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A Preview of Calculus

Describe the tangent problem and how it led to the idea of a derivative.

The Tangent Problem and Differential Calculus

Rate of change is one of the most critical concepts in calculus. We begin our investigation of rates of change by looking at the graphs of the three lines \(f(x)=-2x-3,g(x)=\frac{1}{2}x+1,\) and \(h(x)=2,\) shown in .

As we move from left to right along the graph of \(f(x)=-2x-3,\) we see that the graph decreases at a constant rate. For every 1 unit we move to the right along the x-axis, the y-coordinate decreases by 2 units. This rate of change is determined by the slope (−2) of the line. Similarly, the slope of 1/2 in the function \(g(x)\) tells us that for every change in x of 1 unit there is a corresponding change in y of 1/2 unit. The function \(h(x)=2\) has a slope of zero, indicating that the values of the function remain constant. We see that the slope of each linear function indicates the rate of change of the function.

Compare the graphs of these three functions with the graph of \(k(x)={x}^{2}\) (). The graph of \(k(x)={x}^{2}\) starts from the left by decreasing rapidly, then begins to decrease more slowly and level off, and then finally begins to increase—slowly at first, followed by an increasing rate of increase as it moves toward the right. Unlike a linear function, no single number represents the rate of change for this function. We quite naturally ask: How do we measure the rate of change of a nonlinear function?

We can approximate the rate of change of a function \(f(x)\) at a point \((a,f(a))\) on its graph by taking another point \((x,f(x))\) on the graph of \(f(x),\) drawing a line through the two points, and calculating the slope of the resulting line. Such a line is called a secant line. shows a secant line to a function \(f(x)\) at a point \((a,f(a)).\)

We formally define a secant line as follows:

The accuracy of approximating the rate of change of the function with a secant line depends on how close x is to a. As we see in , if x is closer to a, the slope of the secant line is a better measure of the rate of change of \(f(x)\) at a.

The secant lines themselves approach a line that is called the tangent to the function \(f(x)\) at a (). The slope of the tangent line to the graph at a measures the rate of change of the function at a. This value also represents the derivative of the function \(f(x)\) at a, or the rate of change of the function at a. This derivative is denoted by \({f}^{'}(a).\) Differential calculus is the field of calculus concerned with the study of derivatives and their applications.

Condensed — the full section is in OpenStax Calculus Volume 1.

The Area Problem and Integral Calculus

We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the x-axis over an interval ()?

As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval \([a,b]\) into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles ().

As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of \(f(x)\) and the x-axis over the interval \([a,b].\) Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus is the study of integrals and their applications.

Example

Try it.

Estimate the area between the x-axis and the graph of \(f(x)={x}^{2}+1\) over the interval \([0,3]\) by using the three rectangles shown in .

Solution

The areas of the three rectangles are 1 unit², 2 unit², and 5 unit². Using these rectangles, our area estimate is 8 unit².

Other Aspects of Calculus

So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in . These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.

Key Concepts

Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.

Key Equations

Slope of a Secant Line	\({m}_{\text{sec}}=\frac{f(x)-f(a)}{x-a}\)
Average Velocity over Interval \([a,t]\)	\({v}_{\text{ave}}=\frac{s(t)-s(a)}{t-a}\)

A Preview of Calculus

For the following exercises, points \(P(1,2)\) and \(Q(x,y)\) are on the graph of the function \(f(x)={x}^{2}+1.\)

For the following exercises, points \(P(1,1)\) and \(Q(x,y)\) are on the graph of the function \(f(x)={x}^{3}.\)

For the following exercises, points \(P(4,2)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=\sqrt{x}.\)

For the following exercises, points \(P(1.5,0)\) and \(Q(\phi ,y)\) are on the graph of the function \(f(\phi )=\text{cos}\ (\pi \phi ).\)

For the following exercises, points \(P(-1,-1)\) and \(Q(x,y)\) are on the graph of the function \(f(x)=\frac{1}{x}.\)

For the following exercises, the position function of a ball dropped from the top of a 200-meter tall building is given by \(s(t)=200-4.9{t}^{2},\) where position s is measured in meters and time t is measured in seconds. Round your answer to eight significant digits.

For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is \(h(t)=15t-4.9{t}^{2}.\)

Condensed — the full section is in OpenStax Calculus Volume 1.

Practice (35)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Estimate the slope of the tangent line (rate of change) to \(f(x)={x}^{2}\) at \(x=1\) by finding slopes of secant lines through \((1,1)\) and each of the following points on the graph of \(f(x)={x}^{2}.\)
1. \((2,4)\)
2. \((\frac{3}{2},\frac{9}{4})\)
जवाब दिखाएँ
Use the formula for the slope of a secant line from the definition.
1. \({m}_{\text{sec}}=\frac{4-1}{2-1}=3\)
2. \({m}_{\text{sec}}=\frac{\frac{9}{4}-1}{\frac{3}{2}-1}=\frac{5}{2}=2.5\)
The point in part b. is closer to the point \((1,1),\) so the slope of 2.5 is closer to the slope of the tangent line. A good estimate for the slope of the tangent would be in the range of 2 to 2.5 ().
Estimate the slope of the tangent line (rate of change) to \(f(x)={x}^{2}\) at \(x=1\) by finding the slope of the secant line through \((1,1)\) and the point \((\frac{5}{4},\frac{25}{16})\) on the graph of \(f(x)={x}^{2}.\)

जवाब दिखाएँ

2.25
A rock is dropped from a height of 64 ft. It is determined that its height (in feet) above ground t seconds later (for \(0\le t\le 2)\) is given by \(s(t)=-16{t}^{2}+64.\) Find the average velocity of the rock over each of the given time intervals. Use this information to guess the instantaneous velocity of the rock at time \(t=0.5.\)
1. \([0.49,0.5]\)
2. \([0.5,0.51]\)
जवाब दिखाएँ
Substitute the data into the formula for the definition of average velocity.
1. \({v}_{\text{ave}}=\frac{s(0.5)-s(0.49)}{0.5-0.49}=-15.84\)
2. \({v}_{\text{ave}}=\frac{s(0.51)-s(0.5)}{0.51-0.5}=-16.16\)
The instantaneous velocity is somewhere between −15.84 and −16.16 ft/sec. A good guess might be −16 ft/sec.
An object moves along a coordinate axis so that its position at time t is given by \(s(t)={t}^{3}.\) Estimate its instantaneous velocity at time \(t=2\) by computing its average velocity over the time interval \([2,2.001].\)

जवाब दिखाएँ

12.006001
Estimate the area between the x-axis and the graph of \(f(x)={x}^{2}+1\) over the interval \([0,3]\) by using the three rectangles shown in .

जवाब दिखाएँ

The areas of the three rectangles are 1 unit², 2 unit², and 5 unit². Using these rectangles, our area estimate is 8 unit².
Estimate the area between the x-axis and the graph of \(f(x)={x}^{2}+1\) over the interval \([0,3]\) by using the three rectangles shown here:

जवाब दिखाएँ

17 unit²
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point \(Q(x,y),\) and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
x y \(Q(x,y)\) m_sec
1.1 a. e. i.
1.01 b. f. j.
1.001 c. g. k.
1.0001 d. h. l.

जवाब दिखाएँ

a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000
Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at \(x=1.\)
Use the value in the preceding exercise to find an equation of the tangent line at point P. Graph \(f(x)\) and the tangent line.

जवाब दिखाएँ

\(y=2x\)
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point \(Q(x,y),\) and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
x y \(Q(x,y)\) m_sec
1.1 a. e. i.
1.01 b. f. j.
1.001 c. g. k.
1.0001 d. h. l.
Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at \(x=1.\)

जवाब दिखाएँ

3
Use the value in the preceding exercise to find an equation of the tangent line at point P. Graph \(f(x)\) and the tangent line.
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point \(Q(x,y),\) and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
x y \(Q(x,y)\) m_sec
4.1 a. e. i.
4.01 b. f. j.
4.001 c. g. k.
4.0001 d. h. l.

जवाब दिखाएँ

a. 2.0248457; b. 2.0024984; c. 2.0002500; d. 2.0000250; e. (4.1000000,2.0248457); f. (4.0100000,2.0024984); g. (4.0010000,2.0002500); h. (4.00010000,2.0000250); i. 0.24845673; j. 0.24984395; k. 0.24998438; l. 0.24999844
Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at \(x=4.\)
Use the value in the preceding exercise to find an equation of the tangent line at point P.

जवाब दिखाएँ

\(y=\frac{x}{4}+1\)
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point \(Q(\phi ,y),\) and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
φ y \(Q(\phi ,y)\) m_sec
1.4 a. e. i.
1.49 b. f. j.
1.499 c. g. k.
1.4999 d. h. l.
Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at \(\phi =1.5.\)

जवाब दिखाएँ

π
Use the value in the preceding exercise to find an equation of the tangent line at point P.
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point \(Q(x,y),\) and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
x y \(Q(x,y)\) m_sec
−1.05 a. e. i.
−1.01 b. f. j.
−1.005 c. g. k.
−1.001 d. h. l.

जवाब दिखाएँ

a. −0.95238095; b. −0.99009901; c. −0.99502488; d. −0.99900100; e. (−1;.0500000,−0;.95238095); f. (−1;.0100000,−0;.9909901); g. (−1;.0050000,−0;.99502488); h. (1.0010000,−0;.99900100); i. −0.95238095; j. −0.99009901; k. −0.99502488; l. −0.99900100
Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at \(x=-1.\)
Use the value in the preceding exercise to find an equation of the tangent line at point P.

जवाब दिखाएँ

\(y=\text{-}x-2\)
[T] Compute the average velocity of the ball over the given time intervals.
1. \([4.99,5]\)
2. \([5,5.01]\)
3. \([4.999,5]\)
4. \([5,5.001]\)
Use the preceding exercise to guess the instantaneous velocity of the ball at \(t=5\) sec.

जवाब दिखाएँ

−49 m/sec (velocity of the ball is 49 m/sec downward)
[T] Compute the average velocity of the stone over the given time intervals.
1. \([1,1.05]\)
2. \([1,1.01]\)
3. \([1,1.005]\)
4. \([1,1.001]\)
Use the preceding exercise to guess the instantaneous velocity of the stone at \(t=1\) sec.

जवाब दिखाएँ

5.2 m/sec
[T] Compute the average velocity of the rocket over the given time intervals.
1. \([9,9.01]\)
2. \([8.99,9]\)
3. \([9,9.001]\)
4. \([8.999,9]\)
Use the preceding exercise to guess the instantaneous velocity of the rocket at \(t=9\) sec.

जवाब दिखाएँ

−9.8 m/sec
[T] Compute the average velocity of the runner over the given time intervals.
1. \([1.95,2.05]\)
2. \([1.995,2.005]\)
3. \([1.9995,2.0005]\)
4. \([2,2.00001]\)
Use the preceding exercise to guess the instantaneous velocity of the runner at \(t=2\) sec.

जवाब दिखाएँ

6 m/sec
Sketch the graph of f over the interval \([-1,2]\) and shade the region above the x-axis.
Use the preceding exercise to find the aproximate value of the area between the x-axis and the graph of f over the interval \([-1,2]\) using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.

जवाब दिखाएँ

Under, 1 unit²; over: 4 unit². The exact area of the two triangles is \(\frac{1}{2}(1)(1)+\frac{1}{2}(2)(2)=2.5{\ \text{units}}^{2}.\)
Sketch the graph of f over the interval \([-1,1].\)
Use the preceding exercise to find the aproximate area between the x-axis and the graph of f over the interval \([-1,1]\) using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer.

जवाब दिखाएँ

Under, 0.96 unit²; over, 1.92 unit². The exact area of the semicircle with radius 1 is \(\frac{\pi {(1)}^{2}}{2}=\frac{\pi }{2}\) unit².
Sketch the graph of f over the interval \([-1,1].\)
Approximate the area of the region between the x-axis and the graph of f over the interval \([-1,1].\)

जवाब दिखाएँ

Approximately 1.3333333 unit²

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

imaginary unit
i² = −1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

Euler's number
2.71828…, the base whose exponential is its own derivative.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

y',\ y''

first and second derivative of y
Prime notation for derivatives with respect to x (or t).

C,\ C_1,\ C_2

arbitrary constants
Constants of integration fixed by initial conditions.

How to: A Preview of Calculus

Describe the tangent problem and how it led to the idea of a derivative.
Explain how the idea of a limit is involved in solving the tangent problem.
Recognize a tangent to a curve at a point as the limit of secant lines.
Identify instantaneous velocity as the limit of average velocity over a small time interval.
Describe the area problem and how it was solved by the integral.
Explain how the idea of a limit is involved in solving the area problem.
Recognize how the ideas of limit, derivative, and integral led to the studies of infinite series and multivariable calculus.
Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

अपने आप की कोशिश करें

Parts of this page are adapted from OpenStax Calculus Volume 1 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

अधिक में Calculus

Limits Derivatives Integrals Definite integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests

φ	y	\(Q(\phi ,y)\)	m_sec
1.4	a.	e.	i.
1.49	b.	f.	j.
1.499	c.	g.	k.
1.4999	d.	h.	l.

x	y	\(Q(x,y)\)	m_sec
−1.05	a.	e.	i.
−1.01	b.	f.	j.
−1.005	c.	g.	k.
−1.001	d.	h.	l.

x	y	\(Q(x,y)\)	m_sec
1.1	a.	e.	i.
1.01	b.	f.	j.
1.001	c.	g.	k.
1.0001	d.	h.	l.

x	y	\(Q(x,y)\)	m_sec
1.1	a.	e.	i.
1.01	b.	f.	j.
1.001	c.	g.	k.
1.0001	d.	h.	l.

x	y	\(Q(x,y)\)	m_sec
4.1	a.	e.	i.
4.01	b.	f.	j.
4.001	c.	g.	k.
4.0001	d.	h.	l.