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The calculus section of QuickMath allows you to differentiate and integrate almost any mathematical expression.
What is calculus?
Calculus is a vast topic, and it forms the basis for much of modern mathematics. The two branches of calculus are differential calculus and integral calculus.
Differential calculus is the study of rates of change of functions. At school, you are introduced to differential calculus by learning how to find the derivative of a function in order to determine the slope of the graph of that function at any point.
Integral calculus is often introduced in school in terms of finding primitive functions (indefinite integrals) and finding the area under a curve (definite integrals).
Differentiate
The differentiate command allows you to find the derivative of an expression with respect to any variable. In the advanced section, you also have the option of specifying arbitrary functional dependencies within your expression and finding higher order derivatives. The differentiate command knows all the rules of differential calculus, including the product rule, the quotient rule and the chain rule.
Go to the Differentiate page
The integrate command can be used to find either indefinite or definite integrals. If an indefinite integral (primitive function) is sought but cannot be found for a particular function, QuickMath will let you know. Definite integrals will always be given in their exact form when possible, but failing this QuickMath will use a numerical method to give you an approximate value.
Go to the Integrate page
The Fundamental Theorem of Calculus
Integrals were evaluated in the previous tutorial by identifying the integral with an appropriate area and then using methods from geometry to find the area. This procedure will succeed only for very simple integrals. The main result of this section, the fundamental theorem of calculus, includes a very important formula for evaluating integrals. This theorem shows us how to evaluate integrals by first evaluating antiderivatives. The theorem establishes an amazing relationship between the integral, which may be interpreted as an area, and the antiderivative, which is inversely related to the derivative; that is, it relates area and the derivative.
Let f be a continuous function on [a, b ], and define a function F by
The following theorem is called the fundamental theorem and is a consequence of Theorem 1 . The Fundamental Theorem of Calculus
0 = F (a) = G (a) + c
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Section 3.3 : Differentiation Formulas
For problems 1 – 12 find the derivative of the given function.
- \(f\left( x \right) = 6{x^3} - 9x + 4\) Solution
- \(y = 2{t^4} - 10{t^2} + 13t\) Solution
- \(g\left( z \right) = 4{z^7} - 3{z^{ - 7}} + 9z\) Solution
- \(h\left( y \right) = {y^{ - 4}} - 9{y^{ - 3}} + 8{y^{ - 2}} + 12\) Solution
- \(y = \sqrt x + 8\,\sqrt[3]{x} - 2\,\sqrt[4]{x}\) Solution
- \(f\left( x \right) = 10\,\sqrt[5]{{{x^3}}} - \sqrt {{x^7}} + 6\,\sqrt[3]{{{x^8}}} - 3\) Solution
- \(\displaystyle f\left( t \right) = \frac{4}{t} - \frac{1}{{6{t^3}}} + \frac{8}{{{t^5}}}\) Solution
- \(\displaystyle R\left( z \right) = \frac{6}{{\sqrt {{z^3}} }} + \frac{1}{{8{z^4}}} - \frac{1}{{3{z^{10}}}}\) Solution
- \(z = x\left( {3{x^2} - 9} \right)\) Solution
- \(g\left( y \right) = \left( {y - 4} \right)\left( {2y + {y^2}} \right)\) Solution
- \(\displaystyle h\left( x \right) = \frac{{4{x^3} - 7x + 8}}{x}\) Solution
- \(\displaystyle f\left( y \right) = \frac{{{y^5} - 5{y^3} + 2y}}{{{y^3}}}\) Solution
- Determine where, if anywhere, the function \(f\left( x \right) = {x^3} + 9{x^2} - 48x + 2\) is not changing. Solution
- Determine where, if anywhere, the function \(y = 2{z^4} - {z^3} - 3{z^2}\) is not changing. Solution
- Find the tangent line to \(\displaystyle g\left( x \right) = \frac{{16}}{x} - 4\sqrt x \) at \(x = 4\). Solution
- Find the tangent line to \(f\left( x \right) = 7{x^4} + 8{x^{ - 6}} + 2x\) at \(x = - 1\). Solution
- Determine the velocity of the object at any time t.
- Does the object ever stop changing?
- When is the object moving to the right and when is the object moving to the left?
- Determine where the function \(h\left( z \right) = 6 + 40{z^3} - 5{z^4} - 4{z^5}\) is increasing and decreasing. Solution
- Determine where the function \(R\left( x \right) = \left( {x + 1} \right){\left( {x - 2} \right)^2}\) is increasing and decreasing. Solution
- Determine where, if anywhere, the tangent line to \(f\left( x \right) = {x^3} - 5{x^2} + x\) is parallel to the line \(y = 4x + 23\). Solution
IMAGES
VIDEO
COMMENTS
Integral calculus is often introduced in school in terms of finding primitive functions (indefinite integrals) and finding the area under a curve (definite
... Learn how to do calculus with this basic problem. For more math help to include math lessons, practice problems ...
Can you solve this calculus problem that even stomped a math PhD? The problem is: find all the x such that the function f(x) above is
www.midnighttutor.com Check out the whole collection of tutorials.
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !
TabletClass Math:https://tcmathacademy.com/ How to use calculus to find the area of a triangle. For more math help to include math lessons
Absolute value functions are important in mathematics because the can measure distance. I discuss some examples of how to solve equations of
Calculus Problem solving is a wide topic covering hundreds of possibilities from finding lengths and areas to calculating rates of change and continuity of
You need to use differential equations to solve this problem. a=dv/dt=-2t Then: dv=-2t*dt Applying integral on both sides of the equation, we get: v=-t*t+C
Section 3.3 : Differentiation Formulas · f(x)=6x3−9x+4 f ( x ) = 6 x 3 − 9 x + 4 Solution · y=2t4−10t2+13t y = 2 t 4 − 10 t 2 + 13 t Solution