Calculus

10.1Find a derivative#

Open templates and choose Derivative (8). Type the variable, press ▶, then the expression. You can also type d(expression, variable). A third number is the order: d(f, x, 2) is the second derivative.

Example 10.1

Derivative of sin(x²)

1. Open templates, choose 8, type x and press right.

   templates8x▶

   

2. Type sin(x²) and press enter.

   sin(x^2▶)enter

Example 10.2

Product rule

1. Choose the derivative template and enter x·eˣ.

   templates8x▶x×e^xenter

Example 10.3

Quotient

1. Differentiate ln(x)/x.

   templates8x▶ln(x)÷xenter

Example 10.4

Derivative from the Calculus menu

1. Press menu, 4, 1; type x, right, x^3.

   menu41x▶x^3enter

Example 10.5

Second derivative by typing

$$\frac{d^{2}}{dx^{2}}x^{5}=20x^{3}$$

1. Type d(x^5, x, 2).

   d(x^5▶,x,2)enter

Example 10.6

Partial derivative

1. Type d(x²y, x).

   d(x^2▶×y,x)enter

Example 10.7

Implicit derivative

$$x^{2}+y^{2}=25\Rightarrow\frac{dy}{dx}=-\frac{x}{y}$$

1. Type impDif(x²+y²=25, x, y).

   impdif(x^2▶+y^2▶=25,x,y)enter

10.2Find an integral#

Choose Integral (templates, 9). Type the expression, move right out of any power, then to the variable box. No "+ C" is shown: the answer is one antiderivative.

Example 10.8

Antiderivative of x²

$$\int x^{2}\,dx=\frac{x^{3}}{3}$$

1. Choose the integral template, type x², press right twice, type x.

   templates9x^2▶▶xenter

Example 10.9

Integration by parts

$$\int xe^{x}\,dx=(x-1)e^{x}$$

1. Integrate x·eˣ.

   templates9x×e^x▶▶xenter

Example 10.10

Integral of ln x

1. Integrate ln(x).

   templates9ln(x)▶xenter

Example 10.11

Trig squared

1. Integrate cos(x)².

   templates9cos(x)^2▶▶xenter

Example 10.12

Integral by typing

1. Type integral(x², x).

   integral(x^2▶,x)enter

10.3Definite and improper integrals#

The definite integral template (templates, A) has four boxes: lower limit, upper limit, expression, variable. Use ∞ (∞) for an improper integral.

Example 10.13

Area under a sine

$$\int_{0}^{\pi}\sin x\,dx=2$$

1. Fill 0, π, sin(x), x.

   templatesa0▶π▶sin(x)▶xenter

Example 10.14

Area under a parabola

$$\int_{0}^{1}x^{2}\,dx=\frac13$$

1. Fill 0, 1, x², x.

   templatesa0▶1▶x^2▶▶xenter

Example 10.15

Improper integral

$$\int_{1}^{\infty}\frac{1}{x^{2}}\,dx=1$$

1. Fill 1, ∞, 1/x², x.

   templatesa1▶∞▶1÷x^2▶▶xenter

Example 10.16

Definite integral by typing

1. Type integral(x², x, 0, 3).

   integral(x^2▶,x,0,3)enter

10.4Find a limit#

The limit template (templates, D) takes the variable, the value it approaches, and the expression. By typing: limit(expression, variable, value). Add a fourth argument 1 for a limit from the right or −1 from the left.

Example 10.17

A classic limit

$$\lim_{x\to 0}\frac{\sin x}{x}=1$$

1. Choose the limit template: x, 0, sin(x)/x.

   templatesdx▶0▶sin(x)÷xenter

Example 10.18

A limit at infinity

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{n}=e$$

1. Type limit((1+1/n)^n, n, ∞).

   limit((1+1÷n)^n▶,n,∞)enter

Example 10.19

Cancelling a factor

1. Type limit((x²−1)/(x−1), x, 1).

   limit((x^2▶−1)÷(x−1),x,1)enter

Example 10.20

One-sided limit from the right

1. Type limit(1/x, x, 0, 1).

   limit(1÷x,x,0,1)enter

Example 10.21

One-sided limit from the left

1. Type limit(1/x, x, 0, −1).

   limit(1÷x,x,0,(−)1)enter

Example 10.22

Two-sided limit that does not exist

1. Type limit(1/x, x, 0).

   limit(1÷x,x,0)enter

10.5Sums and products#

The sum template (templates, B) and product template (templates, C) take the counter, start, end and the expression. Use a letter such as n for a general formula, or ∞ for an infinite sum.

Example 10.23

Sum with a symbolic end

$$\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$$

1. Fill k, 1, n, k.

   templatesbk▶1▶n▶kenter

Example 10.24

Sum of squares

1. Fill k, 1, 10, k².

   templatesbk▶1▶10▶k^2▶▶enter

Example 10.25

Product

$$\prod_{k=1}^{5}k=120$$

1. Fill k, 1, 5, k.

   templatesck▶1▶5▶kenter

Example 10.26

Infinite sum

$$\sum_{k=0}^{\infty}\left(\tfrac12\right)^{k}=2$$

1. Fill k, 0, ∞, (1/2)^k.

   templatesbk▶0▶∞▶(1÷2)^k▶▶▶enter

10.6Taylor series#

taylor(expression, variable, order) gives the polynomial about 0 up to the given order.

Example 10.27

Series of e to the x

$$e^{x}\approx 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}$$

1. Type taylor(e^x, x, 4).

   taylor(e^x▶,x,4)enter

Example 10.28

Series of cosine

1. Type taylor(cos(x), x, 4).

   taylor(cos(x),x,4)enter

10.7Numeric derivative, integral, curve length#

nDeriv(f, x, point) gives a decimal derivative at a point. nInt(f, x, a, b) integrates numerically. arcLen(f, x, a, b) gives the length of the curve.

Example 10.29

Numeric derivative

1. Type nDeriv(x³, x, 2).

   nderiv(x^3▶,x,2)enter

Example 10.30

Numeric integral

1. Type nInt(x², x, 0, 1).

   nint(x^2▶,x,0,1)enter

Example 10.31

Arc length

$$L=\int_{a}^{b}\sqrt{1+f^{\prime}(x)^{2}}\,dx$$

1. Type arcLen(x², x, 0, 1).

   arclen(x^2▶,x,0,1)enter

10.8Tangent lines and extreme values#

Example 10.32

Tangent line

$$y=f(3)+f^{\prime}(3)(x-3)=6x-9$$

1. Type tangentLine(x², x, 3).

   tangentline(x^2▶,x,3)enter

Example 10.33

Normal line

1. Type normalLine(x², x, 3).

   normalline(x^2▶,x,3)enter

Example 10.34

Minimum

1. Type fMin(x²−2x, x).

   fmin(x^2▶−2x,x)enter

Example 10.35

Maximum

1. Type fMax(4−x², x).

   fmax(4−x^2▶,x)enter