Tangent Lines and Derivatives: Videos & Practice Problems

The cost function $C(x)$ and the selling price of each article, $p(x)$ are given below. Determine the profit function $P(x)$.

$C(x)=-0.03x^2+70x+75$

$p(x) = 80 - 0.02x$

A tree is planted. The maximum possible height of the tree is $12.5$ feet. The height of the tree is given by the following equation.

$y=\(\frac{500}{40+2500e^{-0.75x}\)}$, where $x$ is in months and $y$ is in feet.

Draw the graph of the function.

Find the value of $f^{\prime }\left(a\right)$ for the function $f\left(x\right)=\sqrt{4x-1}$ when $a=2$.

Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=\frac{6}{\sqrt{x}}$ and $P(36,1)$:

$m_{\text{tan}}=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=\frac{4}{3x-2}$ and $P(0,-2)$:

$m_{\text{tan}}=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$

The function $k$ and point $V$ are given. Determine all points $P$ on the graph of $k$ such that the line tangent to $k$ at $P$ passes through $V$.

$k\left(x\right)=x^2+2x+1$; $V(4,9)$

Simplify the difference quotient $\frac{f\left(x\right)-f\left(a\right)}{x-a}$ for the function $f\left(x\right)=2x^3-\frac{1}{x}$.

A rocket is launched vertically up, and its height above the ground is given by the function $h\left(t\right)=400t-50t^2$, where $h$ is in meters and $t$ is in seconds. The graph below shows $h\left(t\right)$ along with its velocity function $v\left(t\right)=\frac{dh}{dt}$ and the acceleration function $a\left(t\right)=\frac{d^{2}h}{d{t}^2}$ for the time interval $0\le t\le 8$ $\text{s}$.

Using the graph, determine the time when the rocket changes direction.

Determine the tangent and normal lines to the function below at a specified point.

${(x+y)}^2=4x+6$, $(1,3)$

Find the limit by converting it first to a derivative at $x=3$.

$\underset{x\to 3}{\lim }\frac{x^8-6561}{x-3}$

The cost function $C\left(x\right)$ and the selling price of each article, $p\left(x\right)$ are given below. Determine the profit function $P\left(x\right)$.

$C\left(x\right)=-0.05x^2+300x+700$

$p\left(x\right)=300$

The cost function $C\left(x\right)$ and the selling price of each article, $p\left(x\right)$ are given below. Determine the profit function $P\left(x\right)$.

$C\left(x\right)=-0.05x^2+300x+700$

$p\left(x\right)=300-0.04x$

A person drives to the destination on a bicycle. The total distance covered is shown in the following image:

If the journey starts at 00:00, at which time interval is the speed of the bicycle at its minimum? What is the speed of the bicycle?

An electronic company produces 20,000 tablets per year in batches of x items at a time. The total cost (in $) of the production of 20,000 tablets in batches of x item at a time is given by the function below:

Determine and interpret the meaning of the average and marginal costs when x = 10,000.

Energy and power are terms frequently used interchangeably, although they are slightly different. Energy is defined as the ability to provide heat or do work, while power is the rate energy is used. The total energy consumption of a residential building over 24 hours is given by the following function: $E\left(t\right)=200t+4.5t^2-\frac{t^3}{8}$, where t = 0 corresponds to midnight

Provide the graph of the energy function.

At time $t=0$, the drain at the bottom of a cylindrical container full of oil is opened. The volume of the container after t hours is given by the function $V\left(t\right)=50(250-t{)}^2$, where V is in cubic feet. Calculate the time elapsed for the container to be empty.

Evaluate the following limit with the help of the definition of the derivative.

${\lim }_{h\to 0}{\frac{\ln (e^5+h)-5}{h}}_{}$

Evaluate the following limit with the help of the definition of the derivative.

${\lim }_{x\to 3}\frac{7^x-343}{x-3}$

A tangent line to the curve $g$ at the point $(3,-2)$ is given by the equation $y=-3x+7$. What are $g\left(3\right)$ and $g^{\prime }\left(3\right)$?

A line tangent to the curve $h$ at $x=-1$ is given by the equation $y=4x+2$. Determine $h(-1)$ and $h^{\prime }(-1)$.

For a certain function $f$, it is known that $f\left(3\right)=-1$ and $f^{\prime }\left(3\right)=4$. Determine the equation of the line tangent to $f$'s graph at $x=3$.

Calculate the equation of the tangent line to the curve $k\left(x\right)=\sqrt{2x+6}$ at the point $S(4,4)$.

For the function $j\left(x\right)=121x$, calculate $j^{\prime }\left(a\right)$ when $a=0$.

For the function $j\left(x\right)=555x^2$, determine $j^{\prime }\left(a\right)$ when $a=2$.

Find the equation of the tangent line to the curve $h$ at the point $(c,h(c\left)\right)$ where $h\left(x\right)=41x^2$ and $c=1$.

Determine the equation of the line tangent to the curve of function $f$ at the point $(-1,3)$ given that $f^{\prime }(-1)=2$.

A rocket is launched vertically upwards, and its altitude in feet $t$ seconds after launch is given by the function $r\left(t\right)$. Determine the rocket's instantaneous velocity at $t=a$ seconds by using limits.

$r\left(t\right)=-16t^2+96t+256$; $a=4$

Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=3x^2-7$ and $P(2,5)$:

$m_{\text{tan}}=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=-2x^2+4x+3$ and $P(-1,-3)$:

$m_{\text{tan}}=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

For the function $j\left(x\right)=\frac{1}{\sqrt{64x}}$, determine $j^{\prime }\left(a\right)$ when $a=\frac{1}{144}$.