Calculus Quiz Questions And Answers

The derivative of x³ with respect to x is found using the power rule, which states that the derivative of xⁿ is n*xⁿ⁻¹. For x³, n = 3. Applying the power rule, the derivative is 3x². This means that as x increases, the rate of change of the function x³ is proportional to 3x². The derivative provides information about how steep the graph of x³ is at any point. The slope of the curve increases as x moves away from zero, with the function increasing more rapidly for larger values of x.

The integral of 1 with respect to x is x + C, where C is the constant of integration. The integral of a constant, such as 1, with respect to a variable, in this case, x, is simply the constant multiplied by x. Therefore, the integral of 1 with respect to x results in x. The constant C represents an arbitrary constant that can be added to the result because there are infinitely many antiderivatives of a function. This is the fundamental concept in indefinite integration.

The derivative of x² + 1 with respect to x is 2x. Using the power rule, the derivative of x² is 2x, and the constant term 1 has a derivative of 0. This is because the derivative of any constant is zero. Therefore, when differentiating the sum of x² and 1, only the term with x (x²) contributes to the derivative. As a result, the correct derivative of the given function is 2x.

The derivative of a constant function is always 0. In this case, the function f(x) = 1963 is a constant because it does not depend on x. The derivative of a constant function is 0, as there is no change in the value of the function as x changes. The derivative measures the rate of change, and since there is no change in the function, the rate of change is zero. Thus, the derivative of f(x) = 1963 with respect to x is 0.

The derivative of cos(x) with respect to x is -sin(x). This is a standard result from differentiation, derived from the fundamental trigonometric identities. The derivative of the cosine function describes the rate of change of the cosine curve at any point. When you differentiate cos(x), you are essentially calculating how fast the value of cos(x) is changing with respect to x. The result is -sin(x) because the graph of cos(x) slopes downward as x increases, which is why the derivative has a negative sign. This represents the rate of decrease of the cosine function.

The second fundamental theorem of calculus states that if F(x) is defined as the integral of a function f(t) with respect to t from a constant a to a variable x, then the derivative of F(x) with respect to x is equal to the integrand f(x). This theorem establishes a connection between differentiation and integration. It shows that the derivative of the accumulated area under the curve described by f(t) from a to x is simply the function value at x, making differentiation and integration inverse operations.

The integral of 2x with respect to x is solved by applying the power rule for integration, which states that the integral of xⁿ with respect to x is (xⁿ⁺¹)/(n+1) + C. For 2x, n = 1. Applying the rule, the integral is (2x²)/2 + C, which simplifies to x² + C. The constant C represents the constant of integration and accounts for the fact that there are infinitely many antiderivatives of a function, differing by a constant value. Thus, the result is x² + C, where C is arbitrary.

The graph described is a straight line with a positive slope, passing through the origin. The function that produces such a graph when integrated is 2x. When you integrate 2x with respect to x, the result is x², which is a parabolic graph. A linear function with a positive slope like 2x produces a graph where the value of the function increases steadily as x increases. Thus, integrating 2x produces a graph that matches the description of a straight line with a positive slope.

As x approaches infinity, the expression 1/(x - 1) behaves in such a way that the denominator becomes very large, making the entire fraction approach zero. As x increases, the term (x - 1) grows larger, and since the numerator remains constant at 1, the fraction becomes smaller and smaller. Therefore, the limit of 1/(x - 1) as x approaches infinity is 0. This is a standard result for rational functions where the degree of the denominator exceeds that of the numerator.

The expression x² + 1 does not contain the variable y. When differentiating with respect to a variable that does not appear in the expression, such as y, the result is always 0. The derivative represents the rate of change of the function with respect to the variable, and if the function does not depend on that variable, the rate of change is zero. Therefore, the derivative of x² + 1 with respect to y is 0, as there is no change in the function with respect to y.

The limit of sin(x)/x as x approaches 0 is a well-known result in calculus, and its value is 1. This can be shown using L'Hopital's Rule or by using the Taylor series expansion of sin(x). As x approaches 0, both the numerator and denominator approach 0, forming an indeterminate 0/0 form. By applying L'Hopital's Rule, we differentiate the numerator and denominator. The derivative of sin(x) is cos(x), and the derivative of x is 1. Thus, the limit becomes cos(0)/1, which equals 1.

To find the second derivative of 4x⁴ + 3x², we first differentiate the function once. The derivative of 4x⁴ is 16x³, and the derivative of 3x² is 6x. Therefore, the first derivative is 16x³ + 6x. Differentiating this again gives the second derivative. The derivative of 16x³ is 48x², and the derivative of 6x is 6. Thus, the second derivative is 48x² + 6, which describes the rate of change of the rate of change of the function. This second derivative indicates how the curvature of the graph behaves at any point.