence determine the coordinates of the point where the tangents to the curve are parallel to the x-axis. b)Find the equation of the

1. ### calculus

let f be the function f(x) = x^3 + 3x^2 - x + 2 a. the tangent to the graph of f at the point P = (-2,8) intersects the graph of f again at the point Q. Find the coordinates of point Q. b. Find the coordinates of point R, the

2. ### last calc question, i promise!

given the curve x + xy + 2y^2 = 6... a. find an expression for the slope of the curve. i got (-1-y)/(x + 4y) as my answer. b. write an equation for the line tangent to the curve at the point (2,1). i got y = (-1/3)x + (5/3). but i

3. ### Calculus

The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y. What is the equation of the curve if (2, 1) is a point on the curve?

4. ### AP AB Calculus

Linear approximation: Consider the curve defined by -8x^2 + 5xy + y^3 = -149 a. find dy/dx b. write an equation for the tangent line to the curve at the point (4,-1) c. There is a number k so that the point (4.2,k) is on the