MATH 124 Midterm: MATH 124 UW 124 Sp16 Midterm 1 Solution

In Problems 1-4, find all critical points and determine whether they are relative maxima, relative minima, or horizontal points of inflection. 17. Given y 6400x 18xnd the absolute maximum and minimum for y when (a) 0x 50 and (b) 0 s xS 100. In Problems 18 and 19, use the graphs to find the following items. (a) vertical asymptotes (b) horizontal asymptotes (e) lim (d lim) 18 3, f(x) = 1-3x + 3x2-x3 f(x) = +1 In Problems 5-10: (a) Find all critical values, including those at which f(x) is undefined. (b) Find the relative maxima and minima, if any exist. (c) Find the horizontal points of inflection, if any exist. (d) Sketch the graph. y = f(x) d a 6, f(x) = 4x3-x" -2 15 8. y = 5x7-7x5-1 9, y=x2/3-1 10, y = x2/3(x-4)2 19. 11. Is the graph ofy = x4-3x3 + 2x-1 concave up or concave down at x = 2? 2. Find intervals on which the graph ofy = x4-2x3 12x + 6 is concave up and intervals on which it is concave down, and find points of inflection. 13. Find the relative maxima, relative minima, and points 9x+ 10. of inflection of the graph of yx in Problems 14 and 15, find any relative maxima, relative In Problems 20 and 14 n, and points of inflection, and sketch each graph. In Problems 20 an 15, y=2+5x3-3x5 16, Given R and any vertical asymptotes. 3x + 2 2x 21, y = 280x-x, find the absolute maximum ) 0 s x S 200 and and minimum for R when (a (b) 0 x 100

Part A Solve all of the following 5 points each 1. Evaluate (x3 + 4x2-7) at x-2 dr r+1 dx 4, Determine the slope of the normal to the function at x = 2 y=3x3 + 8x2-24x + 10 x((x-1)3 ) 2x2-14x +24 6. yx-4 dy dx find Part C Do ALL 10 points each y=ln(sin(e*) finddy 7, d2 8. y ex.sin(x) find y dx dy 9. y - cos(3x2Hn(7-x2) find sin(3x) y=cos(4x) 10, find

1. 0/6 points LarCalc11 13.1.010. [3864 VIA Evaluate the function at the given values of the independent variables. Simplify the results. fx, y) -4 -x2 - 4y2 (a) ro, 0) for (c) f2, 3) (e) x, o Cal 2. 0/2 points LarCalc11 13.1.019.MI. [386 Evaluate the function at the given values of the independent variables. Simplify the results. x, y)- 6x y2 (a) fix + ax, y)-f(x, y) Ay

3. 1 points LarCalc11 13.2 Find the indicated limit by using the limits limfx, y) 2 and im g(x, y)--4 lim 4. O/1 points LarCalci1 13 2 Find the indicated limit by using the limits lim x, y) 3 and lim (x, y) g(x, y)--5 (a, b) lim x, Y) (x, y) (a, b) g(x, y) 5. y1 points LarCalc11 13.3 Explain whether or not the Quotient Rule should be used to find the partial derivative. Do not differentiate. ay x2 51 O Yes, the function is a quotient. O No, y only occurs in the numerator. 6. 0/2 points LarCalc11 13.3 Find both first partial derivatives oz 7. 0/2 points LarCalc11 13.3 Find both first partial derivatives. hx, y) ex+y) h(x, y)

8. /2 points LarCalc11 13.3.031 Find both first partial derivatives. az dx az ay 9. /1 points LarCalc11 13.4.003. Find the total differential. z 3xSy10 dz 10. 0/2 points LarCalc11 13.5.012 Consider the following. (a) Find dw/dt using the appropriate Chain Rule. dw dt (b) Find dw/dt by converting w to a function of t before differentiating. dw dt 11. 0/1 points LarCalc11 13.6.008. Find the directional derivative of the function at P in the direction of v. rx, y) = x3-y3, Rio, 9), v-V2 (1+j)

12. 0/2 points LarCalc11 13.7.017 (a) Find an equation of the tangent plane to the surface at the given point. x+y+z=9, (2, 5, 2) (b) Find a set of symmetric equations for the normal line to the surface at the given point. ox -2y -5-z-2 13. /3 points LarCalc11 13.8.025 Use a computer algebra system to graph the surface and locate any relative extrema and saddle points. (If an an not exist, enter DNE.) -8x x2 y21 relative minimum (x, y, z) - relative maximum (x, y, z) saddle point 14. 0/1 points LarCalci1 13.10.027 Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. Surface Point Plane: x yz 1 (7,1,1)