Fundamental Theorem of Calculus — Basic Exercises

1Problem 1

Find the derivative of $$G\left(x\right)={\int }_0^x\cos tdt$$G(x)=\int_0^x\cos t\,dt.

1.1Sample Solution

The integrand $\cos t$\cos t is continuous. Part I of the fundamental theorem gives $$G^{\prime }\left(x\right)=\cos x$$G'(x)=\cos x.

1.2Explanation

Increasing the upper endpoint by $h$h adds a signed contribution of width $h$h whose local value is approximately $\cos x$\cos x.

2Problem 2

Find the derivative of $$H\left(x\right)={\int }_x^{x^2}{e}^{t}dt$$H(x)=\int_x^{x^2}e^t\,dt.

2.1Sample Solution

The variable-endpoint formula gives

2.2Explanation

The upper endpoint contributes a positive term, the lower endpoint contributes a negative term, and the chain rule applies to both.

3Problem 3

Evaluate $${\int }_1^2\left(3x^2-2x\right)dx$$\int_1^2\left(3x^2-2x\right)\,dx.

3.1Sample Solution

An antiderivative of $3x^2-2x$3x^2-2x is $x^3-x^2$x^3-x^2. Part II of the fundamental theorem gives

3.2Explanation

An indefinite integral is a family containing a constant of integration. In a definite integral, that constant cancels in the endpoint difference, leaving a single number.

4Problem 4

The position $s\left(t\right)$s(t) has velocity $s^{\prime }\left(t\right)=3t^2-2t$s'(t)=3t^2-2t. Find the total change in position from $t=1$t=1 to $t=2$t=2.

4.1Sample Solution

The total-change formula gives

4.2Explanation

Accumulating a derivative gives the endpoint difference of the original function. If velocity changes sign, this quantity is net displacement rather than total distance traveled.

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