State the constant, constant multiple, and power rules.Indeterminate Forms and L’Hopital’s Rule.Derivatives of Logarithmic and Exponential Functions.Linear Approximations and Differentials.Electronic flashcards for derivatives/integrals.We know by the Pythagorean Theorem that z 2 ( t ) = y 2 ( t ) + x 2 ( t ) and di ↵ erentiating with respect to time in both sides gives 2 z ( t ) dz dt = 2 y ( t ) dy dt + 2 x ( t ) dx dt = ) dz dt = 1 z ( t ) y ( t ) dy dt + x ( t ) dx dt. A C B SOLUTION: Let us denote by z ( t ) the distance at any time between the pedestrians, by y ( t ) the distance between A and the intersection and by x ( t ) the distance between B and the intersection. How fast is the distance from Pedestrian A to Pedestrian B changing when Pedestrian A is 4 miles South of intersection C, and Pedestrian B is 3 miles East of intersection C. Pedestrian A is going North at 2 mph, and Pedestrian B is going East at 3 mph. Pedestrian B is walking away from intersection C. (12 pts.) Pedestrian A is walking towards the intersection C of two streets intersecting at a right angle. (That is find the maximum and minimum value of f ( x ) on the given interval). Find the absolute maximum and absolute minimum of f on the interval. Thus, the full set of critical numbers is x = 0, - 2 / 11, - 1. We see this reduces to the equation 2( x + 1) 3 + 9 x ( x + 1) 2 = 0 This gives ( x + 1) 2 (2( x + 1) + 9 x ) = 0 So x = - 1 is a solution and solving 2( x + 1) + 9 x = 11 x + 2 = 0 tells us that x = - 2 / 11 is another solution. Now, to find the rest of the critical values, we set f 0 = 0. Its derivative is given by f 0 ( x ) = 2( x + 1) 3 x 1 / 3 + 9 x 2 / 3 ( x + 1) 2 = 2( x + 1) 3 + 9 x ( x + 1) 2 x 1 / 3 We see that f 0 is not defined at a = 0 and 0 is in the domain of f. We observe that f is defined for every real number.
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