f(x) : x/ (a) + y/6 = 1
6x + ay = 6a
f(x) = y = -6x/a + 6
g(x) : x/(a) + y/3 =1
g(x) = y = 3 - 3x/a
f(x)/g(x) = (-6x/a + 6) / (3 - 3x/a)
lim x->a
f(x)/g(x) = f(a)/g(a) { f(a) = g(a) }
=1
4. (3 marks) Suppose that you have two linear functions f(x) and g(z) such that f(a)...
9. (4pts) Consider the linear functions f(x) 6-x+3(x-4) and g (x)-3(x+)-5(+1). Solve f(x) g() algebraically, showing all steps. (You may also check graphically) 10. 4pts) Test algebraically whether the function f(x)-4x- is even, odd, or neither even nor odd. Show your work. (You may also check your results graphically.) 11. (4pts) Determine whether the graph of y =-x' + 4x is symmetric with respect to the x-axis, the y-axis, and/or the origin. Use your graphing calculator make a sketch below...
*9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point xo: (e) f,g : R-R given by f(x)and g(x) 1+2r ro0 (f) f, g : (0, oo) → R given by f(r) = In(2) and g(z) = (z-1)3 + In(z): zo = 1. (g) f.g: (0, oo) -R given by f(x)-In(x) and g(x)-(x 1)200 +ln(x); ro 1 x-1)200 *9. For each of the following pairs of functions,...
5 Consider the functions f and g whose graphs are given below. z y = f(x) -4 A3 -2 -1 1 2 3 4 y = 9(2) -4 -3 -2 -1 1 2 3 4 1 + f. Find (3) a. Find f'(-3). b. Find f'(1). g. Suppose p(x) = f(x)g(2). Find p'(-3). c. Find f'(3). h. Suppose q(z) = 5(). Find g(3). d. Find t'(-3). g(2) e. Find g'(1). i. Suppose r(x) = x2 f(x). Find r'(1).
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
7. [5 marks] Suppose that f(x), g(x), and h(x) are functions such that f(x) is O(g(x)) and g(a) is O(h(x)). Prove that f(x) is O(h(x)) 7. [5 marks] Suppose that f(x), g(x), and h(x) are functions such that f(x) is O(g(x)) and g(a) is O(h(x)). Prove that f(x) is O(h(x))
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x/t(x) g(x) f'(x) g'(x) 3 1 16 6 5 4 - 3 3 4 N f(x) g(x), X = 3 Select one: O A. 197 OB. 37 O C. 101 OD. 60 to the integral
3. Below you are given the graphs of the functions f and g. Suppose that: u(x)-f(g(x)), v(x)-f(x) g(x), and w(x)-g(f(x)) Use the graphs to find the indicated derivatives. If the indicated derivative does not exist, write "D.N.E." in the space provided. Be sure to include work that shows how you arrived at your answer. 20 a) u'3) b) v-4) c) wl) 3. Below you are given the graphs of the functions f and g. Suppose that: u(x)-f(g(x)), v(x)-f(x) g(x), and...
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
6. Suppose that f and g are differentiable functions on an interval (a, b), and suppose that for all (a, b) we have f'(z) = g(z) and g'(z) = -f(x). Show that f2+g2 is constant.
1. (6 marks) Provide the domain, target, and range of the following functions (a, b, c, d}3 . For each x E(a, b, cF, fx)-dx. a) fta, b. c}2 b) g: {a, b, c, d)-(a, b, c, d}?. For each x E(a, b, c, d], g(x) (4 marks) Use the ceiling and floor functions to qive a mathematical expression for the following a) Among a random group of 100 people at least 9 must be born in the same month...