Master Calculus 1 Practice Problems with Expert Help

Determine the limit of each function analytically. Show neat and complete solution. (hint: some of the items need to be factored or rationalized for the limits to exist)
${\displaystyle \underset{x\to 2}{lim}\frac{{(3x-2)}^2-{(x+2)}^2}{(x-2)}}$

Find the limits (that is, end behaviors) below.
${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{-5x(x-9)(x+6)}{6-8x^3}}$

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.
${\displaystyle \underset{x\to 0}{lim}\frac{x}{\sqrt{cx+1}-1},c\ne 0}$

Evaluate the following limits algebraically. If neccessary, select the most informative answer from infinity, negative infinity, and DNE (Does Not Exist)
${\displaystyle \underset{x\to 0}{lim}\frac{3}{x^2}}$

Find the limits
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\sqrt{\frac{8x^2-3}{2x^2+x}}}$

Use the LHospitals

Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions
Use integration by parts to integrate functions.
Integrate products of powers of trigonometric functions.
Evaluate
${\displaystyle \int {\sin }^4\theta {\cos }^2\theta d\theta }$

Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity.
${\displaystyle \underset{x\to 9}{lim}\frac{2t-18}{5t-45}}$

Analizing the polynomial
${\displaystyle y=(x-1)(3x-2)(x+3)}$
a) Determine the roots of the polinomial
b) Determine the end behaviors
c) Determine how many turning points

Solve the following limit.
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\left(\frac{3x^3-4x+2}{7x^3+5}\right)}$

Find functions fand g such that
${\displaystyle h=g\circ f}$
(Note: The answer is not unique. Enter your answers as a comma-separated list of functions. Use non- identity functions for fand g.)
$h\left(x\right)={(5x^2-8)}^{-3}$

Solve the equation:
${\displaystyle \int 2e^{-5x}dx}$

Describe the First Derivative Test in your own words.

Without integrating, explain why
${\displaystyle {\int }_{-2}^{2}x{(x^2+1)}^{2}dx=0}$

Find the critical points of the following functions.
${\displaystyle f(x,y)=y{e}^x-e^y}$

List the various rules that allow us to differentiate constant functions,
power functions, polynomials, exponential functions, rational functions, and certain combinations of them,
simply and directly, without having to take limits each time?

Analyzing the Motion of a Projectile A projectile is fired
at an inclination of ${\displaystyle {45}^{\circ }}$ to the horizontal, with a muzzle
velocity of 100 feet per second. The height h of the projectile
is modeled by ${\displaystyle h\left(x\right)=\frac{-32x^2}{{100}^2}+x}$
where x is the horizontal distance of the projectile from the firing point
(a) At what horizontal distance from the firing point is the height of the projectile a maximum?
(b) Find the maximum height of the projectile.
(c) At what horizontal distance from the firing point will the projectile strike the ground?
(d) Using a graphing utility, graph the function h, ${\displaystyle 0\le x\le 350}$ .

Analyzing critical points Use the Second Derivative Test to classify the critical points of ${\displaystyle f(x,y)=xy(x-2)(y+3)}$.

What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions?
The inverse trigonometric functions are
${\displaystyle {\sin }^{-1}x}$, ${\displaystyle {\cos }^{-1}x}$, ${\displaystyle {\tan }^{-1}x}$, ${\displaystyle {\csc }^{-1}x}$, ${\displaystyle {\sec }^{-1}x}$, ${\displaystyle {\cot }^{-1}x}$

Plot the graphs of the given functions
$f(x)=\{\begin{array}{ll}|x|& forx<2\\ -x+1& forx\ge 2\end{array}$
$m(x)=\{\begin{array}{ll}3& for-4<x<-1\\ -x& for-1\le x<3\\ \sqrt{x-3}& forx\ge 3\end{array}$