Exercises: Definition of the derivative of a function at a point

Using the definition of derivative by means of the limit $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, calculate the value of $f'(2)$ for the function $f(x) = x^2 + 3$.

Find the instantaneous rate of change (derivative) of the function $f(x) = 3x - 5$ at the point with x-coordinate $x = 1$, applying the limit of the difference quotient.

The distance covered by an object in free fall on a fictitious planet is given by $s(t) = 4t^2$, where $s$ is measured in metres and $t$ in seconds. Calculate the instantaneous velocity at $t = 3$ s applying the definition of derivative as a limit.

Given the function $f(x) = 2x^2 - x$, use the alternative expression of the definition $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ to find the derivative at the point $a = -1$.

Prove, by explicitly calculating the one-sided limits of the difference quotient, that the absolute value function $f(x) = |x - 3|$ is not differentiable at the point $x = 3$.

A cyclist in a velodrome covers a distance given by the function $d(t) = t^2 + 5t$ (measured in metres, with $t$ in seconds). Calculate their instantaneous speed at $t = 2$ seconds using strictly the definition of derivative by limits.

Find the equation of the tangent line to the curve $f(x) = x^2$ at the point with x-coordinate $x = 2$. To do this, first calculate the gradient $m = f'(2)$ using the definition of derivative by means of the limit.

Calculate the derivative of the constant function $f(x) = 7$ at the point $x = 5$ applying the formal definition with the limit. Briefly explain the geometrical meaning of the result obtained.

Seán wants to calculate the slope of the tangent line to the graph of $f(x) = 2x^2 - 3x$ at the point with abscissa $a = 1$ using the formal definition of the derivative via the limit of the difference quotient. In his notebook, he writes the following working:

And he concludes that the slope of the tangent line is $-3$, so its equation is $y - (-1) = -3(x - 1)$.

1. a) Identify and explain in detail the algebraic error that Seán made when setting up and expanding the expression $f(1+h)$.
2. b) Correct the calculation by properly solving the limit to find the true value of $f'(1)$.
3. c) Using the corrected value, write the equation of the tangent line to the graph of $f$ at the point $(1, f(1))$.

To calculate the derivative of the rational function $f(x) = \frac{1}{x+2}$ at the point with abscissa $a = 1$, you have two equivalent formulas based on the limit of the difference quotient:

• Method A (increment $h$): $f'(1) = \displaystyle\lim_{h \to 0} \frac{f(1+h) - f(1)}{h}$

• Method B (approach to the abscissa): $f'(1) = \displaystyle\lim_{x \to 1} \frac{f(x) - f(1)}{x - 1}$

1. a) Set up the algebraic expression for the fraction obtained in the numerator before evaluating the limit in each of the two methods.
2. b) Compare both approaches: describe what indeterminate form appears in each one and what algebraic manipulations (combining fractions, simplifying factors) each method requires. Justify with reasoning which of the two seems more straightforward or less prone to sign errors to you.
3. c) Evaluate the limit using your chosen method and explain whether the sign of the obtained slope is consistent with the fact that the function is decreasing around $x = 1$.

Using the definition of derivative by means of limits, calculate $f'(3)$ for the rational function $f(x) = \frac{1}{x - 1}$.

Find the derivative of the radical function $f(x) = \sqrt{x + 5}$ at the point $x = 4$, applying the definition of derivative as a limit and multiplying by the conjugate to resolve the indeterminate form.

In a technology component factory, the total cost of producing $q$ units is given by the function $C(q) = 100 + 2q + 0.1q^2$ in euros. Determine the marginal cost (the derivative of the cost function) when exactly $q = 20$ units are produced, by solving the limit of the formal definition of derivative.

Determine whether the following piecewise function is differentiable at $x = 2$, first checking its continuity and then applying the definition of one-sided derivatives:

$f(x) = \begin{cases} x^2 & \text{if } x \le 2 \\ 4x - 4 & \text{if } x > 2 \end{cases}$

Find the equation of the tangent line to the function $f(x) = \frac{x}{x + 2}$ at the origin. You must calculate the gradient by evaluating the limit of the definition of derivative at $x = 0$.

The altitude of an agricultural observation drone follows the equation $h(t) = -5t^2 + 20t + 2$, with height $h$ expressed in metres and time $t$ in seconds. Using the definition of derivative by means of the limit for a general point $t$, obtain the velocity function $v(t) = h'(t)$ and determine at what exact instant the drone reaches its maximum height (when its velocity is zero).

The attached graph shows a continuous function that has a sharp corner. Define analytically an absolute value function that corresponds to this graph and prove rigorously, by calculating the one-sided limits of the difference quotient, why the function is not differentiable at $x = 2$.

Calculate, using the definition of derivative with the limit for a general point $x$, the derivative function $f'(x)$ of $f(x) = x^3 - 3x$. Then, use the result obtained to find the points on the curve where the tangent line is completely horizontal.

Consider the function $f(x) = x^3 - 4x$ and the point with abscissa $a = 2$.

1. a) Before calculating the exact derivative, make a numerical estimate of $f'(2)$. To do this, calculate the average rate of change (slope of the secant line) over two intervals close to the point: one to its right, $[2; 2.05]$, and another to its left, $[1.95; 2]$. From both results, propose an estimated integer value for $f'(2)$.
2. b) Calculate the exact value of $f'(2)$ rigorously using the definition of derivative via the limit of the difference quotient:

1. c) Compare your initial estimate with the exact result. Explain the geometric relationship that exists between the slopes of the two secant lines calculated in part a) and the slope of the tangent line obtained in part b).

According to records from a Met Éireann weather station on a spring day, the ambient temperature $T$ (in degrees Celsius, $^{\circ}\text{C}$) evolves between $7:00$ ($t = 0$) and $13:00$ ($t = 6$) according to the model:

where $t$ is the time measured in hours elapsed since $7:00$.

1. a) Write the expression for the difference quotient measuring the average rate of change of temperature between $9:00$ ($t = 2$) and any subsequent instant $2+h$ hours.
2. b) Calculate the instantaneous rate of change of temperature at $9:00$ ($t = 2$) by evaluating the limit of the difference quotient as $h \to 0$. Express the numerical result along with its physical units and interpret its meaning.
3. c) Find the equation of the tangent line to the graph of $T(t)$ at the point corresponding to $t = 2$.
4. d) Use the tangent line to estimate the temperature at $9:15$ ($t = 2.25$) and evaluate the accuracy of this approximation by comparing it with the actual value predicted by the quadratic model $T(2,25)$.

Determine the exact values of the parameters $a$ and $b$ so that the piecewise function

$f(x) = \begin{cases} ax^2 + b & \text{if } x < 1 \\ \frac{1}{x} & \text{if } x \ge 1 \end{cases}$

is differentiable at $x = 1$. You must justify your answer by applying the definition of continuity and evaluating the limits of the one-sided derivatives using the difference quotient.

Calculate the derivative of the function $f(x) = \sqrt{x^2 + 3}$ at the point $x = 1$ using exclusively the definition of derivative as a limit. Show all the algebraic steps necessary to resolve the indeterminate form.

Suppose $f(x)$ is an even function (that is, it satisfies $f(-x) = f(x)$ for all real numbers $x$) and is known to be differentiable at the point with x-coordinate $x = 0$. Prove analytically, using the definition of derivative by means of the limit $\lim_{h \to 0} \frac{f(h) - f(0)}{h}$, that $f'(0) = 0$ must necessarily hold.

The temperature $T$ (measured in degrees Celsius) of a computer processor as a function of time $t$ (in minutes) after starting an intensive calculation process is given by the expression $T(t) = \frac{4t}{t^2 + 1} + 30$. Calculate the instantaneous rate of heating (the derivative) at exactly $t = 2$ minutes, evaluating step by step the limit of the definition of derivative.

Find the equations of all the tangent lines to the curve $f(x) = x^3 - x$ that are strictly parallel to the line with equation $y = 2x + 1$. To solve this, you must first obtain the general expression for the gradient $f'(x)$ applying the definition of derivative by limits.

Given the function $f(x) = x|x|$, prove using the definition of derivative by calculating the one-sided limits of the difference quotient that the function is smooth (differentiable) at $x = 0$, and find the exact value of $f'(0)$.

Using the general definition of derivative $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, prove step by step that the derivative function of $f(x) = \frac{1}{\sqrt{x}}$ is exactly $f'(x) = -\frac{1}{2x\sqrt{x}}$ for any value of $x > 0$.

The quarterly profit of a renewable energy company as a function of its investment $x$ (in millions of euros) follows a quadratic model of the form $B(x) = -ax^2 + bx$. Given that maximum profit is reached with an exact investment of $10$ million euros and that this maximum profit amounts to $500$ million, deduce the values of the constants $a$ and $b$. Subsequently, prove using the definition of derivative by limits that the instantaneous rate of change at that point satisfies $B'(10) = 0$.

Consider the following limit, which represents the derivative of an unknown function at a point:

1. a) Comparing this expression with the definition $f'(a) = \displaystyle\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$, identify with justification the function $f(x)$ and the value of the abscissa $a$. Then, resolve the indeterminate form to determine the exact value of the limit $L$.
2. b) Construct a rational function of the form $g(x) = \frac{k}{x - c}$ (where $k$ and $c$ are real constants that you must determine) satisfying two conditions at $x = 4$:

1. $g(4) = f(4)$.

2. The tangent line to the graph of $g$ at $x = 4$ is parallel to the tangent line to the graph of $f$ at that same point.

Calculate the exact values of $k$ and $c$ by obtaining the derivative $g'(4)$ via the limit of its difference quotient.

1. c) Write the equation of the common tangent line to both curves at the point with abscissa $x = 4$.

Aoife must study the differentiability at $x = 1$ of the piecewise function:

On her exam, she writes the following justification:

"I compute the derivative of each branch separately: the derivative to the left ($x < 1$) is $2x + 2$, which when evaluated at $x = 1$ equals $2\times 1 + 2 = 4$. The derivative to the right ($x > 1$) is the constant $4$. Since both one-sided derivatives coincide ($4 = 4$), I state that the function is differentiable at $x = 1$ and its derivative is $f'(1) = 4$."

1. a) Point out Aoife's fundamental conceptual error in taking for granted an essential property before analyzing differentiability. Prove whether that property holds or not at $x = 1$.
2. b) Apply the formal definition of derivative using the one-sided limits of the difference quotient:

Calculate both limits in detail and state what algebraic result each one yields.

1. c) Conclude with reasoning whether $f$ is differentiable at $x = 1$ and explain why the rule of thumb of "differentiating the branches and plugging in" leads to a wrong conclusion when the underlying definitions are not respected.