Backlinks
Table of Contents
1 Problem 1
Differentiate (with respect to x)
1.1 \((a)\)
\[
\begin{aligned} y &= x^2 + x^{74} - \ln{x} - \log_{3}{x} + 51^x - e^x + \sin{x} - \cos{x} \\ \frac{d}{dx}[y] &= 2x + 74x^{73} - \frac{1}{x} - \frac{1}{x\ln{(3)}} + ln{(51)}*51^x - e^x + \cos{x} + \sin{x} \\ \end{aligned}\]
1.2 \((c)\)
\[
\begin{aligned} f(x) &= 7 + x^2 + 6x^3 + 3\sqrt[4]{x} + \frac{1}{x} - \ln{x} + 5^x \\ \frac{d}{dx}[f(x)] &= 2x + 18x^2 + \frac{3}{4\sqrt[4]{x^3}} - \frac{1}{x} + \ln{(5)}5^x \\ \end{aligned}\]
2 Problem 2
Sketch the function \(f(x) = 2x^5 - 10x^4 - 70x^3\), and label \((x, y)\) of intercepts, maxima, and minima.
3 Problem 5
Find antiderivatives
4 \((a)\)
\[
\begin{aligned} \int x^4 + 3x^8 - 12x^7 + 14 \,dx \\ &= \int x^4 \,dx + \int 3x^8 \,dx - \int 12x^7 \,dx \\ &= \frac{1}{5}x^5 + \frac{1}{3}x^9 - \frac{3}{2}x^8 + C \\ \end{aligned}\]
5 \((d)\)
\[
\begin{aligned} \int 323(4x^3 + 3x^2)(x^4 + x^3)^{322} \,dx &= (x^4 + x^3)^{323} + C \\ \end{aligned}\]
6 Problem 6
\(f(x) = 2x^5 - 10x^4 - 70x^3\)
6.1 \((a)\)
Area underneath function from \(x=-4\) to \(x=-1\): \[
\begin{aligned} f(x) &= 2x^5 - 10x^4 - 70x^3 \\ \int_{-4}^{-1} f(x) \,dx &= \int_{-4}^{-1} 2x^5 - 10x^4 - 70x^3 \,dx \\ &= [\frac{1}{3}x^6 - 2x^5 - \frac{2}{35}x^4]_{-4}^{-1} \\ &= -\frac{23775}{7} \\ \end{aligned}\]