Calculus I

Department of mathematics:

Calculus I, Fall 2013

Midterm Examination: (Show your work for full credit)

1- Find the composite functions f o g and g o f. Find the domain oi Pach
composite function. Are the two composite functions equal?
f(x) = ifx – 2 and 906) = x2
2- Show that f and 9 are inverse fzunctions (a) analytically and (b) graphically
f(x): Vx+3andg(x) =x -3, x20
3- Find the limit (if it exists), if it does not exist, explain Why.
lim -“’” b) lim L“
a x46- x2-36 ‘ x..25+ x-zs’
2)2 x < 2
lm x ,where x ___{(x
C) 11.sz f() 2_x‘ x>2
x+3
4- Let f be a function defined by f (x) – -x2_3x-18.
a) Find the x-values (if any) at which f is not continuous. Which of the
discontinuities are removable?
b) Find the vertical Asymptote if any of the graph of the function.
c) Find the horizontal asymptote if any of the graph of the function.
5- Find the value of c such that the function is continuous on the entire real
number line.
f(x) _ {ex2 -2x, x < -1
ex“ + 3, x _>_ -1
6- Find the derivative of the function by the limit process.

a)f(x)=x3-2x+1, g(x)= I
7- Find the slope of the graph of the function at the given pomt-
f (x) = 5 cos x – 9x. (0. 5)