Let g(z) = z-i z+i' (a) Evaluate g(i) and g(1). (b) Evaluate the limits lim g(z), and lim g(z). 2-12 (c) Find the image of the real axis under g. (d) Find the image of the upper half plane {z: Iz > 0} under the function g.
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- (ii) Make a contour plot of the function z(x, y) = -xy in the xy-plane, by drawing a collection of curves z(x, y) = c for different (positive and negative) values of the constant c. Try and sketch a graph of the function in the space of xyz coordinates (as a check you may want to compare that sketch with a computer/Python graph).A small aircraft starts its descent from an altitude of h =- miles, 4 milles west of the runway (see figure). (a) Find the cublc function f(x) = ax3 + bx2 + cx + d on the Interval [-4, 0] that describes a smooth glide path for the landing. f(x) = (b) The function in part (a) models the glide path of the plane. When would the plane be descending at the greatest rate? X =#8.) Determine the slope of the tangent at point (1,4) on the graph of f(x) = (9-x²) ²3/ #10.) a) Determine the values of x where the graph of each function has a horizontal tangent i> y=(x²-435 -ii) y= (x³ -x)² to x + 5y -10 = D. #12.3 A tangent to the parabola y=3x²-7x +5 is perpendicular Determine the equation of the tangent. #13.) The line y = 8x tb tangent to the curve y² 2x². Determine the point of tangency and the value of b #29. Find numbers a, b, and c so that the graph of f(x) = ax² + bx tc has and a tangent with Slope 16 where x = 2 x-intercepts at (0,0) and (8.0)
- 2. Consider the function f(x) = 4 – x². (a) Draw a sketch of the graph of the function. (b) Find the slope of the secant line connecting the points P(1, 3) and Q(2,0) on the graph. (c) Find the slope of the secant line connecting the points P(1,3) and Q(x,4 – x²). (d) Use algebra to simplify your formula from part (c) as much as possible, assuming that P and Q are distinct points. (e) What value do you get when you plug x = 1 into the simplified expression you found in part (d)? (f) Assume that your answer in part (e) is the slope of the tangent line to the graph y = 4 – x2 at the point P(1,3). Use that information to write an equation for the tangent line to the graph y = 4 – a² at the point P(1,3).Using the function f: (-1,2] → R, ƒ (x) = 4x³ – 3x² – 6x + 2, a. Determine all the stationary points within the domain restriction and the nature of each of these turning points b.-Sketch the graph of the function in the space below, indicating the endpoints, y- intercepts, and the turning points. Show calculations for the start and end point in the space below too. ORMITUniversity Page 4 of 7 MM Quiz (2021)5. For the following function: f(x) =x² - 7x +10 a) Find the coordinates for the y-intercept. b) Find the coordinates for the x-intercept(s) c) Test the function for symmetry about the axes or origin. If symmetric, provide the symmetry type or indicate that the function is not symmetric about the axes or the origin.
- Let f(x) = x2 + 3x- 2 f(x) – f(1) X - 1 a) Find lim x+1 b) Use your answer in part (a) to find an equation of the tangent line to the curve y = x +3x - 2 at the point (1,2).Consider the function f(x) = 4x - x² and the point P(1, 3) on the graph of f. (a) Graph fand the secant lines passing through P(1, 3) and Q(x, f(x)) for x-values of 2, 1.5, 0.5. Find the slope of each secant line. line passing through Q(2, f(x)) line passing through Q(1.5, f(x)) line passing through Q(0.5, f(x)) (b) Use the results of part (a) to estimate the slope of the tangent line to the graph of fat P(1, 3). MY NOTES Describe how to improve your approximation of the slope. Define the secant lines with points farther away from P. Choose secant lines that are nearly horizontal. O Define the secant lines with points closer to P. Choose secant lines that are nearly vertical. ASK YOUR TEACHERParameterize the following portions of curves: (a) e The straight line between (-5, 3) and (11, 11). (b) A portion of the function f(x) = r* + 1 from (1, 2) to (-2,-7) Three-quarters of the unit circle traversed counter-clockwise, starting at (1,0).
- Consider the function y=-2(1/2(x-1))^2+5. Find the x and y-intercepts. Round to one decimal place where necessary. i) State the mapping notation formula. ii) Transform the key points of y=x^2 using mapping notation. i) (x/k + d, ay + c) Graph the function using a) and b) above. ALL PART OF ONE QUESTION. SO PLS DO IT1) Let f(x) be a function with Domain [-6,-2] and Range [-10,-4]. Find the domain and range of the following: a) y=f(x+4)-1 Use interval notation.a) Sketch the graph of the function f (x) = {2x – 3 if x 2 %3D then evaluate lim f(x), lim f(x) at a = 2. X --> a+ X --> a- b) Find the derivative of the following function f (x) = 3x5 – 2x4 + 3x2 - 2x + 1 %3D c) Find the derivative of the following function g (x) = (1+x) e d) Find the derivative of the following function h (x) = (x + ) %3D