Single Variable Calculus Review These problems are great for reviewing limits and derivatives from Calculus I in preparation for the upcoming chapter 1. Evaluate the limits if they exist. A. limx xtan(x) H. limx0 f(x) if + 1 : if x is rational ( ) 1 : if x is irrational L. limx( (x2 + 1) x) M. limx(( x2 + 1 ) x) 1 O. limx0 arctane1/x P. limx0+ arctane1/x 2. Find the values of a such that the limit lim exists. 3. Given that limx2(2x3) = 1, find the values of (in the precise definition of a limit) that corresponds to 4. Prove each limit using the , definition of a limit. A. limx2(2x 3) = 1 B. limx2(x2 2x) = 0 C. limx4 x = 2 E. limx2 ex = e2 5. Explain why the function is discontinuous at the given number. A. : 1 if x 6= 2 : if x = 2 at a = 2 B. : if x 6= 2 : if x = 2 1 at a = 2 6. Show that f is continuous on (,) if ) ) : if x < /4 : if x /4 2 , and 7. Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval: A. 3 (x) = 1 x on the interval (0,1). B. arctanx = 1 x on the interval (0,1). 8. Find the derivative of each function using the limit definition: A. f(x) =(x + 1) B. x2 2x C. 9. Show why the function f(x) = 3( x) is not differentiable at x = 0. 10. Differentiate each: A. 3x5 2x3 + 4x A. B. xe + ex C. 11. Find an equation for the tangent line to the curve at the given point: A. y = at the point (3,2) 3 B. y = x + xex at the point (1,e + 1). 12. Consider +7 (x) = x4 ax2 + b : if x < 1 : if x 1 Find the values of a and b such that this function is differentiable everywhere. 4