Calculate the derivatives of the following functions explicitly:
\[ f(x)= \ln\left(3x^2-x\right) \qquad g(x) = \ln(\tan(x)) \qquad h(x) = x^2\ln(x) \]
\[ j(x) = \left(\ln(x)\right)^5 \qquad k(x) = \frac{1}{\ln(x)} \qquad \ell(x) = \ln\left(\frac{1}{x}\right) \]
Calculate the derivatives of the following functions explicitly:
\[ \newcommand{\ex}{{\mathrm{e}}}
f(x) = \frac{1}{3}\left(\ex^x\right)^2
\qquad
g(t) = \ex^{\sin(t)}
\qquad
h(\theta) = \tan\left(2\ex^\theta\right)
\]
\[\newcommand{\ex}{\mathrm{e}}
j(x) = x^2\ex^{5x^3}
\qquad
k(x) = \frac{\ex^x}{x} + 17\mathrm{e}^2
\qquad
\ell(x) = 3\ex^{2x}\sec(x)
\]
Demonstrate how to evaluate the following integrals.
\[\newcommand{\ex}{\mathrm{e}}
\int \left(\pi\ex^x + \ex\right) \,\mathrm{d}x
\qquad
\int x^2\ex^{5x^3} \,\mathrm{d}x
\qquad
\int\limits_1^2 \frac{\ex^{\frac{1}{x}}}{x^2} \,\mathrm{d}x
\]
Calculate the following indefinite integrals.
Having seen this pattern will be helpful later
when we integrate general rational functions.
\[
\int \frac{1}{x-1} \,\mathrm{d}x
\qquad
\int \frac{3}{2x+5} \,\mathrm{d}x
\]
\[
\int \frac{3x}{2x^2+3} \,\mathrm{d}x
\qquad
\int \frac{x+4}{x^2+8x-1} \,\mathrm{d}x
\]
Consider the cubic function as \(g(x) = x^3 + 2x^2 + 3x + 4\).
Convince yourself that \(g\) is an invertible function,
What is the value of \(\left(g^{-1}\right)'(4)\)?
What about \(\left(g^{-1}\right)'(10)\)?
Can a formula for the inverse of \(g\) be written down?
Sketch, and calculate the area of,
the region enclosed between the curves:
\[
y = \frac{1}{x}\ln(x)
\qquad\text{ and }\qquad
y = \frac{1}{x}\left(\ln(x)\right)^2
\]
Recall that \(f^{(n)}(x)\) is the notation
for the \(n\)th derivative of a function \(f\).
If we let \(y = f(x)\), there’s a simpler notation
where \(\dot y \) denotes the first derivative,
\(\ddot y \) denotes the second derivative,
and so on.
For \(f(x) = \mathrm{e}^{3x}\),
what is a formula for \(f^{(n)}(x)\)?
Can you think of a function \(y = f(x)\)
such that \(\dot{y} = 7y\)?
Can you think of a function \(y = g(x)\)
such that \(\ddot{y} = 7y\)?
Compare the graphs of the two curves
\( y = \mathrm{e}^x\) and \( y = 5^x\).
Prove that \(5^x\) is really just
a horizontal rescaling of \(\mathrm{e}^x.\)
I.e. \(5^x = \mathrm{e}^{kx}\) for some constant \(k\).
Compare the graphs of the two curves
\( y = \ln(x)\) and \( y = \log_5(x)\),
and recalling the base-change formula for logarithms
show that \(\log_5(x)\) is really just
a vertical rescaling of \(\ln(x).\)
I.e. \(\log_5(x) = c\ln(x)\) for some constant \(c\).
Consider the two curves
\(y = 3^x\) and \(y = 5^x\).
Let \(\mathrm{R}\) be the region in the plane
bound between these two curves and the lines \(x=0\) and \(x=1\).
Find the area of \(\mathrm{R}\).
Find the volume of the solid that results
from revolving \(\mathrm{R}\) about the \(x\)-axis.
Can you find the volume of the solid that results
from revolving \(\mathrm{R}\) about the \(y\)-axis?
Calculate the derivatives of the following functions explicitly:
\[
f(x) = \log_6(x) + 6^{x}
\qquad \qquad
g(x) = \log_6(13x) + 6^{17x}
\]
\[
h(x) = \log_6(13x)\times6^{17x}
\qquad \qquad
k(x) = \log_{10}(\sin(x))
\]
\[
j(x) = 7^{\csc(x)}
\qquad \qquad
\ell(\theta) = \log_{42}\left(7^{\tan\theta}\right)
\]
Remember that you can check your calculations using
WolframAlpha.
Calculate the derivatives of the following functions explicitly:
\[
\mu(x) = x^x
\qquad
\nu(x) = x^{\ln(x)}
\qquad
\rho(x) = \cos(x)^{\sin(x)}
\]
\[
\sigma(x) = \sqrt{x}^x
\qquad
\tau(x) = x^{\sqrt{x}}
\qquad
\omega(x) = x^{x^x}
\]
Hint: recall that if there are variables in both the base
and in the exponent, you’ve gotta use the technique called
logarithmic differentiation.
Remember that you can check your calculations using
WolframAlpha.
Calculate the following integrals explicitly:
\[
\int \frac{7}{6x+5} \,\mathrm{d}x
\qquad
\int t^7 + 7^t + 7^7 \,\mathrm{d}t
\]
\[
\int 3^{\cos(x)}\sin(x) \,\mathrm{d}x
\qquad
\int \pi^x\tan\left(\pi^x\right) \,\mathrm{d}x
\]
Remember that you can check your calculations using
WolframAlpha.
What are formulas for the following derivatives?
\[
\frac{\mathrm{d}}{\mathrm{d}x} \arccos\left(\mathrm{e}^x\right)
\qquad
\frac{\mathrm{d}}{\mathrm{d}x} 3\arctan(x^2-7)
\qquad
\frac{\mathrm{d}}{\mathrm{d}x} \arcsin\left(\frac 1 x \right)
\]
Remember that you can check your calculations using
WolframAlpha.
Calculate the exact value of these definite integrals.
\[
\int\limits_{1/4}^{\sqrt{2}/4} \frac{1}{\sqrt{1-(2x)^2}} \,\mathrm{d}x
\qquad\quad
\int\limits_{\sqrt{3}}^{3} \frac{1}{x^2+9} \,\mathrm{d}x
\qquad\quad
\int\limits_{\sqrt{\frac 1 3}}^{\sqrt{\frac 2 3}} \frac{1}{x\sqrt{9x^4-1}} \,\mathrm{d}x
\]
Hint: stare at the list of derivatives of the inverse trig functions,
and try to make the correct substitution to transform the integrand
into one of those expressions.
Remember that you can check your calculations using
WolframAlpha.
How do you write each of the following
as an algebraic function of \(x\)?
Hint: Recall the double-angle
and half-angle trigonometric formulas.
\[
\sin\left(2\arccos(x)\right)
\qquad
\cos\left(2\mathrm{arcsec}(x)\right)
\qquad
\sin\left(\frac 1 2 \arctan(x)\right)
\]
It’s midnight.
You are standing on a relatively straight California beach,
staring at a lighthouse on an island three miles out,
your line-of-sight perpendicular to the shoreline,
as the beam of the lighthouse beacon sweeps
over you, sweeps past you, disappears,
then sweeps towards you and over you again,
capturing you in its patina every \(24\) seconds.
How fast is the illuminated portion of the shoreline moving
for the brief moment you are in it?
How fast is the illuminated portion of shoreline moving
the moment it is one mile past you down the beach?
Demonstrate how to calculate the values of the following limits.
\[
\lim\limits_{x \to 0} \frac{1-\cos(x)}{x^2}
\qquad
\lim\limits_{x \to 0} \frac{\mathrm{e}^x}{\mathrm{e}^x-1} - \frac{1}{x}
\qquad
\lim\limits_{x \to \infty} \frac{\sqrt[3]{x}}{\ln(x)}
\]
\[
\lim\limits_{x \to 2} \frac{x^4 - 4^x}{\sin(\pi x)}
\qquad
\lim\limits_{x \to 0} \big(\cos(x)\big)^{\csc(x)}
\qquad
\lim\limits_{x \to 0} \sin(x)\ln(x)
\]
Challenges
Prove the “rules of logarithms”:
\[
\ln\left(b^a\right) = a\ln(b)
\qquad
\ln(ab) = \ln(a) + \ln(b)
\]
Prior to this class, you’d have proven these
by appealing to the analogous rules for exponents
and transforming those into these two identities.
Here though we’ve defined \(\ln\) as an integral.
How do you prove those two “rules” hold
based on that definition of \(\ln\) as an integral?
I.e. how do we prove that this function \(\ln\)
actually behaves like the logarithm we already know and love?
Prove the “rules of exponentials”:
\[
\exp(a+b) = \exp(a)\exp(b)
\qquad
\exp(ab) = \exp(a)^b
\]
I.e. prove that the function \(\exp(x)\) actually behaves
like the exponential function \(\mathrm{e}^x\)
as we’ve been assuming.
We defined the constant \(\mathrm{e}\) as the unique number
such that \[\int\limits_1^\mathrm{e} \frac{1}{x} \,\mathrm{d}x = 1.\]
But what is this number explicitly?
Can you write an algorithm to approximate the decimal presentation
of the number \(\mathrm{e}\)?
As a reference, to check your algorithm, past mathematicians
have computed
\[\mathrm{e} \approx 2.718281828459045235360287471352662497757247\,.\]