Exam 10: An Introduction to Calculus: Limits, Derivatives, and Integrals

A coin is dropped from a window. Find the coin's (a) average speed during the first 4 sec of fall and (b) instantaneous speed at t=4 .

Draw the graph of $f(x)=\frac{1}{2} x$ over the interval [0,4] . On the graph, show and shade the rectangles that would be used to approximate the area under the curve by the right rectangle approximation method using 4 subintervals.

Find the instantaneous rate of change of $f(x)=5 x^{2}$ at x=3 .

What is the instantaneous rate of change of $f(x)=2 x^{2}$ at x=1 ?

Draw the graph of f(x)=-2(x-1)(x-4) over the interval [1,4] . On the graph, show and shade the rectangles that would be used to approximate the area under the curve f(x) over [1,4] by the right rectangle approximation method using 6 subintervals. Compute the estimation.

Determine the following limits, if they exist. (a) $\lim _{x \rightarrow-2} x(x-2)^{2}$ (b) $\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x^{2}-4 x+3}$ (c) $\lim _{x \rightarrow \infty} 5 x-2$ (d) $\lim _{x \rightarrow 0} \frac{\sin 5 x}{x^{2}-x}$ (e) $\lim _{x \rightarrow 3} \frac{x-3}{x^{2}+5}$ (f) $\lim _{x \rightarrow 0} \frac{2}{x}$

For the function f(x)=4+3 x-2 x² : (a) Use the definition of the derivative at a point to find the slope of the tangent line of the function at x=1 . (b) What is the equation of the tangent line in part a?

Determine $\lim _{x \rightarrow 0} \frac{\cos x-1}{x^{2}}$ if the limit exists.

Find the instantaneous rate of change of $f(x)=-3 x^{2}$ at x=4 .

At what points c in the domain of $f(x) \text { does } \lim _{x \rightarrow c} f(x)$ exist? $f(x)=\left\{\begin{array}{l}-x^{2} & -2 \leq x<1\\-2&x=1 \\3 x+5 & 1 < x \leq 3\end{array}\right.$

For the function $f(x)=\frac{-3}{x+2}$ (a) Find the slope of the function at x=2 using the definition of the derivative. (b) What is the equation of the tangent line to the curve f(x) at x=2 ?

$\lim _{x \rightarrow 3} \frac{x^{3}-27}{x-3}$ (a) Explain why direct substitution cannot be used to find the limit. (b) Find the limit algebraically, if it exists.

Explain how to find the area under the graph of $f(x)=|x-3|$ from x=1 to x=5 by computing the geometric area.

Partition the interval [1,5] into eight equal subintervals. List the eight subintervals.

A baseball is dropped from a window. Find the baseball's (a) average speed during the first 2 sec of fall and (b) instantaneous speed at t=2 .

Determine the following limits, if they exist. (a) $\lim _{x \rightarrow 4} x(x+4)^{2}$ (b) $\lim _{x \rightarrow 1} \frac{x^{2}+3 x-4}{x^{2}-6 x+5}$ (c) $\lim _{x \rightarrow-\infty} 6 x+2$ (d) $\lim _{x \rightarrow 0} \frac{\sin 5 x}{x}$ (e) $\lim _{x \rightarrow 4} \frac{x-4}{x^{2}+2}$ (f) $\lim _{x \rightarrow 0} \frac{2}{x^{2}}$

For the function $f(x)=\frac{3}{x-4}$ : (a) Find the slope of the function at x=1 using the definition of the derivative. (b) What is the equation of the tangent line to the curve f(x) at x=1 ?