With all these problems you will need to know how they relate to trigonometric functions($ cosx, sinx, tanx$, etc), functions involving $ e^x $ and $ ln x $. The class was Calculus 1 and this was the actual final exam. We learned about second and third derivatives and concavity and how to find inflection points. In this video I go over a final exam from Harvard University. We also learned implicit differentiation, and along with this we learned how to solve related rates problems (as the diameter of a snowball decreases as it melts, how much does it's circumference decrease). The Mean Value Theorem and problems involving it. Problems involving the tangent line to a curve and finding the equation of a tangent line and a normal line (line perpendicular to the tangent line).Įvaluating limits involving infinity, and 0, and indeterminate forms using $L'Hospitals$ rule.ĭifferential equations and problems using differential equations. Using the derivative to find the max and min of a function. How to take $ n $ number of derivatives of a function using the chain rule, the power rule, the product rule, and the quotient rule. Instead, there are a number of properties that limits have which allow you. The second derivative test is a method for determining a function's maxima, minima, and points of inflection by using its first and second derivatives.I'm just finishing a Calculus 1 class this semester and here's the main types of problems we have learned to solve: A definite integral is defined as a limit of Riemann sums. Newton's method is an iterative method for numerically finding a root of a function.Ī Riemann sum is an estimate, using rectangles, of the area under a curve. is the smallest value attained by that object. Calculus, originally called infinitesimal calculus or 'the calculus of infinitesimals', is the mathematical study of continuous change, in the same way that. The mean-value theorem states that if f( x) is differentiable on the open interval ( a, b) and continuous on the closed interval, there is at least one point c in ( a, b) such that ( a - b) f( c) = f( a) - f( b). is the largest value attained by that object. If the function is not continuous, the limit could be different from the value of the function at that point. The intermediate value theorem states that if f is continuous on a closed interval, and c is any number between f( a) and f( b) inclusive, then there is at least one number x in such that f( x) = c.Ī limit is the value a function approaches as the variable approaches some point. Integrals and derivatives are the fundamental objects of calculus. Implicit differentiation is the procedure of differentiating an implicit equation (one which has not been explicitly solved for one of the variables) with respect to the desired variable, treating other variables as unspecified functions of it.Īn indefinite integral, also known as an antiderivative, is an integral without upper and lower limits.Īn inflection point is a point on a curve at which the concavity changes.Īn integral is a mathematical object that can be interpreted as an area or a generalization of area. The fundamental theorems of calculus are deep results in analysis that express definite integrals of continuous functions in terms of antiderivatives. The limit of f f at x3 x 3 is the value f f approaches as we. We start with the function f (x)x+2 f (x) x +2. To understand what limits are, lets look at an example. This simple yet powerful idea is the basis of all of calculus. The first derivative test is a method for determining the maximum and minimum values of a function using its first derivative. Limits describe how a function behaves near a point, instead of at that point. The extreme value theorem states that a continuous function on a closed interval has both a maximum and minimum value. The chain rule is a formula for the derivative of the composition of two functions in terms of their derivatives.Ī continuous function is function with no jumps, gaps, or undefined points.Ī critical point is a point in the graph of a function where the derivative is either zero or undefined.Ī definite integral is an integral with upper and lower limits.Ī derivative is the infinitesimal rate of change in a function with respect to one of its parameters.Ī discontinuity is a point at which a function jumps suddenly in value, blows up, or is undefined. To learn more about a topic listed below, click the topic name to go to theĬalculus is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects.
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