The curve y= ax^3 -3x^2 -2x + 1 has the gradient 7 when x = 1. Find the value of a.
gradient = slope = dy/dx
dy/dx = 3ax^2 - 6x - 2
using the given:
7 = 3a(1) -6(1) - 2
7 = 3a - 8
3a = 15
a = 5
Why did the curve turn into a detective all of a sudden? It's on a mission to find the elusive value of "a". Let's unravel this mystery, shall we?
Now, we know that the gradient of a curve is the derivative of the equation. So, let's find the derivative of the curve:
dy/dx = 3ax^2 - 6x - 2
Since we're given that the gradient is 7 when x = 1, we can plug these values into our derivative equation:
7 = 3a(1)^2 - 6(1) - 2
Simplifying this equation, we have:
7 = 3a - 6 - 2
Now, let's bring those numbers together:
7 = 3a - 8
To find the value of "a", let's isolate it:
3a = 7 + 8
3a = 15
a = 15/3
a = 5
So, after all the detective work, we've finally found that the value of "a" is 5. Case closed!
To find the value of 'a' when the curve y = ax^3 - 3x^2 - 2x + 1 has a gradient of 7 when x = 1, we need to differentiate the equation and then substitute the given values.
The first derivative of the equation is found by differentiating each term separately:
dy/dx = 3ax^2 - 6x - 2
Now, substitute x = 1 and set the derivative equal to 7:
7 = 3a(1)^2 - 6(1) - 2
Simplifying the equation further:
7 = 3a - 6 - 2
7 = 3a - 8
Add 8 to both sides:
15 = 3a
Divide both sides by 3 to isolate 'a':
a = 5
Therefore, the value of 'a' is 5.
To find the value of "a" in the equation y = ax^3 - 3x^2 - 2x + 1 given that the gradient is 7 when x = 1, we need to calculate the derivative of the equation and plug in the given values.
The derivative of a function represents its gradient at any given point. Therefore, we'll differentiate the equation y = ax^3 - 3x^2 - 2x + 1 with respect to x to find the derivative dy/dx.
Taking the derivative of each term, we get:
dy/dx = 3ax^2 - 6x - 2
Now, we need to evaluate the derivative at x = 1. Substituting x = 1 into the derivative, we have:
dy/dx = 3a(1)^2 - 6(1) - 2
dy/dx = 3a - 6 - 2
dy/dx = 3a - 8
Given that the gradient is 7 when x = 1, we can set dy/dx equal to 7 and solve for "a":
7 = 3a - 8
Adding 8 to both sides:
7 + 8 = 3a
15 = 3a
Dividing both sides by 3:
15/3 = a
5 = a
Therefore, the value of "a" in the equation y = ax^3 - 3x^2 - 2x + 1 is 5.