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A: solve by partial differentiation
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- The origin is a critical point for the function f(x, y) = 15 − x^2y^2 and ∆(x, y) = 0 there, i.e.,the Second Derivative Test fails. Use what you know about shapes of functions to decide ifthere is a local minimum, local maximum, or saddle point for this function at (0, 0).From the partial differential eqiuation by eliminating the arbitrary function z=(x2+y2+z2)Verify Eular's Theorem for the function xy+yz+zx
- The linearization of ex at x = 0 Derive the linear approximation ex = 1 + x at x = 0.Find the critical point of the function f(x,y)=6x−2y^2−ln(|x+y|). c=Use the Second Derivative Test to determine whether it isA. a saddle pointB. a local maximumC. test failsD. a local minimumSuppose ƒ is differentiable on (- ∞, ∞) andƒ(5.01) - ƒ(5) = 0.25. Use linear approximation to estimate the value of ƒ'(5).
- express the extreme value theorem for functions of two variables . find the absolute maximum and minimum values of f(x,y)=2x2-4x+y2-4y+1 on the triangle bounded by x=0, y=2 and y=2x.A differentiable function f has only one critical number at x = 5. What can we say about this point (regarding local extrema) if we know that f ′ (4.3) = −2.5 and f ′ (5.9) = 3.2?Demonstrate the use of the method with reflections on the use of numerical methods, find the minimum for the function below: F(x,y)= Ax^2 - Bxy- cy^2= x -y (Xo=4, Yo=4) A=2 B=-2 C=1 Identify the minimum again using the Newton’s method with dynamic . However, use this time numerical derivatives instead of . When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, . Here equals some very small number. For each step , solve first the and optimal using the condition . When taking the derivative of , please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form .
- Suppose ƒ is differentiable on (- ∞, ∞), ƒ(1) = 2, and ƒ'(1) = 3.Find the linear approximation to ƒ at x = 1 and use it to approximate ƒ(1.1).The functions f(x, y) = x 2 + y 2 and g(x, y) = x 2 - y 2 both have acritical point at (0, 0). How is the behavior of the two functions at the critical point different?find the extreme values (absolute and local) of thefunction over its natural domain, and where they occur. y = ex - e-x