Practice: derivatives and investigation

Exponential and logarithmic functions, grade 12 (5 units) Lesson 19 of 23Practice

Practice: derivatives and investigation

Differentiate. In part e, find the second derivative.

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The function $\displaystyle {f(x)}={x\ln x}$ is given.

Calculate the slope of the tangent to the graph at the point where $\displaystyle {x}={e}$ .

$\displaystyle f(x)=x\ln x$

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$\displaystyle {f(e)}={e}$ .
Write the equation of the tangent to the graph at the point $\displaystyle (e,e)$ .

\displaystyle f(x)=x\ln x

\displaystyle (2.718,2.718)

The function $\displaystyle {f(x)}={(x+1)e^{-x}}$ is given.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=(x+1)e^{-x}$

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Find the extremum point of the function, and its type.

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=(x+1)e^{-x}$

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Find where the function is concave up.

Mark the extremum point and the inflection point on the graph.

\displaystyle f(x)=(x+1)e^{-x}

The function $\displaystyle {f(x)}={\ln(x^2-1)}$ is given.

Find the domain.

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=\ln(x^{2}-1)$

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Show that the function is concave down throughout its domain.
Is the function even? Give a reason.

Add a row for each of the vertical asymptotes.

\displaystyle f(x)=\ln(x^{2}-1)

The functions $\displaystyle {g(x)}={e^{x^2-2x}}$ and $\displaystyle {h(x)}={\ln(x^2-2x+2)}$ are given.

Mark the extremum point of each function on the graph.

\displaystyle g(x)=e^{x^{2}-2x}

\displaystyle h(x)=\ln(x^{2}-2x+2)

Explain, without differentiating, why both functions have an extremum point at the same $\displaystyle x$ , and why both are minima.

The graph below shows the derivative of a function $\displaystyle f$ : $\displaystyle {f'(x)}={g(x)}={(x-2)e^x}$ .

On the graph of $\displaystyle f'$ , mark the point whose $\displaystyle x$ is the x-coordinate of the extremum point of $\displaystyle f$ , and the point whose $\displaystyle x$ is the x-coordinate of the inflection point of $\displaystyle f$ .

\displaystyle g(x)=(x-2)e^{x}

Find where $\displaystyle f$ increases.

Find the type of the extremum point of $\displaystyle f$ , and where $\displaystyle f$ is concave up. Give the reasons from the graph of $\displaystyle f'$ .

The function $\displaystyle {f(x)}={x^2+a\ln x}$ is given, with $\displaystyle a$ a parameter.
The function has an extremum point where $\displaystyle {x}={2}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{2}+a\ln x$

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Find $\displaystyle a$ .

Substitute the $\displaystyle a$ you found, and calculate the y-coordinate of the extremum point.

$\displaystyle f(x)=x^{2}-8\ln x$

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Mark the extremum point on the graph.

\displaystyle f(x)=x^{2}-8\ln x

The concentration of a substance in the blood, $\displaystyle t$ hours after it was taken, is $\displaystyle {C(t)}={50te^{-0.25t}}$ milligrams per litre ( $\displaystyle t\geq0$ ).
When is the concentration highest, and what is it?

Find $\displaystyle C'(t)$ .

$\displaystyle C(t)=50te^{-0.25t}$

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Solve the equation $\displaystyle {C'(t)}={0}$ .

Calculate the highest concentration, rounded to two decimal places.

$\displaystyle C(t)=50te^{-0.25t}$

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Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$