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4.08 The fundamental theorem of calculus

Interactive practice questions

The function $f$f has an antiderivative $F$F, and $F\left(3\right)=4$F(3)=4.

a

Express $\int_3^xf\left(t\right)dt$x3f(t)dt in terms of $F$F and $x$x.

b

Now, calculate $\frac{d}{dx}\int_3^xf\left(t\right)dt$ddxx3f(t)dt.

Easy
6min

Consider the function $f\left(t\right)=-4t$f(t)=4t.

Easy
4min

Consider the function $f\left(t\right)=12t+9$f(t)=12t+9.

Easy
3min

Calculate $\frac{d}{dx}\int_2^x\left(6t^2-8t+3\right)dt$ddxx2(6t28t+3)dt.

Easy
1min
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Outcomes

3.2.15

examine the concept of the signed area function F(x)=∫ {from a to x} f(t)dt

3.2.16

apply the theorem F'(x)=d/dx ( ∫ {from a to x} f(t)dt)=f(x), and illustrate its proof geometrically

3.1.10

use the increments formula: δy≅dy/dx × δx to estimate the change in the dependent variable y resulting from changes in the independent variable x

3.2.17

develop the formula ∫ {from a to b} f(x)dx= F(b)−F(a) and use it to calculate definite integrals

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