9.4.3 The Fundamental Theorem of Calculus, Part I

Question 1

The Fundamental Theorem of Calculus, Part I

Answer 1

• Understand The Fundamental Theorem of Calculus, Part I, which links areas under curves with derivatives.
• Apply The Fundamental Theorem of Calculus, Part I to differentiate a complicated function defined by an integral.

Question 2

note

Answer 2

• The Fundamental Theorem of Calculus, Part I states that if a function f(x) is continuous on a closed interval [a, b] and then F(x) is continuous and differentiable on [a, b], and F´(x) = f(x).
• The Fundamental Theorem of Calculus is the fundamental link between areas under curves and derivatives of functions.
• This example demonstrates the power of The Fundamental Theorem of Calculus, Part I. To differentiate the given complicated function F(x) directly requires first performing the integration, which itself requires a u substitution. Once integrated, taking the derivative requires using the Chain Rule.
• All of this can be bypassed by using The Fundamental Theorem of Calculus, Part I. The derivative of the given
  function is simply the integrand evaluated at x.

Question 3

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x) = ∫^x t+1/t-1 dt, [1,5]

Answer 3

F’(x) = is not defined over all of [1,5]

Question 4

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫ x100 cost/√t−99 dt, [100,200]

Answer 4

F′(x)=cosx/√x−99 on [100,200]

Question 5

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x17 cost/√t−17 dt, [17,40]

Answer 5

F′(x) is not defined over all of [17,40]

Question 6

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−2f(t)dt,[−2,2],where f is continuous on [−2,2].

Answer 6

F′(x)=f(x) on [−2,2]

Question 7

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x2 5t^−1e^t dt, [2,8]

Answer 7

F′(x)=5x^−1e^x on [2,8]

Question 8

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−5 4t^−1 dt,[−5,−1]

Answer 8

None of the above

Question 9

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x1 lnt/t dt,[1,4]

Answer 9

F′(x)=lnx/x on [1,4]

Question 10

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−3 (−4t^2+3t^−1 −7) dt,[−3,7]

Answer 10

F′(x) is not defined over all of [−3,7]

Question 11

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x0 sint dt,[0,2π]

Answer 11

F′(x)=sinx on [0,2π]

Question 12

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x1 3dt, [1,1000]

Answer 12

F′(x)=3 on [1,1000]