Tangent Lines and Linear Approximations
Student Saturday Session
Approximating Derivatives
On the AP Calculus Exams, students are often asked to approximate derivatives given function values in a table. In these problems, students are expected to approximate the slope of the tangent line at the indicated points using the slope of a secant line that is close to that point. For their approximations, students should select points that are as close as possible to the desired point of tangency. Often the point of tangency is not included in the given table.
Difference Quotients & Average Rate of Change
You can approximate the slope of the tangent line at x = cby using the forward, the backward, and the symmetric difference quotients. These difference quotients all represent slopes of different secant lines but all approximate the slope of the tangent line at x = c. Here are some pictures to illustrate:
“Forward” Difference Quotient/
“Backward” Difference Quotient
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“Symmetric” Difference Quotient
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On the AP Exam, you usually aren’t given points that are symmetric about the point of tangency. Instead, you can use the closest points on either side of the point of tangency to approximate the derivative. In other words, use the average rate of change!
Average rate of changefrom x = a to x = b.
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Recall that the equation of a line passing through a point with slope m can be found by the point-slope formula . Replacing with and m with we have…
or. Which is equivalent to or
This formula is commonly known as the “Linear approximation of f near x = a” or the “Local Linearization of f near x = a”. Don’t think of it as another formula to memorize. It is simply the equation of the tangent to the graph of at x = a. We can use this linear function to approximate the value of provided that a is sufficiently close to c and we write . It is also important to know if the linear approximation is an over- or under-approximation…
if the graph of is concave upward near This will occur if near / over-approximationif the graph of is concave downward near This will occur if near / under-approximation
Multiple Choice Questions:
1. (2003 AB24 – no calc)
Let f be the function defined by . Which of the following is an equation of the line tangent to the graph of f at the point where x = -1?
A.
B.
C.
D.
E.
2. (1998 AB18 – no calc)
An equation of the line tangent to the graph of at the point is
A.
B.
C.
D.
E.
3. (1998 AB87 – calc permitted)
Which of the following is an equation of the line tangent to the graph of at the point where ?
A.
B.
C.
D.
E.
4. (1997 AB10 – no calc)
An equation of the line tangent to the graph of at is
A.
B.
C.
D.
E.
5. (2003 AB89 – calc permitted)
Let f be a differentiable function with and , and let g be the function . Which of the following is an equation of the line tangent to the graph of g at the point where ?
A.
B.
C.
D.
E.
6. (1997 AB77 - calc permitted)
Let f be the function given by . For what value of x is the slope of the line tangent to the graph of f at equal to 3?
A.0.168
B.0.276
C.0.318
D.0.342
E.0.551
7. (1998 AB77 - calc permitted)
Let f be the function given by and let g be the function given by . At what value of x do the graphs have parallel tangent lines?
A.-0.710
B.-0.567
C.-0.391
D.-0.302
E.-0.258
8. (1997 AB14 – no calc)
Let f be a differentiable function such that and . If the tangent line at is used to find an approximation to a zero of f, that approximation is
A.0.4
B.0.5
C.2.6
D.3.4
E.5.5
9. (1997 AB12 – no calc)
At what point on the graph of is the tangent line paralel to the line ?
A.
B.
C.
D.
E.
10.(1969 AB36 – calc permitted)
The approximate value of at , obtained from the tangent to the graph at is
A.2.00
B.2.03
C.2.06
D.2.12
E.2.24
Free Response Questions:
2001 AB2ac – calc permitted
2002 AB6b – no calc
1996 AB6abc – no calc
National Math and Science Initiative 2012