Exercises
Build confidence in core calculus concepts with this Introduction to Calculus Quiz. Test your understanding of evaluating limits, including trigonometric and infinite limits; finding first and second derivatives; determining tangent-line slopes; and calculating basic indefinite integrals and antiderivatives. Questions also cover polynomial functions, constant functions, and the derivative of the natural logarithm. Ideal for students beginning calculus or reviewing essential differentiation and integration skills before an exam.
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The limit of a polynomial function can be evaluated by direct substitution. Substitute x = 2 into the function: 3(2)^2 + 5(2) = 12 + 10 = 22, so the correct limit is 17.
The power rule for differentiation states that the derivative of x^n is nx^(n-1). Applying this to x^3, we get the derivative as 3x^2.
To solve for the integral of 2x, apply the power rule in reverse: increase the exponent by 1 and divide by the exponent. The result is x^2 + C.
This is a standard trigonometric limit and is known to equal 1. As x approaches 0, sin(x) becomes approximately equal to x, making the limit 1.
The anti-derivative of 6x + 4 is calculated by integrating each term: ∫6x dx = 3x^2 and ∫4 dx = 4x. Add the integration constant C for generality.
To find the second derivative, differentiate f(x) = x^4 twice. First derivative: 4x^3, second derivative: 12x^2.
Divide all terms by x^2 (the highest power in the denominator). The limit simplifies to (3 + 0)/(4 + 0) = 3/4.
The derivative y' = 2x - 4 gives the slope of the tangent line. Substituting x = 3, y' = 6 - 4 = 2.
For a constant function, the anti-derivative is the constant multiplied by x plus the constant of integration C.
The derivative of ln(x) with respect to x is known to be 1/x.
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