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II. Find the equation of the tangent line to the curve z2+V#j+y² = 19 at the point (1, 4).

II. Find the equation of the tangent line to the curve z2+V#j+y² = 19 at the point (1, 4).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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Question 6II
6. Based on our class discussions, recall that some functions are defined implicitly by a relation between z and y such as z² + y² = 25. In such cases, differentiating with respect to z both sides helps us compute . We should be careful, however, with the Chain Rule that needs to be applied in the way. That is, if y is assumed to be a function of z, then (1²) = 2y(3') 1 (VD" = (5 (e")' = (e");/ and so on. This process is called implicit differentiation. I. Apply implicit differentiation to compute for: z²+ /#j + y° = 19 II. Find the equation of the tangent line to the curve z² + VEj +y = 19 at the point (1, 4). Note: Recall that, in general, the slope of the tangent line to f(x) at the point (a, f(a)) is f'(a). In this case, since y is implicitly assumed to be a function of x, the slope of the tangent line becomes y, or , calculated at z= a and y = f(a). 7. Now, recall that we have already seen some of the limits discussed below in the previous practice and midterm exams. You must explain each and every step in order to receive credit. Only when appropriate, i.e. in the cases or , we may also apply L'Hospital's rule, which makes the computation of certain limits a little bit easier. L'Hospital's rule states that S(z) a g(2) lim = lim (2) a g'(z) a) lim %3D b) 1 lim -z-4 c) z+4 lim i z+2 d) lim
Transcribed Image Text:6. Based on our class discussions, recall that some functions are defined implicitly by a relation between z and y such as z² + y² = 25. In such cases, differentiating with respect to z both sides helps us compute . We should be careful, however, with the Chain Rule that needs to be applied in the way. That is, if y is assumed to be a function of z, then (1²) = 2y(3') 1 (VD" = (5 (e")' = (e");/ and so on. This process is called implicit differentiation. I. Apply implicit differentiation to compute for: z²+ /#j + y° = 19 II. Find the equation of the tangent line to the curve z² + VEj +y = 19 at the point (1, 4). Note: Recall that, in general, the slope of the tangent line to f(x) at the point (a, f(a)) is f'(a). In this case, since y is implicitly assumed to be a function of x, the slope of the tangent line becomes y, or , calculated at z= a and y = f(a). 7. Now, recall that we have already seen some of the limits discussed below in the previous practice and midterm exams. You must explain each and every step in order to receive credit. Only when appropriate, i.e. in the cases or , we may also apply L'Hospital's rule, which makes the computation of certain limits a little bit easier. L'Hospital's rule states that S(z) a g(2) lim = lim (2) a g'(z) a) lim %3D b) 1 lim -z-4 c) z+4 lim i z+2 d) lim
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