find tangent line equation to the curve at the given point:
y= 7e^x cos(x) P=(0,7)
dy/dx = -7 e^x sin x + 7 e^x cos x
= 7 e^x (cos x - sin x)
at (0,7)
dy/dx = m the slope = 7 e^0 (cos 0 - sin 0)
= 7
so y = 7x+b
at(0,7)
7 = 7(0) + b
so b = 7
and
y = 7 x + 7
or
y = 7(x+1)
Well, well, well! Looks like we have a calculus problem on our hands. Don't worry, I'm here to clown around with math.
To find the equation of the tangent line to a curve at a given point, we need two things: the slope of the tangent line and the point where it touches the curve.
First, let's find the slope. We can do that by taking the derivative of the curve equation. Here, we have y = 7e^x cos(x). Are you ready for some math magic?
Differentiating the equation, we get:
dy/dx = (7e^x)(-sin(x)) + (7cos(x))(e^x)
Now, let's plug in the x-coordinate of the given point (0) into the derivative to find the slope:
dy/dx = (7e^0)(-sin(0)) + (7cos(0))(e^0)
dy/dx = (-7)(0) + (7)(1)
dy/dx = 7
So, the slope of the tangent line is 7. Now, let's use this slope and the given point (0, 7) to find the equation of the tangent line.
The equation of a straight line is given by y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.
Plugging in the values, we have:
y - 7 = 7(x - 0)
Simplifying, we get:
y - 7 = 7x
And voila! The equation of the tangent line is y = 7x + 7.
So, there you have it - the tangent line equation to the curve at the point (0, 7). Keep clowning around with math and have a laugh-tastic time!
To find the equation of the tangent line to the curve at the given point, we need to find the derivative of the function at that point.
The derivative of the function y = 7e^x cos(x) can be found using the product rule and the chain rule.
Let's differentiate the function step by step:
1. Use the product rule to differentiate the product of two functions:
(d/dx)(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
2. Differentiate the first function: f(x) = 7e^x
- The derivative of e^x is e^x
- Multiply it by 7 to get the derivative of 7e^x: 7e^x
3. Differentiate the second function: g(x) = cos(x)
- The derivative of cos(x) is -sin(x)
4. Apply the product rule:
- f'(x)g(x) = (7e^x)(cos(x))
- f(x)g'(x) = (7e^x)(-sin(x))
5. Putting it all together:
- The derivative of y = 7e^x cos(x) is: y' = (7e^x)(cos(x)) + (7e^x)(-sin(x))
- Simplifying this gives: y' = 7e^x (cos(x) - sin(x))
Now that we have the derivative, we can find the equation of the tangent line at point P(0,7).
6. Substitute the x-coordinate of the point (0) into the derivative equation to find the slope of the tangent line:
- Plug x = 0 into y' = 7e^x (cos(x) - sin(x))
- y' = 7e^0 (cos(0) - sin(0))
- y' = 7(1)(1 - 0)
- y' = 7
The slope of the tangent line at P(0,7) is 7.
7. We know the slope of the line and a point it passes through (P(0,7)). We can use the point-slope form of the equation to find the tangent line equation:
- The equation of the tangent line is: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point
- Plugging in the values: y - 7 = 7(x - 0)
- Simplifying: y - 7 = 7x
- Rearrange to slope-intercept form: y = 7x + 7
Therefore, the equation of the tangent line to the curve y = 7e^x cos(x) at the point (0,7) is y = 7x + 7.
To find the equation of the tangent line to the curve at the given point, you need to determine the slope of the tangent line first.
The slope of a tangent line to a curve at any point can be found by taking the derivative of the equation of the curve with respect to x.
Let's find the derivative of y = 7e^x * cos(x).
Using the product rule, we can differentiate this function term by term:
dy/dx = (7e^x * cos(x))'
= (7e^x)' * cos(x) + 7e^x * (cos(x))'
Differentiating both terms:
(7e^x)' = 7(e^x)'
(cos(x))' = -sin(x)
Simplifying:
dy/dx = 7 * e^x * cos(x) - 7 * e^x * sin(x)
Now, substitute the x-coordinate of the point P into the derivative to find the slope at that point.
Let x = 0:
m = dy/dx |_x=0
= 7 * e^0 * cos(0) - 7 * e^0 * sin(0)
= 7 * 1 * 1 - 7 * 1 * 0
= 7
So, the slope of the tangent line at the point P(0, 7) is 7.
Now we can use the point-slope equation of a line to find the equation of the tangent line.
Using the point-slope form:
(y - y1) = m(x - x1), where (x1, y1) is the point P and m is the slope.
Substituting the values:
(y - 7) = 7(x - 0)
y - 7 = 7x
y = 7x + 7
Therefore, the equation of the tangent line to the curve y = 7e^x * cos(x) at the point P(0, 7) is y = 7x + 7.