Applications of Differential Calculus: One Mark Questions, Book Back Answers and Notes

About Applications of Differential Calculus

Hard ~240 min study

In Class 11 you learnt how to differentiate. This unit is about what the derivative tells you. If a quantity changes with time, its derivative is the rate of change at that instant: the velocity of a particle, the rate at which the volume of a balloon grows, the rate at which a hot object cools. Euler remarked that nothing happens in the world without some maximum or minimum behind it, and the unit ends with exactly that, finding the largest or smallest value of a quantity.

The topics follow a clear order. First come rates of change and tangents and normals. Then Rolle's theorem and the Lagrange mean value theorem, which describe how a function behaves over a whole interval. These lead to Taylor and Maclaurin series, and to L'Hôpital's rule for limits of the form $0/0$ and $\infty/\infty$. The first and second derivative tests then give local extrema, and everything comes together in curve tracing.

Many questions here are word problems. You have to read the situation, write the quantity as a function of one variable, differentiate, and interpret the answer. In MCQs, watch for the conditions of the theorems: a function that is not continuous on the closed interval, or not differentiable in the open interval, does not satisfy Rolle's theorem even if the end values are equal. Concavity and points of inflection also come up regularly, so be clear about what the sign of the second derivative means.

What you'll learn

• Find velocity, acceleration and related rates from a given function.
• Write the equations of tangents and normals and find the angle between two curves.
• State Rolle's theorem and the Lagrange mean value theorem, check their conditions and find c.
• Expand functions using Taylor and Maclaurin series.
• Evaluate indeterminate limits using L'Hôpital's rule.
• Solve maximum and minimum problems using the first and second derivative tests.

Before you start

• Limits and continuity of functions.
• Rules of differentiation, including the chain rule.
• Properties of trigonometric, exponential and logarithmic functions.

Applications of Differential Calculus explained

Topics in This Unit

Rate of Change and Motion

If $s(t)$ is the position of a particle at time $t$, then its velocity is $v(t) = s'(t)$ and its acceleration is $a(t) = v'(t)$. A particle is momentarily at rest when $v(t) = 0$ and changes direction when $v(t)$ changes sign. The same idea handles related rates. If the radius of a spherical balloon increases at a known rate, differentiate $V = \frac{4}{3}\pi r^3$ with respect to $t$ and substitute the given radius to get the rate of change of volume. A common slip is substituting the value of $r$ before differentiating.

Tangents, Normals and Angle Between Curves

The derivative at a point is the slope of the tangent there. With slope $m$ at $(x_1, y_1)$, the tangent is $y - y_1 = m(x - x_1)$. The normal is perpendicular to the tangent, so its slope is $-1/m$. The angle between two curves at a point of intersection is the angle between their tangents at that point. If the product of the two slopes is $-1$, the curves cut orthogonally.

Mean Value Theorems

Rolle's theorem says: if $f$ is continuous on $[a, b]$, differentiable on $(a, b)$ and $f(a) = f(b)$, then $f'(c) = 0$ for at least one $c$ in $(a, b)$. Geometrically, there is a horizontal tangent somewhere between the ends. The Lagrange mean value theorem drops the condition $f(a) = f(b)$ and concludes that $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c$ in $(a, b)$. In words, at some instant the instantaneous rate equals the average rate.

Series Expansions and Indeterminate Forms

The Taylor series writes a function as a power series in $(x - a)$ using its derivatives at $a$. When $a = 0$ it is called the Maclaurin series, which gives the familiar expansions of $e^x$, $\sin x$, $\cos x$ and $\log(1 + x)$. L'Hôpital's rule evaluates limits of the form $0/0$ or $\infty/\infty$: differentiate the numerator and the denominator separately, not the fraction as a whole, and take the limit again. Other forms such as $0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $0^0$ and $\infty^0$ must first be rewritten as a quotient, often by taking logarithms.

Maxima, Minima and Optimisation

A critical number is a point in the domain where $f'(x) = 0$ or $f'(x)$ does not exist. The first derivative test looks at the sign of $f'$ on either side: positive to negative means a local maximum, negative to positive means a local minimum. The second derivative test is quicker when $f'(c) = 0$: $f''(c) < 0$ gives a maximum and $f''(c) > 0$ gives a minimum. If $f''(c) = 0$ the test fails and you go back to the first derivative test. For absolute extrema on a closed interval, compare the values at the critical numbers and at the end points. Optimisation problems, such as the least surface area of a closed cylinder of fixed volume, use exactly these steps.

Symmetry, Asymptotes and Curve Tracing

To sketch a curve, collect its domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, local extrema, concavity and points of inflection. A vertical asymptote usually occurs where the denominator is zero and the numerator is not. A horizontal asymptote comes from the limit of $f(x)$ as $x \to \pm\infty$, and for a rational function a slant asymptote appears when the degree of the numerator is one more than that of the denominator. Together these give an accurate rough graph.

Common mistakes to avoid

• Applying Rolle's theorem or the mean value theorem without checking continuity on the closed interval and differentiability on the open interval.
• Using the quotient rule on the whole fraction when applying L'Hôpital's rule, instead of differentiating numerator and denominator separately.
• Missing the end points of a closed interval when finding the absolute maximum or absolute minimum.
• Reading the sign of the second derivative the wrong way round: a negative value of f''(c) means a local maximum, not a minimum.
• Missing a vertical asymptote by not checking every value where the denominator becomes zero.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

How do I find velocity and acceleration in motion problems?

Differentiate the position function with respect to time to get velocity, and differentiate again to get acceleration. Then substitute the given value of t. If the question asks when the particle is at rest, solve v(t) = 0.

When can I use L'Hôpital's rule?

Only when direct substitution gives 0/0 or ∞/∞. Differentiate the numerator and the denominator separately and take the limit again. If the new limit is still of the same form, you can apply the rule again. For forms like 0 · ∞ or 1^∞, rewrite the expression as a quotient first.

How do I find absolute extrema on a closed interval?

Find all critical numbers inside the interval, that is, points where the derivative is zero or does not exist. Evaluate the function at each critical number and at both end points. The largest of these values is the absolute maximum and the smallest is the absolute minimum.

What is a point of inflection?

It is a point on the curve where the concavity changes, from concave up to concave down or the other way. Candidates are points where the second derivative is zero or does not exist, but you must check that the sign of the second derivative actually changes there. For y = x^4, the second derivative is zero at 0, yet there is no inflection.

What is the difference between Taylor and Maclaurin series?

Both write a function as a power series using its derivatives at one point. The Taylor series is about any point a, in powers of (x − a). The Maclaurin series is the special case a = 0, in powers of x.

What is a stationary point?

A point where the first derivative is zero, so the tangent is horizontal. Every stationary point is a critical number, but a critical number may also be a point where the derivative does not exist, such as x = 0 for |x|.