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Compare dy and delta y for y = x^2 minus 4 at x = negative 1 with delta x = d x ...
Jul 15, 2024
Compare dy and delta y for y = x^2 minus 4 at x = negative 1 with delta x = d x = negative 0.07. Give your answers to four decimal places.
Generated Graph
Solution by Steps
step 1
Differentiate y=x2−4y = x^2 - 4 with respect to xx: dydx=2x\frac{dy}{dx} = 2x
step 2
Evaluate dydx\frac{dy}{dx} at x=−1x = -1: dydx∣x=−1=2(−1)=−2\frac{dy}{dx} \bigg|_{x=-1} = 2(-1) = -2
step 3
Calculate dydy using dy=dydx⋅dxdy = \frac{dy}{dx} \cdot dx: dy=−2⋅(−0.07)=0.14dy = -2 \cdot (-0.07) = 0.14
step 4
Calculate Δy\Delta y using y=x2−4y = x^2 - 4: y(x)amp;=x2−4y(x+Δx)amp;=(x+Δx)2−4y(−1+(−0.07))amp;=(−1−0.07)2−4amp;=(−1.07)2−4amp;=1.1449−4amp;=−2.8551 \begin{align*} y(x) &= x^2 - 4 \\ y(x + \Delta x) &= (x + \Delta x)^2 - 4 \\ y(-1 + (-0.07)) &= (-1 - 0.07)^2 - 4 \\ &= (-1.07)^2 - 4 \\ &= 1.1449 - 4 \\ &= -2.8551 \end{align*} Δy=y(x+Δx)−y(x)=−2.8551−(−3)=0.1449 \Delta y = y(x + \Delta x) - y(x) = -2.8551 - (-3) = 0.1449
Answer
dy=0.14dy = 0.14, Δy=0.1449\Delta y = 0.1449
Key Concept
Differentiation and small changes in xx
Explanation
dydy represents the instantaneous rate of change, while Δy\Delta y represents the actual change over a small interval.
Compare dy and delta y for y = 3x^4 plus 3 at x = negative 1 with delta x = d x = 0.09. Give your answers to four decimal places.
Generated Graph
Solution by Steps
step 1
Differentiate y=3x4+3y = 3x^4 + 3 with respect to xx: dydx=12x3\frac{dy}{dx} = 12x^3
step 2
Evaluate dydx\frac{dy}{dx} at x=−1x = -1: dydx∣x=−1=12(−1)3=−12\frac{dy}{dx} \bigg|_{x=-1} = 12(-1)^3 = -12
step 3
Calculate dydy using dy=dydx⋅dxdy = \frac{dy}{dx} \cdot dx: dy=−12⋅0.09=−1.08dy = -12 \cdot 0.09 = -1.08
step 4
Calculate Δy\Delta y using y=3x4+3y = 3x^4 + 3:
step 5
Evaluate y(x)y(x) at x=−1x = -1: y(−1)=3(−1)4+3=6y(-1) = 3(-1)^4 + 3 = 6
step 6
Evaluate y(x+Δx)y(x + \Delta x) at x=−1x = -1 and Δx=0.09\Delta x = 0.09: y(−1+0.09)=3(−0.91)4+3y(-1 + 0.09) = 3(-0.91)^4 + 3
step 7
Simplify y(−0.91)y(-0.91): y(−0.91)≈3(0.68574961)+3=5.05724883y(-0.91) \approx 3(0.68574961) + 3 = 5.05724883
step 8
Calculate Δy\Delta y: Δy=y(−0.91)−y(−1)=5.05724883−6=−0.94275117\Delta y = y(-0.91) - y(-1) = 5.05724883 - 6 = -0.94275117
Answer
dy=−1.08dy = -1.08, Δy=−0.9428\Delta y = -0.9428
Key Concept
Differentiation and Increment Calculation
Explanation
The problem involves finding the differential dydy and the actual change Δy\Delta y for a given function at a specific point and increment. Differentiation provides the rate of change, while evaluating the function at the incremented point gives the actual change.
Find the differential dy of the function y = x sin(5x).
Generated Graph
Solution by Steps
step 1
Differentiate the function y=xsin⁡(5x)y = x \sin(5x) with respect to xx:
step 2
Using the product rule, we get: ddx(xsin⁡(5x))=sin⁡(5x)+5xcos⁡(5x)\frac{d}{dx}(x \sin(5x)) = \sin(5x) + 5x \cos(5x)
Answer
dydx=sin⁡(5x)+5xcos⁡(5x)\frac{dy}{dx} = \sin(5x) + 5x \cos(5x)
Key Concept
Product Rule
Explanation
The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Find the differential d y of the function y = negative x^2 minus 3x minus 1.
Generated Graph
Solution by Steps
step 1
To find the differential dydy of the function y=−x2−3x−1y = -x^2 - 3x - 1, we first need to differentiate the function with respect to xx
step 2
The derivative of yy with respect to xx is given by dydx\frac{dy}{dx}
step 3
Differentiating y=−x2−3x−1y = -x^2 - 3x - 1 with respect to xx: dydx=ddx(−x2)+ddx(−3x)+ddx(−1) \frac{dy}{dx} = \frac{d}{dx}(-x^2) + \frac{d}{dx}(-3x) + \frac{d}{dx}(-1) dydx=−2x−3 \frac{dy}{dx} = -2x - 3
step 4
Therefore, the differential dydy is: dy=(−2x−3)dx dy = (-2x - 3)dx
Answer
dy=(−2x−3)dxdy = (-2x - 3)dx
Key Concept
Differentiation
Explanation
Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In this case, we differentiated the function y=−x2−3x−1y = -x^2 - 3x - 1 to find its differential dydy.
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