Exam 1 Practice - Limits and Derivatives

Part I

(1) Calculate the following limits:

(a)

$$\lim_{x \to 1} \frac{x^2-7x+6}{x^2-2x+1}$$

(b)

$$\lim_{x \to 0} \frac{\sin x}{x^2}$$

(c)

$$\lim_{x \to 3} \frac{\sqrt{1+x}-2}{x-3}$$

(2) Find the derivatives of the following functions:

(a) $f(x) = {(x^3+3\sin x +4)}^8$

(b) $g(x) = x^3\csc x$

(c) $h(x) = \frac{x+1}{x+5}$

Part 2

(3) Consider $f(x)= \frac{x^3+1}{x^2-5x-6}$.

(a) Where is f(x) not continuous?

(b) Identify each discontinuity as removable or nonremovable.

(c) Where are the vertical asymptotes?

(4) Calculate the derivative of $f(x)= \frac{1}{x+1}$ using the definition of the derivative.

(5) Find the equation of the tangent line to the curve $y=2\sin x+3$ at the point $(\pi,3)$.

(6) Find the derivative of y with respect to x for the equation:

y³+3xy-2(x²y)=sin (y).

(7) Find the velocity at impact of a bowling ball dropped from a building which is 400 feet high, given that the ball has position equation

s(t)=-16t²+400.