Topics and practice problems for LSU's credit exam for Differential and Integral Calculus (Math 1550)

Evaluate each limit.

1. \(\displaystyle \lim_{x\to0} \bigl(x^2-3x+1\bigr)\)
2. \(\displaystyle \lim_{x\to1} \sqrt{x^2+2x+4}\)
3. \(\displaystyle \lim_{x\to2} \frac{x-5}{x^2+4}\)
4. \(\displaystyle \lim_{x\to0} \frac{x\,e^{-2x+1}}{x^2+1}\)
5. \(\displaystyle \lim_{x\to0} \frac{\sin x}{\tan x}\)
6. \(\displaystyle \lim_{x\to0} \frac{\tan 2x}{x}\)
7. \(\displaystyle \lim_{x\to1} \frac{x-1}{\sqrt{x}-1}\)
8. \(\displaystyle \lim_{x\to-1} f(x) \quad\text{where}\quad f(x) = \begin{cases}2x+1, & \text{if \(x < -1\)}\\3, & \text{if \(-1 < x < 1\)}\\2x+1, & \text{if \(x > 1\)}\end{cases}\)
9. \(\displaystyle \lim_{x\to4^+} \sqrt{16-x^2}\)

Determine each limit. Answer with a number, or with \(\infty\) or \(-\infty\), or with “does not exist”.

1. \(\displaystyle \lim_{x\to1^-} \frac{1-2x}{x^2-1} \)
2. \(\displaystyle \lim_{x\to1} \frac{1-2x}{x^2-1} \)
3. \(\displaystyle \lim_{x\to1^+} \frac{1-2x}{x^2-1} \)
4. \(\displaystyle \lim_{x\to2} \frac{4-x}{(x-2)^2} \)
5. \(\displaystyle \lim_{x\to2^-} \frac{-x}{\sqrt{4-x^2}} \)
6. \(\displaystyle \lim_{x\to-1^-} \bigl(x^2-2x-3\bigr)^{-2/3} \)

Use l'Hôpital's rule to evaluate each limit.

1. \(\displaystyle \lim_{x\to2} \frac{x-2}{x^2-4} \)
2. \(\displaystyle \lim_{x\to0} \frac{x^3}{\sin x - x} \)
3. \(\displaystyle \lim_{x\to1} \frac{x-1}{\ln x} \)
4. \(\displaystyle \lim_{x\to0} \frac{e^x-1}{\cos x - 1} \)
5. \(\displaystyle \lim_{x\to0} \frac{x^2}{\cos x - x} \)
6. \(\displaystyle \lim_{x\to1} \frac{\ln(\ln x)}{\ln x} \)

Determine each limit. Answer with a number, or with \(\infty\) or \(-\infty\), or with “does not exist”.

1. \(\displaystyle \lim_{x\to-\infty} \frac{-x}{\sqrt{4+x^2}} \)
2. \(\displaystyle \lim_{x\to\infty} \frac{x^3-2x+5}{3x^2+4x-1} \)
3. \(\displaystyle \lim_{x\to\infty} \frac{x^2-\sin x}{x^2+4x-1} \)
4. \(\displaystyle \lim_{x\to\infty} \bigl( \sqrt{x^2+3} - x \bigr) \)
5. \(\displaystyle \lim_{x\to\infty} e^{2x} \)
6. \(\displaystyle \lim_{x\to\infty} \sin 2x \)
7. \(\displaystyle \lim_{x\to\infty} \bigl( e^{-3x} \cos 2x \bigr) \)
8. \(\displaystyle \lim_{x\to\infty} \ln 2x \)

Find all discontinuities of \(f(x)\).

1. \(\displaystyle f(x) = \frac{x-1}{x^2-1} \)
2. \(\displaystyle f(x) = \frac{4x}{x^2+4} \)
3. \(\displaystyle f(x) = \begin{cases}2x, & \text{if \(x < 1\)}\\x^2, & \text{if \(x\ge1\)}\end{cases} \)

Determine the intervals on which \(f(x)\) is continuous.

4. \(\displaystyle f(x) = \sqrt{x+3} \)
5. \(\displaystyle f(x) = \sin\bigl(x^2+2\bigr) \)

Find the derivative of each function.

1. \(\displaystyle f(x) = x^3 - 2x + \pi \)
2. \(\displaystyle f(t) = 3t^3 - 2\sqrt{t} \)
3. \(\displaystyle f(x) = \frac3x - 8x + e \)
4. \(\displaystyle h(x) = \frac{10}{\sqrt{x}} - 2x \)
5. \(\displaystyle f(s) = 2s^{3/2} - 3s^{-1/3} \)
6. \(\displaystyle f(x) = \frac{3x^2-3x+1}{2x} \)

Find the derivative of each function.

1. \(\displaystyle f(x) = (x^2+3)(x^3-3x+1) \)
2. \(\displaystyle f(x) = \bigl(\sqrt{x}+3x\bigr)\bigl(5x^2-\frac3x\bigr) \)
3. \(\displaystyle f(x) = \bigl(x^{3/2}-4x\bigr)\bigl(x^4-\frac{3}{x^2}+2\bigr) \)

Find the derivative of each function.

1. \(\displaystyle f(x) = \frac{3x-2}{5x+1} \)
2. \(\displaystyle f(x) = \frac{x-2}{x^2+x+1} \)
3. \(\displaystyle f(x) = \frac{3x-6\sqrt{x}}{5x^2-2} \)
4. \(\displaystyle f(x) = \frac{(x+1)(x-2)}{x^2-5x+1} \)
5. \(\displaystyle f(x) = (x^2-1)\left(\frac{x^3+3x^2}{x^2+2}\right) \)

Find the derivative of each function.

1. \(\displaystyle f(x) = 4\sin x - x \)
2. \(\displaystyle f(x) = \tan x - \csc x \)
3. \(\displaystyle f(x) = x\,\cos x \)
4. \(\displaystyle f(x) = 4\sqrt{x}-2\sin x \)
5. \(\displaystyle f(x) = \sin x \, \sec x \)
6. \(\displaystyle f(x) = \frac{\cos x - 1}{x^2} \)
7. \(\displaystyle f(x) = 2\sin x\, \cos x \)
8. \(\displaystyle f(x) = 4x^2\tan x \)
9. \(\displaystyle f(x) = 4\sin^2 x + 4\cos^2 x \)

Find the derivative of each function.

1. \(\displaystyle f(x) = x\,e^x \)
2. \(\displaystyle f(x) = x+2^x \)
3. \(\displaystyle f(x) = 2e^{x+1} \)
4. \(\displaystyle f(x) = \bigl(\tfrac13\bigr)^x \)
5. \(\displaystyle f(x) = 4^{-x+1} \)

Find the derivative of each function.

1. \(\displaystyle f(x) = \ln 2x \)
2. \(\displaystyle f(x) = \ln x^3 \)
3. \(\displaystyle f(x) = \frac{\ln x}{x} \)
4. Find an equation of the tangent line to \(y=x^2\ln x\) at \(x=1\).

Find the derivative of each function.

1. \(\displaystyle f(x) = \sqrt{x^2+4} \)
2. \(\displaystyle f(x) = (x^3+x-1)^5 \)
3. \(\displaystyle f(x) = \sin(2x^2+3) \)
4. \(\displaystyle f(x) = \sin^4 x \)
5. \(\displaystyle f(x) = \tan^2 x \)
6. \(\displaystyle f(x) = e^{1/3x} \)
7. \(\displaystyle f(x) = \ln(x^3+3x) \)
8. \(\displaystyle f(x) = x^2\sin 4x \)
9. \(\displaystyle f(x) = \frac{\sin x^2}{x^2} \)
10. \(\displaystyle f(x) = \sin\bigl(\ln(\cos x^3)\bigr) \)
11. \(\displaystyle f(x) = \sqrt{\sin x^2} \)
12. \(\displaystyle f(x) = \bigl(\ln(x^2+1)\bigr)^8 \)
13. \(\displaystyle f(x) = \cos\left(\frac{4x}{x^2+1}\right) \)

Compute the slope of the tangent line at the given point.

1. \(\displaystyle x^2+4y^2=8 \quad\text{at}\quad (2,1) \)
2. \(\displaystyle y-3x^2y=\cos x \quad\text{at}\quad (0,1) \)

Find the derivative \(y'(x)\) implicitly.

3. \(\displaystyle x^2y^2+3y=4x \)
4. \(\displaystyle \sqrt{xy} - 4y^2 = 12 \)
5. \(\displaystyle \frac{x+y}{y}=4x+y^2 \)
6. \(\displaystyle e^{x^2y}-e^y=x \)
7. Find an equation of the tangent line at \((2,1)\) for \(x^2-4y^2=0\).

Find the indicated derivative.

1. \(\displaystyle f''(x) \quad\text{for}\quad f(x)=x^4-3x^3+2x^2-x-1 \)
2. \(\displaystyle f'''(x) \quad\text{for}\quad f(x)=x\,e^{2x} \)
3. \(\displaystyle f''(x) \quad\text{for}\quad f(x)=\tan x \)

1. Assume that the infected area of an injury is circular. If the radius of the infected area is \(3\) mm and growing at a rate of \(1\) mm/hr, at what rate is the infected area increasing?
2. An oil tanker has an accident and oil pours out at the rate of \(150\) gallons per minute. Suppose that the oil spreads onto the water in a circle at a thickness of \(0.1\) inch. Given that \(1\) ft³ equals \(7.5\) gallons, determine the rate at which the radius of the oil spill is increasing when the radius reaches \(500\) feet.

Find all critical numbers and determine whether each represents a local maximum, local minimum, or neither.

1. \(\displaystyle f(x) = x^3-3x+1 \)
2. \(\displaystyle f(x) = x^4-3x^3+2 \)
3. \(\displaystyle f(x) = \sin x\,\cos x \), on the interval \( [0,2\pi] \)
4. \(\displaystyle f(x) = \frac{x}{x^2+1} \)
5. \(\displaystyle f(x) = \frac12\bigl(e^x+e^{-x}\bigr) \)
6. \(\displaystyle f(x) = x^{4/3} + 4x^{1/3} + 4x^{-2/3} \)

Find the absolute extrema of the given function on the indicated interval.

7. \(\displaystyle f(x) = x^4-8x^2+2 \), on the interval \( [-3,1] \)
8. \(\displaystyle f(x) = \sin x + \cos x \), on the interval \( [0,2\pi] \)

Determine the intervals where the function is increasing and where it is decreasing and the intervals of concavity.

1. \(\displaystyle f(x) = x^3-3x^2+4 \)
2. \(\displaystyle f(x) = x+\frac1x \)
3. \(\displaystyle f(x) = x^{3/4}-4x^{1/4} \)
4. \(\displaystyle f(x) = (x^2+1)^{2/3} \)
5. \(\displaystyle f(x) = \frac{x^2}{x^2-9} \)
6. \(\displaystyle f(x) = \frac{x^2-5x+4}{x} \)

1. A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. The enclosed area is to equal \( 1800 \) ft². Find the minimum perimeter and the dimensions of the corresponding enclosure.
2. A box with no top is to be built by taking a \(6\)-inch by \(10\)-inch sheet of cardboard and cutting \(x\)-inch squares out of each corner and folding up the sides. Find the value of \(x\) that maximizes the volume of the box.
3. A water line runs east–west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. One development is \(3\) miles south of the existing line; the other development is \(4\) miles south of the existing line and \(5\) miles east of the first development. Find the place on the existing line to make the connection to minimize the total length of new line.

Compute each integral exactly.

1. \(\displaystyle \int_0^1 \bigl( x\sqrt{x} + x^{-1/2} \bigr)\,\mathrm{d}x \)
2. \(\displaystyle \int_0^{\pi/4} \sec x\, \tan x\, \mathrm{d}x \)
3. \(\displaystyle \int_0^3 \bigl( 3e^{2x} - x^2 \bigr)\,\mathrm{d}x \)
4. \(\displaystyle \int_0^3 \bigl( x^3 - \sin x \bigr)\,\mathrm{d}x \)

Find the position function \( s(t) \) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds.

5. \( v(t)=40-\sin t \), where \( s(0)=2 \)
6. \( v(t)=25(1-e^{-2t}) \), where \( s(0)=0 \)
7. \( a(t)=4-t \), where \( v(0)=8 \) and \( s(0)=0 \)

Find the function \( f(x) \) satisfying the given conditions.

8. \(\displaystyle f'(x) = 4x^2-1 \), where \( f(0)=2 \)
9. \(\displaystyle f'(x) = 3e^x+x \), where \( f(0)=4 \)

Evaluate the indicated integral.

1. \(\displaystyle \int \bigl(2x+1\bigr)\bigl(x^2+x\bigr)^3 \mathrm{d}x \)
2. \(\displaystyle \int \bigl(\cos x\bigr)\bigl(\sqrt{\sin x + 1}\bigr)\,\mathrm{d}x \)
3. \(\displaystyle \int x e^{x^2+1} \mathrm{d}x \)
4. \(\displaystyle \int \frac{1}{x\ln\sqrt x} \mathrm{d}x \)

Evaluate the definite integral.

5. \(\displaystyle \int_0^2 x \sqrt{x^2+1} \mathrm{d}x \)
6. \(\displaystyle \int_{\pi/2}^\pi \frac{4\cos x}{\bigl(\sin x+1\bigr)^2} \mathrm{d}x \)

Find the area between the curves on the given interval.

1. \( y=x^3 \), \( y=x^2-1 \), \( 1 \le x \le 3 \)
2. \( y=e^x \), \( y=x-1 \), \( -2 \le x \le 0 \)

Find the area of the region determined by the intersection of the curves. Choose the variable of integration so that the area is written as a single variable.

3. \( y=x^2-1 \), \( y=7-x^2 \)
4. \( y=x \), \( y=2-x \), \( y=0 \)
5. \( x=3y \), \( x=2+y^2 \)

Compute the volume of the solid formed by revolving the given region about the given line.

1. Region bounded by \( y=\sqrt x \) on the interval \( [0,4] \) about the \(x\)-axis.
2. Region bounded by \( y=2-\dfrac12x^2 \) from \( x=0 \) to \( x = 2 \) about the \(y\)-axis.
3. Region bounded by \( y=x^2 \), \(x=0\), and \(y=1\) about
   1. the \(y\)-axis;
   2. the \(x\)-axis;
   3. the line \(y=2\).
4. Region bounded by \( y=4-x^2 \) and \(y=0\) about
   1. the \(y\)-axis;
   2. the line \(y=-3\);
   3. the line \(y=7\);
   4. the line \(x=3\).
5. Region bounded by \(y=x\) and \(y=x^2\) in the first quadrant revolved about the \(y\)-axis.
6. Region bounded by the graphs of \(y=4-x^2\) and the \(x\)-axis about the line \(x=3\).
7. Region bounded by the graphs of \(y=x\), \(y=2-x\), \(y=0\) and revolved about
   1. the line \(y=2\);
   2. the line \(y=-1\);
   3. the line \(x=3\).

1. A force of \(5\) pounds stretches a spring \(4\) inches. Find the work done, in foot-pounds, in stretching the spring \(6\) inches beyond its natural length.
2. A concrete block weighing \(250\) pounds is lifted a distance of \(20\) inches. How much work was done in foot-pounds?
3. If you have to lift a \(100\) pound television set up \(3\) feet onto a table, how much work have you done?

Find the average value of the function on the given interval.

1. \(\displaystyle f(x)=2x+1 \), on the interval \([0,4]\)
2. \(\displaystyle f(x)=x^2-1 \), on the interval \([1,3]\)
3. \(\displaystyle f(x)=\cos x \), on the interval \([0,\pi/2]\)