Chapter 5, Problem 27RE

7. The number of visible sun spots varies rhythmically from 10 to 110 per year over a period of eleven years. The 110 max last occurred in the year 2014. a.) b.) Write a sinusoidal function N (t) to model that number of sun spots per year, where t is in years past the year 2014. What number of sun spots does your function predict for the year 2018?

2. Describe the transformations (in point form) required to transform the function y= cos x to the function. y- (x+

#6

Determine whether the pair of functions is linearly independent or linearly dependent on the real line. f(x) = e* cos x g(x) = e-* sin x

d,e,f

Let f be the function given by f(x) = 4 cos x. As shown below, the graph of f crosses the y-axis at point P (0, 4) and the x-axis at point Q (1,0).

d-e

The figure shows how a function f(x) and its linear approximation (i.e., its tangent line) change value when a changes from co to xo + dx. Suppose f(x)=x²-3x, xo = 3 and dx = 0.02. Your answers below need to be very precise, so use many decimal places. (a) Find the change Af = f(xo + dx)-f(xo). Af=0 (b) Find the estimate (i.e., the differential) df = f'(xo) dx. df = (c) Find the approximation error | Af-dfl. Error = f(xo+dx) f(ro) Af = f(ro+dr)-f(ro) TO dar (Click on graph to enlarge) y = f(x) Error= |Af-df xo+da Tangent df = f'(x)dx

3. a) Use the Chain Rule and the fact that |x| = Vx2 to show that d %3D dx |x| b) Sketch the graphs of the function f(x) = |x| and its derivative. %3D c) Use the result of part (a) to differentiate the function g(x) = x|x|. Write the derivative as a simplified monomial. %3D